Reactive Power Control and Optimization of Large Scale Grid
Connected Photovoltaic Systems in the Smart Grid
Dissertation
Presented in Partial Fulfillment of the Requirements for the Degree
Doctor of Philosophy in the Graduate School of The Ohio State
University
By
Zhongkui Wang, M.S., B.S.
Graduate Program in Electrical and Computer Engineering
The Ohio State University
2014
Dissertation Committee:
Kevin M. Passino, Advisor
Jin Wang
Wei Zhang
c Copyright by
Zhongkui Wang
2014
Abstract
In future smart grid, DC-AC inverter based renewable energy sources will greatly
participate in not only the real power generation but also reactive power compensation. Among all the renewable energy sources, grid-connected photovoltaic (PV)
systems have received much attention by engineers and researchers. Grid-connected
PV systems with DC-AC inverters are able to supply real power to the utility grid as
well as reactive power. The real power extracted by the DC-AC inverters is usually
at the maximum power point (MPP) of the attached PV arrays and the reactive
power is used to compensate the grid demand. However, the considerable number of
DC-AC inverters and the complicated operating environment result in challenges in
the control and optimization of both real and reactive power.
For large scale grid-connected PV systems with multiple DC-AC inverters, due
to the limited apparent power transfer capability of each inverter, the reactive power
needs to be allocated among the DC-AC inverters in a proper way. In this thesis, an optimal strategy is proposed for the reactive power allocation in large scale
grid-connected PV systems. The proposed method achieves the maximum reactive
power transfer capability of the entire system by applying classic Lagrange multiplier
method. The sufficient conditions of the optimal reactive power allocation strategy
are provided and mathematically proved. Then, the optimal solutions of the reactive power allocation in the large scale PV systems are developed into a distributed
ii
optimization algorithm by using dual theory. The cost function of the optimization
problem is proved to be convex and the original optimization problem is decomposed
into multiple sub-problems. Each inverter only needs to solve the sub-problem subjected to it and then the optimal solutions of each sub-problem lead to the optimal
solutions of the entire system. Later, control schemes are proposed by applying the
balancing strategies for the reactive power. Reactive power balancing strategies have
been discussed for uniform distribution and optimal distribution based on derived
optimal solutions. Invariant sets are presented to denote the desired distribution of
reactive power among inverters and stability analysis is conducted for the invariant
sets for different conditions by using Lyapunov stability theory. Finally, the proposed
optimal reactive power allocation strategies, the distributed optimization algorithm,
and the reactive power balancing strategies are validated in case studies against a
sample large scale grid-connected PV system.
iii
This is dedicated to my parents and family.
iv
Acknowledgments
First, I give thanks to my parents, who have always been supportive and understanding, even when I am studying in United States and have no chance to visit them.
Any success that I have today could not have been possible without them.
My adviser, Professor Kevin M. Passino, deserves many thanks not only for his
wisdom, knowledge, and guidance but also for his unending patience and warmhearted
helps. Through him, I have not only learned control engineering, power systems,
biomimicry, and research methods, but I have become a self-motivated Ph.D. student,
a better writer, and overall thinker. He has become my first mentor in United States
and has taught me a different perspective of studying and researching. Without his
patient guidance, I could not finish my thesis and degrees.
Also, I thank Professor Jin Wang who is an expert in power area. He helped me
decide my research directions and answered a lot of my stupid questions about the
power systems, renewable energy sources, and the smart grid. Without his help, I was
definitely not able to accomplish anything. Finally, I thank Professor Wei Zhang for
taking the time to be a member of my Ph.D. candidacy and dissertation committee.
As he is an expert in control theory and engineering, I am eager to hear his responses
to my work.
v
Vita
June 25, 1984 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Born - Zhengzhou, China
2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S. Control Engineering, Wuhan University of Technology, China
2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.S. Electrical and Computer Engineering, The Ohio State University,
USA
2010-present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Research/Teaching Associate,
The Ohio State University.
Publications
Research Publications
Z. Wang, K. M. Passino, J. Wang “Optimal Reactive Power Allocation in Large
Scale Grid-Connected Photovoltaic Systems”. Submitted to Journal of Optimization
Theory and Applications, 2013.
Fields of Study
Major Field: Electrical and Computer Engineering
vi
Table of Contents
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
Chapters
1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
1.2
1.3
2.
Control and Optimization of Real and Reactive Power in the Smart
Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Control and Optimization of Real Power . . . . . . . . . . .
1.1.2 Control and Optimization of Reactive Power . . . . . . . . .
Reactive Power Control and Optimization of Distributed Geneartion
1.2.1 Reactive Power Control in Large Scale PV Systems . . . . .
Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimal Reactive Power Allocation in Large Scale Grid-connected Photovoltaic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
2.2
Introduction . . . . . . . . . . . . . . . . . . . . . . . .
Real and Reactive Power of Grid-connected PV Systems
2.2.1 Grid-connected PV Systems . . . . . . . . . . . .
2.2.2 Real and Reactive Power Capability Functions .
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2.3
2.4
2.5
3.
Distributed Optimization of Reactive Power Allocation in Large Scale
Grid-connected Photovoltaic Systems . . . . . . . . . . . . . . . . . . . .
3.1
3.2
30
32
32
34
34
35
37
43
Reactive Power Control Based on Balancing Strategies . . . . . . . . . .
48
4.1
4.2
48
49
50
51
52
4.3
4.4
5.
18
20
20
21
23
23
24
28
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Parallel Algorithm Based on the Decomposition of the Optimization
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Convexity Analysis . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 The Decomposition of the Optimization Problem . . . . . .
3.2.3 Communication Network . . . . . . . . . . . . . . . . . . . .
3.2.4 Parallel Algorithms . . . . . . . . . . . . . . . . . . . . . . .
3.2.5 Simulation: A Case Study . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
4.
2.2.3 Reactive Power Allocation for the Grid-connected PV Systems
Optimal Reactive Power Allocation Strategies . . . . . . . . . . . .
2.3.1 Optimal Allocation Strategy for Small Reactive Power Demand
2.3.2 Optimal Allocation Strategy for Large Reactive Power Demand
Implementation: Algorithm and Case Study . . . . . . . . . . . . .
2.4.1 Algorithm for Optimal Reactive Power Allocation . . . . . .
2.4.2 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The System Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 The Grid-connected PV Systems . . . . . . . . . . . . . . .
4.2.2 Communication Network . . . . . . . . . . . . . . . . . . . .
4.2.3 Operation Schemes . . . . . . . . . . . . . . . . . . . . . . .
Stable Distributed Reactive Power Control Based on Balancing Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Uniformly Distributed Reactive Power . . . . . . . . . . . .
4.3.2 Uniformly Distributed Reactive Power with Inverter Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Optimally Distributed Reactive Power . . . . . . . . . . . .
4.3.4 Simulation: A Case Study . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
53
53
61
69
79
98
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.1
5.2
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
viii
5.3
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Appendices
A.
Proofs of Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
A.1 Optimal Reactive Power Allocation Strategies . . . . . . . . . . . . 103
A.2 Reactive Power Control Based on Balancing Strategies . . . . . . . 111
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
ix
List of Tables
Table
2.1
Page
Data of the inverters in the sample grid-connected PV system . . . .
x
25
List of Figures
Figure
2.1
Page
System diagram of the grid-connected PV systems with one single DCAC inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
System diagram of the grid-connected PV systems with multiple DCAC inverters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3
Solar irradiation profile. . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.4
Total power transfer capability: shown from t = 1 h to t = 7 h,
expressed in kVA. The top curve is the power transfer capability for
the optimal allocation algorithm and the bottom one is for the flexible
even distribution algorithm. . . . . . . . . . . . . . . . . . . . . . . .
27
2.2
2.5
Total real power transferred. The top curve is for the optimal allocation
algorithm and the bottom one is for the strict even distribution algorithm. 28
3.1
Optimal reactive power obtained by the distributed reactive power
allocation algorithm when the desired reactive power QD = −3 MVar.
Left: reactive power of all type 1 inverters; right: reactive power of all
type 2 inverters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Total optimal reactive power of the system obtained by the distributed
reactive power allocation algorithm when the desired reactive power
QD = −3 MVar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
40
3.3
Lagrange multiplier λ when the desired reactive power QD = −3 MVar. 40
3.4
Individual dual function when QD = −3 MVar. Left: dual function of
type 1 inverters; right: dual function of type 2 inverters. . . . . . . .
xi
41
3.5
3.6
3.7
3.8
3.9
Total dual function of the system when the desired reactive power
QD = −3 MVar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimal reactive power obtained by the distributed reactive power allocation algorithm when the desired reactive power QD = −4.4 MVar.
Left: reactive power of all type 1 inverters; right: reactive power of all
type 2 inverters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Total optimal reactive power of the system obtained by the distributed
reactive power allocation algorithm when the desired reactive power
QD = −4.4 MVar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
43
44
Lagrange multipliers µi and µi+N , i = 1, . . . , 30 (type 1 inverters) when
QD = −4.4 MVar. Left: µi ; right: µi+N . . . . . . . . . . . . . . . . .
45
Lagrange multipliers µi and µi+N , i = 31, . . . , 55 (type 2 inverters)
when QD = −4.4 MVar. Left: µi ; right: µi+N . . . . . . . . . . . . .
45
3.10 Lagrange multiplier λ when the desired reactive power QD = −4.4 MVar. 46
3.11 Individual dual function when QD = −4.4 MVar. Left: dual function
of type 1 inverters; right: dual function of type 2 inverters. . . . . . .
3.12 Total dual function of the system when the desired reactive power
QD = −4.4 MVar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
46
47
The ring topology of the communication system of the DC-AC inverter
network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
Uniform reactive power distribution without saturated inverters. The
solid line of each subplot: reactive power of each inverter; the dashed
line of each subplot: lower bound of the reactive power. . . . . . . . .
81
Total error of inverter reactive power and the uniform reactive power
level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
4.4
The minimum reactive power at each time step. . . . . . . . . . . . .
83
4.5
Uniform reactive power distribution: the emergence of inverter islands
due to saturated inverters and limited connection topology. . . . . . .
84
4.2
4.3
xii
4.6
4.7
4.8
4.9
A complete graph of the DC-AC inverter network. It only shows the
connections of inverter 1. . . . . . . . . . . . . . . . . . . . . . . . . .
84
Uniform reactive power distribution with saturated inverters for a fully
connected graph. The solid line of each subplot: reactive power of each
inverter; the dashed line of each subplot: lower bound of the reactive
power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
Optimal reactive power distribution with unsaturated inverters for a
ring connection topology. The solid line of each subplot: reactive power
of each inverter; the dashed line of each subplot: lower bound of the
reactive power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
The ratio of optimally distributed reactive power to real power with
unsaturated inverters for a ring connection topology. The solid line of
each subplot: the ratio of reactive power to real power; the dashed line
of each subplot: the ratio of reactive power lower bound to real power.
88
4.10 The current margin of the system for the optimal reactive power distribution with unsaturated inverters. . . . . . . . . . . . . . . . . . .
89
4.11 Optimal reactive power distribution with saturated inverters (partially
shaded conditions) for a ring connection topology. The solid line of
each subplot: reactive power of each inverter; the dashed line of each
subplot: lower bound of the reactive power. . . . . . . . . . . . . . .
91
4.12 The ratio of optimally distributed reactive power to real power with
saturated inverters for a ring connection topology. The solid line of
each subplot: the ratio of reactive power to real power; the dashed line
of each subplot: the ratio of reactive power lower bound to real power.
92
4.13 The current margin of the system for the optimal reactive power distribution with saturated inverters (partially shaded conditions) for a
ring connection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.14 The ratio of optimally distributed reactive power to real power with
saturated inverters for a fully connected graph. The solid line of each
subplot: the ratio of reactive power to real power; the dashed line of
each subplot: the ratio of reactive power lower bound to real power. .
94
xiii
4.15 The current margin of the system for the optimal reactive power distribution with saturated inverters (partially shaded conditions) for a
fully connected graph. . . . . . . . . . . . . . . . . . . . . . . . . . .
95
4.16 System topologies in which each inverter has a different number of
neighbors (only shows the connections of inverter 1). . . . . . . . . .
97
4.17 Settling time of uniform reactive power distribution for different system topologies. Every data point represents 200 simulation runs with
varying initial conditions. The error bar are standard deviations for
these runs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
4.18 Settling time of optimal reactive power distribution for different system topologies. Every data point represents 200 simulation runs with
varying initial conditions. The error bar are standard deviations for
these runs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
xiv
Chapter 1: Introduction
1.1
Control and Optimization of Real and Reactive Power in
the Smart Grid
In current power grid, electricity which is also considered as real power is generated
by different kinds of generating units, transferred by complex transmission systems,
and distributed through distribution systems to considerable loads. Many challenging
problems of optimally controlling and planning real power have arisen due to the
complex hierarchical structure of the power grid, extremely large amount of real power
demand, the limited capability of the grid, and considerable electric components. For
instance, as indicated by Bergen and Vittal [12] and Machowski et al. [51], unbounded
real power load increasing will result voltage “collapse” at certain point of the grid.
Hence, The injection of certain amount of reactive power provides the capability to
take more real power load without a voltage collapse. In current power grid, the
control of voltage levels, which allow real power to be transferred, is accomplished
by controlling the generation, absorption, and flow of reactive power [45]. Real and
reactive power control and optimization are always two major topics that have been
studied for years for existing power grid. However, current power grid is not able to
fully address critical issues such as generation diversification, optimal deployment of
1
expensive assets, demand response, energy conservation, and reduction of the use of
carbon based fuel. Also, it is a unidirectional system with little fault-tolerant, selfmonitoring, and self-healing capability [29]. The so-called smart grid (intelligent grid )
has recently received much attention. As indicated by Amin and Wollenberg [8], the
components of the future smart grid will have independent processors and have the
capability to cooperate and compete with others with a plug-and-play feature. Also,
due to the wireless sensing network of the smart grid, two-way communication based
distributed control scheme is applicable [29]. All these new components and features
add both new techniques and new challenges to the future smart grid especially to
the control and optimization of real and reactive power.
1.1.1
Control and Optimization of Real Power
Conventional and Intelligent optimization and control techniques are applied in
many problems in the power systems. One of the most significant problems for the
real power control and optimization is to maximize the benefits obtained from the
generation of real power but to minimize the cost of it with the energy conservation
between the generation units and loads. Sometimes, it refers to the problem of socalled “economic dispatch”.
Optimal Economic Dispatch
As described by Bergen and Vittal [12], in a large power system several generators can supply the power to the grid simultaneously while any particular generator
may have different features, such as power generation capability, generation cost, and
constraints on output power. As long as the total amount of the generated power
2
and the grid demand power are in balance, different profiles exist among those generation units. The basic economic dispatch problem addresses the minimization of
the total generation cost while keeps the power balance consistent. Conventional optimization techniques such as Lagrange multiplier method [13] can solve the simple
optimal economic dispatch problems without consideration of power loss and constraints on generation units. However, in order to obtain a global optimal solution,
intelligent optimization methods are applied to most of other optimal economic dispatch problems. In [56, 80], and [9], particle swarm optimization (PSO), evolutionary
programming (EP), and sequential quadratic programming (SQP) are employed for
the economic dispatch problem with nonsmooth cost functions, respectively. The
PSO and EP technique is also used to solve the problem with the consideration of
generator constraints in [34, 39] and nonconvex cost functions in [21, 57, 71]. Similar
intelligent optimization techniques have been applied to another new challenge of real
power control in the smart grid, i.e., demand response.
Demand Response
In traditional power grid, optimization and control of generation have been given
lots of efforts, while in the future smart grid, engineers focus on the scheduling and
management on the load side rather than generation side. Since in current electric power networks, the control of real power demand has become an important
challenge. High demand peaks impose constraints on the operation of the electrical grid, which may cause a power supply failure under certain circumstances [36].
These infrequent demand peaks require large energy generation, transmission, and
distribution capabilities from the power grid, which may lead to an oversized and
over-dimensioned power grid. Hence, the future smart grid needs modernized load
3
management to schedule power in optimal ways and respond to a wide range of
power demand conditions [36]. Due to the emerging advanced metering infrastructure
(AMI), it is possible to realize complex intelligent functions such as two-way communications, self-monitoring and self-healing, coordinated distributed generation, and
especially demand-side management (DSM) in the smart grid [29, 64]. Demand-side
management, also known as load management, commonly refers to the management
implemented by the utility companies to control the electrical energy consumption of
the customers [53]. It has been practiced since the early 1980s and aims to adjust
the load rather than the power station output to balance electricity generation of
the grid with electrical consumption. There are two general approaches for energy
consumption management: reducing consumption and shifting consumption [53–55].
The former is achieved among customers by encouraging energy-aware consumption
patterns and there is also a need for practical solutions to shift the high-power electrical energy consumption to off-peak hours. One specific technique used in traditional
DSM is the so-called direct load control (DLC) [35, 68]. In DLC, based on an agreement between the utility company and the customers, the utility or an aggregator,
which is managed by the utility, can remotely control the operations and energy
consumption of certain load demands. However, when it comes to residential load
control and home automation, the privacy of customers is a major concern and even
a barrier in implementing DLC. Also, traditional DSM is mainly implemented on the
basis of system reliability when the decision is made to curtail load. The utility in a
sense “owns the switch” and sheds loads only when the stability or reliability of the
electrical system is threatened and the customers do not have too much consensus
about the activities of utility and flexibility to control its own load. Hence, in order to
4
protect the privacy of customers, improve the flexibility of load usage, and decrease
the energy cost for each customer while at the same time to achieve the functions
of traditional DSM, the so-called smart pricing technique, which includes real-time
pricing (RTP), day-ahead pricing (DAP), time-of-use pricing (TOUP), critical-peak
pricing (CPP), is proposed by exploiting the advantages of AMI in the smart grid.
These newly proposed pricing strategies offer an opportunity for the customers to capture the “on-off switch” to control their local demand in response to the time-varying
prices. Appropriate “demand response” techniques for residential load management
are foreseen to be crucial as the numbers of plug-in hybrid electrical vehicles (PHEVs)
are increasing rapidly [55]. This new type household “appliance”, combined with traditional household appliances such as washers, dryers, air conditioners, dishwashers,
and so on, raises residential energy consumption significantly. Hence, many demand
response techniques and DSM strategies have been proposed for residential energy
consumption management [22, 23, 25, 40, 50, 53–55, 69]. Mohsenian-Rad et al. [53],
[55] present an autonomous and distributed DSM strategy that considers the minimization of peak-to-average ratio (PAR) in load demand and minimization of the
energy costs of residential customers. A game theoretic approach is applied to formulate an energy consumption scheduling game among all the customers and it is
indicated that the global optimal point of total energy cost minimization is achieved
at the so-called Nash equilibrium. A trade-off between minimization of electricity
payment and minimization of waiting time for the operation of each appliance in the
household is studied in [54]. Li et al. [50] consider the differences among household
appliances including PHEVs and batteries and propose a demand response strategy
based on utility maximization and energy cost minimization. It is shown that there
5
exist time-varying prices that can align individual optimality with social optimality, i.e., under such prices, when household customers selfishly optimize their own
benefits, they automatically maximize the social welfare as well. Real-time demand
response problems are considered in [23, 25, 40, 69]. Chen et al. [23] and Conejo et al.
[25] use a robust optimization tool to deal with the real-time prices and uncertain
prices of the following hours of the day. In [23], a stochastic optimization approach
is proposed for the uncertainties of electricity prices. The uncertainties of electricity
supply from renewable energy sources such as wind and solar are considered in [40].
Pedrasa et al. [60] let the available distributed energy resources (DERs) of end users
participate in the energy consumption scheduling to maximize the benefits and they
employ the PSO technique to solve the optimization problem.
All these control and optimization techniques for both generation side and load
side provide considerable approaches to obtain maximum benefits from real power.
However, as indicated above reactive power controls the voltage levels then allows real
power to be transferred from the generation side to the load side. Hence, without
reactive power it is impossible to connect the generation with load and all the solutions
for real power are useless.
1.1.2
Control and Optimization of Reactive Power
In alternative-current (AC) power grid, the phase difference between the voltage
and the current leads to the occurrence of reactive power. As reactive power cannot
be transmitted over long distances, special devices have to be dispersed throughout
the system [45]. Historically, a synchronous condenser is able to be added to a point
6
in the power grid to provide or absorb reactive power to nearby loads [12]. Also, certain types of shunt capacitor banks with specific sizes are determined to be placed at
predetermined locations [10, 11]. Traditional reactive power generation units also consist of static var compensators (SVC) and static compensators (STATCOM) [52, 70].
SVCs and STATCOMs are also parts of the so-called flexible AC transmission system
(FACTS) which is a framework based on power electronics and static equipment, and
developed by the IEEE to provide control of power system parameters to enhance the
controllability and power transfer capability [27].
In order to plan these reactive power generation units in an effective and efficient
manner, many optimization and control techniques (similar to the ones applied for real
power) have been employed to conquer different allocation and control challenges [83].
Delfanti et al. [26] and Pudjianto et al. [62] formulate the problem as the minimization
of the cost of reactive power sources (also known as Var sources) that is modeled by a
linear function of the generated reactive power. The combined objectives of Var cost
minimization and real power loss are considered in [5, 46, 76]. Similarly, Lee and Yang
[47] and Lee et al. [48] consider the minimization of both Var cost and generation
fuel cost instead of real power loss. Another important problem in reactive power
planning is the voltage stability related optimization. Voltage stability of a bus in
the power grid is usually represented by a power-voltage (P-V) curve [12, 51]. There
is a nose point of the P-V curve which is called the point of collapse (PoC), where the
voltage drops rapidly with an increased real power load [83]. The instability is usually
local area voltage problems due to lack of reactive power. Hence, reactive power
planning problems sometimes include the maximization of the stability margin (SM)
defined as the distance between the PoC and the operation condition [7, 20]. In order
7
to solve these problems, conventional and intelligent methods have been employed.
Conventional methods include linear programming [47, 62], nonlinear programming
[46, 62], and mixed integer programming [26]. Intelligent approaches consist of genetic
algorithm [26, 47, 48, 76], fuzzy control [5], EP [46, 47], and so on.
However, the control and optimization approaches proposed by the literature listed
above for both real and reactive power are all computer based numeric algorithms
and programmings, analytical solutions are rarely given due to the complexity of the
problems. Also, these solutions focus on the system level but not inside the generation
sources. The reactive power generation sources are viewed as a single unit and the
optimization problems inside the generation units are not considered. In the future
smart grid distributed generation units will essentially participate in both real and
reactive power generation. This introduces new challenges and and problems and the
existing control and optimization techniques cannot fully solve them.
1.2
Reactive Power Control and Optimization of Distributed
Geneartion
In the future smart grid, considerable renewable energy sources will be used for energy generation. Most renewable energy sources are considered as distribution generation which can generate both real and reactive power throughout the associated smart
direct-current-alternative-current (DC-AC) inverters. The so-called voltage-source inverters (VSI) have been widely applied for distributed generations [16, 24, 61, 75]. As
indicated by Blaabjerg et al. [16] and Coelho et al. [24], the VSI has a classical pulsewidth-modulation (PWM) controller with an inner current loop and an outer voltage
loop, both using proportional-integral (PI) controllers. In this controller structure,
the oscillating currents and voltages of the grid in abc frame are obtained through a
8
phase-lock-loop (PLL) and transformed into a direct-quadrature (dq) reference frame.
Then, the real and reactive power outputs of the inverter are calculated and sent
to a droop controller to derive the reference frequency and voltage magnitude of the
inverter. The reference signals are then used to generate the PWM waveform to
control the VSI. Although certain standards [37] do not permit inverter-based DGs
to participate in the local voltage regulation, more and more research is focusing on
reactive power generation of DGs with smart inverters, especially grid-connected PV
systems.
1.2.1
Reactive Power Control in Large Scale PV Systems
The prior problem of PV systems is the tracking of maximum power point (MPP)
because of the nonlinear current-voltage (I-V) and power-voltage (P-V) characteristics of PV panels. Such a nonlinearity makes one single PV panel have only one
MPP under uniform irradiation condition. For different irradiation conditions, the
changing of I-V and P-V characteristics make the MPP vary. Many MPPT methods have been developed and implemented for a single PV panel. In [28], 19 MPPT
methods are reviewed and compared. Hill-climbing [42, 43, 78] and perturb and observe (P&O) [38, 49] are two simple and widely used methods. Other popular MPPT
methods include incremental conductance, fractional open-circuit voltage, fractional
short-circuit current, fuzzy logic, neural network and ripple correlation control (RCC)
[28]. As the maximum power of a single PV panel usually varies from several watts
to kilowatts, in large scale grid-connected PV systems, many PV panels are usually
connected together to in series to form a PV string and PV strings are connected
in parallel to form a large PV array to obtain high enough voltage and current as
9
well as output power. As the spatial size of a large PV array can be huge, the solar irradiation on different areas of the PV array will be different because of partial
shading conditions caused by buildings, trees, flock of birds, planes, and mainly the
movement of clouds. Partial shading conditions were studied in [59]. Under partially
shaded conditions, the P-V characteristics of the PV array exhibits multiple peaks
which could cause the failure of the traditional MPPT techniques. Hence, distributed
control techniques were developed to solve the multiple peaks problem under the partially shading condition by applying multiple DC-DC converters inside the PV array,
such that each converter is able to guarantee the maximum real power of the PV
string connected to it [30]. Additional problems associated with real power control of
grid-connected PV systems have been also discussed. In [65], a dynamic programming
(DP) optimization approach is used to minimize the cost of peak shaving service of
the grid-connected PV systems with energy storage elements, and the structure of a
power supervisor based on an optimal predictive power scheduling algorithm is also
proposed.
Similar to other distributed generation sources, grid-connected PV systems consist
multiple DC-AC inverters that can not only transfer real power and but also generate
reactive power. A variety of literature such as [19, 72, 74] addresses the control and
optimization problems of reactive power for gird-connected PV systems with a single
DC-AC inverter. In [72], several reactive power control methods and different PV
inverters’ working modes to support reactive power have been compared. Different
challenges of reactive power control by PV inverters are discussed in [74] and control
schemes associated with the problems are presented. In [19], an online optimal control
strategy to minimize the energy losses of grid-connected PV inverters is proposed.
10
A decentralized nonlinear autoadaptive controller is designed for such an objective.
Corresponding to the distributed control techniques for the real power of large scale
PV systems, a similar approach has been also applied to the reactive power control
for large-scale grid-connected PV systems, i.e., inside a large-scale grid-connected PV
system, multiple DC-AC inverters are used for the connection of PV strings with
the grid. Another active research problem is the reactive power control of multiple
distributed PV generators in a distribution network [44, 73, 81]. These optimization
problems deal with the minimization of either voltage deviation of, or the power
loss between distribution feeders. Adaptive control scheme is developed to solve
the problem in [81] and numerical methods are employed in [73] and [44]. In [79],
a distributed cooperative control algorithm is developed to regulate the real and
reactive power outputs of multiple PV generators in a distribution network. However,
although much attention has been given to the research of reactive power control of
PV systems, the reactive power generation, allocation, and control problems in the
MW level large scale PV systems with multiple PV inverters have not been studied
much.
1.3
Outline of the Thesis
This thesis begins with an overview of the optimization and control techniques that
have been applied to the real and reactive power management of the power systems
and the smart grid. Among these studies the reactive power control and optimization
of distributed generation especially PV systems have been given attentions. Under
the maximum real power transfer conditions, the problem of allocating reactive power
among the DC-AC inverters in large scale PV systems has been discussed and an
11
optimization is formulated by considering the maximization of the power transfer
capability of the entire system. Analytical solutions and algorithms are provided
to the optimization problem. Then, a distributed optimization algorithm is derived
from the analytical solutions by using dual theory. Based on the optimal solutions,
reactive power control schemes are designed by using the balancing strategies for
different operation conditions of the PV systems. A Lyapunov stability analysis is
conducted for the balancing strategy based reactive power control to show that the
uniqueness of the optimal solutions under certain conditions and the convergence rate
to the optimal solutions.
The organization of this thesis is as follow:
• Chapter 2 describes the basis PV system models for a single inverter and multiple inverters and then formulates the reactive power optimization problem by
maximizing a cost function of the total power transfer capability of the entire
large scale PV system. Lagrange multiplier method is employed to solve this
optimization problem. A theorem is presented to study when the constraints
of the optimization are active and when not. Optimal solutions for both active constraint and inactive constraint cases are derived analytically. Then, a
centralized algorithm is developed and validated in a case study.
• In Chapter 3, a convexity analysis is conducted for the cost function of the optimization problem in Chapter 2. For such a convex cost function, a distributed
optimization algorithm is obtain from the analytical optimal solutions derived
in Chapter 2 by using the separable principle and decomposing the original
problem into multiple sub-optimization problems. Also, this distributed optimization algorithm is tested in a case study.
12
• Chapter 4 proposes control schemes for the optimal reactive power allocation
in the large scale PV systems. A communication network is formulated and
modeled by a directed graph and balancing strategies are employed to design
the control schemes for different operation modes and objectives. Then, an
invariant set which represents the desired reactive power distribution is shown
to be asymptotically stable without and with the consideration of the inverter
constraints. Also, the invariant set is shown to be exponentially stable under
certain assumptions. In the case study, a sample system with 8 DC-AC inverters
is used to validate the convergence of the reactive power passing strategies as
well as to study the effects of the inverter constraints. The evaluations of the
communication topologies are investigated in the case study as well.
• Finally, the conclusions of this dissertation and further research directions are
presented in Chapter 5.
13
Chapter 2: Optimal Reactive Power Allocation in Large
Scale Grid-connected Photovoltaic Systems
2.1
Introduction
In the future smart grid, distributed generation (DG) will provide a large amount
of real power generation and participate in reactive power generation as well. Among
all of the distributed generation resources, grid-connected PV electricity and energy
generation systems have a significant increase worldwide. At the end of 2011 the
worldwide total capacity of installed solar PV systems reached 70,000 MW. Germany
with a capacity of 24,678 MW installed PV generation leads all other countries [3].
These installed PV energy conversion systems contain a large portion of large scale
grid-connected PV systems which are being developed to hundreds of MW level.
The largest installed large scale grid-connected PV plant in the world is the Agua
Caliente Solar Project in Arizona, United States with an installed capacity of 247
MW and it will be built to have a total capacity of 397 MW. There are also large
scale PV plants with larger capacity that are under construction, such as the Desert
Sunlight Solar Farm (550 MW) [1] in Riverside County, California, United States,
the Topaz Solar Farm (550 MW) [4] in San Luis Obispo County, California, United
States, the Golmud Solar Park (200 MW installed, 370 MW being constructed) [2]
14
in Qinghai, China, and so on. In large scale grid-connected PV systems, due to the
limited efficiency and power conversion capability of a single PV panel, multiple PV
panels are connected together in series to form a PV string and multiple PV strings
are connected in parallel to form a large PV array. The connection to the utility
grid of large-scale PV plants is realized by inverters. Typical centralized PV inverters
have a power rating under 500 kW, some PV inverters with large capacity may have
a power rating up to 700 kW [66, 67]. For the application of PV inverters in MW
level large scale grid-connected PV plants, one centralized PV inverter is not able to
handle the connection of the entire PV system with the grid. Hence, several inverters
are connected in parallel as the interface between the large scale PV system and the
utility grid [6, 17, 66, 67].
The new topology of large scale grid-connected PV systems imposes new challenges on existing real power generation of large scale PV systems as well as reactive
power generation. In this chapter, we propose an optimization strategy for the reactive power allocation of a system with multiple PV inverters. Under such an optimal
allocation strategy, these PV inverters intend to cooperatively provide reactive power
support to the grid while simultaneously achieving maximum power transfer capability of the entire system. We provide the analytical form of the allocation strategy
and mathematically prove the strategy is optimal by using optimization tools. We
develop the optimal strategy for reactive power allocation into an algorithm and test
such an algorithm in a case study.
15
2.2
Real and Reactive Power of Grid-connected PV Systems
Consider a grid-connected PV system with N ∈ N+ DC-AC inverters. For each
inverter, there is one PV string connected to the utility grid through it. Let the
continuous variable Qi ∈ R, i ∈ {1, . . . , N }, be the amount of reactive power of the
P
ith DC-AC inverter. Suppose that N
i=1 Qi = QD , where QD is the reactive power
demand from the utility grid, which is a constant.
2.2.1
Grid-connected PV Systems
Let us consider one single DC-AC inverter with the PV string connected to it
as shown in Figure 2.1. In Figure 2.1 the PV string is connected to the utility grid
through the three-phase DC-AC inverter. We assume that there is a MPPT control
for the PV string. Due to the DC-AC inverter the system shown in Figure 2.1 is able
to supply not only real power but also reactive power to the utility grid. As we do not
focus on the circuit level of such a system, we assume the inverter is ideal, i.e., there
is no power loss on it. We also assume that the PV array generates the maximum
power to the grid and the system is able to supply/absorb reactive power to/from
the grid because of the DC-AC inverter. Figure 2.1 only shows one inverter in the
gird-connected PV systems. Now we consider the entire grid-connected PV system
with multiple DC-AC inverters. The system topology diagram is shown in Figure 2.2.
2.2.2
Real and Reactive Power Capability Functions
Consider the ith DC-AC inverter in the system shown in Figure 2.2. The inverter
has limited capability to transfer real and reactive power. Suppose that Ci > 0 is a
16
PV string
3-phase
with MPPT
DC-AC
inverter
control
Grid
inverter control
Vgrid , Igrid
Figure 2.1: System diagram of the grid-connected PV systems with one single DC-AC
inverter
constant that we use to represent the current limit of the ith inverter. We assume
that the size of the ith DC-AC inverter is optimally designed, i.e., the value of Ci is
optimally selected based on the rating of the real power of the PV string attached
to the ith inverter such that the ith inverter does not have much additional curent
margin. Such design for the ith inverter reduces cost but it makes the power transfer
capability of the ith inverter limited. Let |V | > 0 denote the voltage magnitude of
the grid voltage and si be the power transfer capability function for the ith inverter,
which is expressed as
si = C i −
where the term
p
Pi2 + Q2i
3|V |
(2.1)
p
Pi2 + Q2i /(3|V |) stands for the current of the ith inverter and Pi
is the real power generated by the PV array. It is obvious that the current of the
inverter cannot exceed the limit Ci . This requires the capability function si ≥ 0.
In Equation (2.1), the capability function si actually calculates the current margin
of the ith inverter, i.e., how much more current the inverter is able to take. Hence,
Equation (2.1) provides a method to define the power transfer capability of the ith
17
PV string
three-phase
with MPPT
control
inverter with
control
1st
b
b
b
b
b
b
PV string
three-phase
with MPPT
control
inverter with
control
N th
Grid
Figure 2.2: System diagram of the grid-connected PV systems with multiple DC-AC
inverters.
inverter. As the PV array is connected to the grid through the DC-AC inverter, the
real power transferred by the inverter is assumed to be at the MPP of the PV array.
Another candidate capability function is
si =
p
Pi2 + Q2i
3|V |Ci
(2.2)
The capability function in Equation (2.2) represents the ratio between the inverter
current and the current limit which is similar to the one given by Equation (2.1). In
this paper, we use Equation (2.1) to show our results (by using Equation (2.2) we
would obtain similar results).
2.2.3
Reactive Power Allocation for the Grid-connected PV
Systems
In Figure 2.2, there are a total number of N DC-AC inverters that are capable of
transferring real and reactive power to the grid. The total power capability function
18
is defined as
sT =
N
X
si =
i=1
N X
i=1
Ci −
p
Pi2 + Q2i
3|V |
(2.3)
In Equation (2.3), sT represents the total current margin of the entire system. As Pi
is the MPP of the ith PV string which is known, we need to seek an optimal allocation
method for Qi , i ∈ {1, . . . , N }, such that the total power capability of the system is
maximized.
As discussed above since the power capability function si needs to be non-negative
and |V | and Ci are positive constants, we have
q
Pi2 + Q2i − 3|V |Ci ≤ 0, i = 1, . . . , N
(2.4)
The value of the ith inverter’s reactive power Qi can be both positive and negative.
Positive Qi means the ith inverter is providing reactive power to the grid and a
negative value of Qi means it is absorbing the reactive power from the grid. So we
can expand Equation (2.4) as
q
Qi − 9|V |2 Ci2 − Pi2 ≤ 0, i = 1, . . . , N
q
− 9|V |2 Ci2 − Pi2 − Qi ≤ 0, i = 1, . . . , N
Combining the reactive power balance equation
allocation problem is formulated as follows,
min − sT = −
s.t. h(Q) =
N X
i=1
N
X
Ci −
p
PN
Pi2 + Q2i
3|V |
i=1
(2.5)
Qi = QD the reactive power
Qi = QD
(2.6)
i=1
q
gi (Qi ) = Qi − 9|V |2 Ci2 − Pi2 ≤ 0, i = 1, . . . , N
q
gi+N (Qi ) = − 9|V |2 Ci2 − Pi2 − Qi ≤ 0, i = 1, . . . , N
19
2.3
Optimal Reactive Power Allocation Strategies
In Equation (2.6), the total reactive power generated by all the DC-AC inverters
is equal to the grid demand QD . Hence, the allocation strategy of the reactive power
for those inverters partially depends on QD . Also, Equation (2.5) shows the limits,
i.e., upper and lower bounds, of the reactive power of the ith inverter Qi . Now let us
consider two cases.
2.3.1
Optimal Allocation Strategy for Small Reactive Power
Demand
For relatively small reactive power demand QD , we assume that the optimally
allocated reactive power Q∗i , i = 1, . . . , N , satisfies Equation (2.5). The next theorem
provides values for the so-called “relatively small” reactive power demand QD and
if the reactive power demand of the grid satisfies the values, the optimal allocation
strategy given by the theorem applies.
Theorem 2.3.1. For i = 1, . . . , N , the reactive power profile
Pi
Q∗i = PN
i=1
Pi
QD
(2.7)
is the optimal reactive power allocation strategy of Equation (2.6) , whenever QD
satisfies
max X
N
N
X
Qi
Qmin
i
Pi ≤ QD ≤ min
Pi
max
i=1,...,N
i=1,...,N
Pi i=1
Pi i=1
p
p
= − 9|V |2 Ci2 − Pi2 and Qmax = 9|V |2 Ci2 − Pi2 .
where Qmin
i
(2.8)
Proof of Theorem 2.3.1. See Appendix A.
As indicated by Theorem 2.3.1, when Equation (2.8) holds, the optimal reactive
power of the ith inverter is proportional to its real power. The power capability
20
function of the entire system sT is
sT =
N
X
i=1
2.3.2
Ci −
qP
2
2
( N
i=1 Pi ) + QD
(2.9)
3|V |
Optimal Allocation Strategy for Large Reactive Power
Demand
When Equation (2.8) does not hold, we say that the reactive power demand QD
is “relatively large”. Obviously, we cannot use Theorem 2.3.1 to find the optimal
reactive power for each inverter. For this case, the optimal reactive power of some
inverters will reach the limits. To find the optimal reactive power allocation strategy
for this case, we need to know which inverters’ reactive power will reach the limits.
Clearly, Equation (2.5) gives the upper and lower bounds of Qi , i = 1, . . . , N . We
consider the problem for two cases, i.e., QD > 0 and QD < 0. Then we have the
following theorem.
Theorem 2.3.2. If the reactive power demand QD > 0, suppose there are r ∈ N+ ,
r ≤ N , inverters with reactive power Qi that hits the upper bound Qmax
, then these
i
inverters are the first r inverters in the order that
Qmax
Qmax
Qmax
1
≤ 2 ≤ ··· ≤ N
P1
P2
PN
(2.10)
where
r = arg min r : PN
Pi
i=r+1
Pi
QD −
r
X
Qmax
i
i=1
<
Qmax
,
i
i = r + 1, . . . , N
(2.11)
Similarly, if the reactive power demand QD < 0, suppose there are t ∈ N+ , t ≤ N ,
inverters with reactive power Qi that hits the lower bound Qmin
i , then these inverters
are the first t inverters in the order that
Qmin
Qmin
Qmin
1
≥ 2 ≥ ··· ≥ N
P1
P2
PN
21
(2.12)
where
Pi
t = arg min t : PN
i=t+1 Pi
QD −
t
X
Qmin
i
i=1
>
Qmin
i ,
i = t + 1, . . . , N
(2.13)
Proof of Theorem 2.3.2. See Appendix A.
Theorem 2.3.2 provides a method to seek those inverters of which the reactive
power hit the upper bounds (lower bounds) if QD > 0 (QD < 0) when the reactive
power demand is relatively large. Based on Theorem 2.3.2, we have the optimal
allocation strategy for the reactive power when the reactive power demand QD does
not satisfy Equation (2.8).
Theorem 2.3.3. If Equation (2.8) does not hold, for the case that QD > 0 suppose all
the inverters are in the order given in Equation (2.10) and there are r ∈ N+ , r < m,
inverters with reactive power Qi that hits the upper bound, then for i = 1, . . . , N , the
reactive power profile
Q∗i = Qmax
, i = 1, . . . , r
i
r
X
Pi
max
∗
, i = r + 1, . . . , N
QD −
Qi
Qi = PN
i=r+1 Pi
i=1
(2.14)
is the optimal reactive power allocation strategy of Equation (2.6); for the case that
QD < 0 suppose all the inverters are in the order given in Equation (2.12) and there
are t ∈ N+ , t < N , inverters with reactive power Qi that hits the lower bound, then
for i = 1, . . . , N , the reactive power profile
Q∗i = Qmin
i , i = 1, . . . , t
t
X
Pi
min
∗
, i = t + 1, . . . , N
QD −
Qi
Qi = PN
P
i
i=t+1
i=1
is the optimal reactive power allocation strategy of Equation (2.6).
Proof. See Appendix A.
22
(2.15)
Theorem 2.3.3 indicates that if the reactive power demand QD does not satisfy
Equation (2.8), the optimal reactive power profile will make some inverters’ reactive
powers hit their limits, and for other inverters that the reactive power do not reach
the limits, the optimal strategy will allocate the reactive power to be proportional to
the real power.
2.4
Implementation: Algorithm and Case Study
In this section, we will apply the optimal reactive power allocation strategy proposed in section 2.3 for a sample grid-connected PV system. First, we introduce
an optimal reactive power allocation algorithm for grid-connected PV systems, then
we provide a sample grid-connected PV system and test the algorithm against it in
simulation.
2.4.1
Algorithm for Optimal Reactive Power Allocation
The optimal reactive power allocation strategy proposed above is developed into
an algorithm as follows:
1. Evaluate if Equation (2.8) holds.
2. If Equation (2.8) holds, the reactive power of all the inverters is calculated by
Equation Equation (2.7).
3. If Equation (2.8) does not hold (it means the reactive power of some inverters
reaches the limits),
• If QD > 0, sort all the inverters in the order given by Equation Equation (2.10).
23
1
(a) The reactive power of the 1st inverter in the new order is (Qmax
1′ ) .
(b) For i = 2′ . . . , N ′ , evaluate
QD −
i−1
X
j=1′
Qj ≤ min
j=i,...,N ′
′
N
X
Qmax
j
Pj
Pj j=i
(2.16)
(c) If Equation (2.16) holds, then
Pi
Qi = PN ′
j=i
Pj
QD −
i−1
X
j=1′
Qj
(2.17)
(d) If Equation (2.16) does not hold, then Qi = Qmax
.
i
• If QD < 0, sort all the inverters in the order given by Equation (2.12).
(a) The reactive power of the 1st inverter in the new order is (Qmin
1′′ ).
(b) For i = 2′′ . . . , N ′′ , evaluate
QD −
i−1
X
j=1′′
Qj ≥ max ′′
j=i,...,N
′′
N
X
Qmin
j
Pj
Pj j=i
(2.18)
(c) If Equation (2.18) holds, then
Pi
Qi = PN ′′
j=i Pj
QD −
i−1
X
j=1′′
Qj
(2.19)
(d) If Equation (2.16) does not hold, then Qi = Qmin
i .
2.4.2
Case Study
In order to validate the optimal reactive power allocation algorithm, we will compare the algorithm we presented above with algorithms based on an even allocation
strategy to show its optimality.
Now consider a 10 MW grid-connected PV system with two types of DC-AC
inverters included. The required data of these inverters for the case study is shown
24
Table 2.1: Data of the inverters in the sample grid-connected PV system
Maximum output power
Nominal output voltage
Nominal output current
Nominal output frequency
Number of inverters
Type 1 inverter
250 kW
480 V
301 A
60 Hz
30
Type 2 inverter
100 kW
480 V (AC, line to line)
121 A
60 Hz
25
in Table 2.1. In this system, we have a total of N = 55 of DC-AC inverters. Assume
that for i = 1, . . . , 30, Ci = 301 A and Ci = 121 A for i = 31, . . . , 55. The nominal
√
output voltage is 480 V AC, line to line. Then, |V | = 480/ 3 = 277.1 V.
Let us consider the condition that all the solar panels in the system have same
solar irradiation level. The solar irradiation profile we employed in the simulation
is shown in Figure 2.3, which provides the solar irradiation of 8 hours during a day.
From t = 2 h to t = 5 h, it becomes cloudy such that the solar irradiation level is lower
than it under a clear condition. The entire solar profile is noisy with a small random
noise added. For such a given solar irradiation profile, we compare the algorithm
based on optimal reactive power allocation strategy with two algorithms based on an
evenly distributed reactive power allocation strategy.
The first algorithm is so-called flexible even distribution of the reactive power:
1. The reactive power demand of the grid is evenly divided into N parts, each part
is QD /N .
max
2. If max{Qmin
}, the reactive power of each inverter is
i } <= QD /N <= min{Qi
i
i
QD /m.
1
The subscript 1′ means it is the first inverter in the new order.
25
Solar irradiation profile
2
Solar irradiation level (kW/m )
1
0.9
0.8
0.7
0.6
0.5
0.4
0
1
2
3
4
5
6
7
8
Time (h)
Figure 2.3: Solar irradiation profile.
3. If the amount of reactive power QD /N exceeds the limits of some inverters,
then for these inverters, the reactive power equals the limit (i.e., upper limit if
QD > 0 lower limit if QD < 0); other inverters will evenly allocate the remaining
reactive power demand.
In the simulation, we set the reactive power demand of the grid to be QD = −3 MW.
The total power transfer capability obtained from simulation for the algorithm based
on optimal allocation strategy and the flexible even distribution allocation algorithm
are shown in Figure 2.4. We observe from Figure 2.4 that the optimal allocation
algorithm is capable of transferring more power than the flexible even distribution
algorithm. The flexible even distribution algorithm indicates that if the reactive power
of some inverters reaches the limits, then other inverters will take more reactive power.
26
Total power transfer capability when Q D = −3 MW
A p p arent p ower tran sfer cap ab i l i ty (kVA )
600
500
400
300
200
Optimal allo cation strategy
100
flexible even distribution
0
1
2
3
4
5
6
7
Time (h)
Figure 2.4: Total power transfer capability: shown from t = 1 h to t = 7 h, expressed
in kVA. The top curve is the power transfer capability for the optimal allocation
algorithm and the bottom one is for the flexible even distribution algorithm.
Next, we will compare the algorithm for the optimal allocation strategy with the
so-called strict even distribution algorithm. For the strict even distribution algorithm,
each inverter takes exactly QD /N of reactive power. The inverters of which the reactive power reaches the limits do not allocate extra reactive power on other inverters
but sacrifice certain amount of transferred real power instead. For a solar profile
given in Figure 2.3, the total transferred real power for both algorithms is shown in
Figure 2.5. It is obvious that the optimal allocation strategy algorithm transfers more
real power than the strict even distribution algorithm when the solar irradiation level
is high. It is worth pointing out that the total power transfer capability of the strict
even distribution algorithm is higher than the one of the optimal allocation strategy
27
algorithm at some points. However, it is because that the strict even distribution
algorithm sacrifices much real power instead.
Total real power transferred by the inverters when Q D = −3 MW
10
Real power (MW)
9
8
7
6
5
Optimal allo cation strategy
Strict evenly distributed strategy
4
0
1
2
3
4
5
6
7
8
Time (h)
Figure 2.5: Total real power transferred. The top curve is for the optimal allocation
algorithm and the bottom one is for the strict even distribution algorithm.
2.5
Summary
In this chapter, based on the power capability function we defined for the gridconnected PV systems with multiple DC-AC inverters, we proposed the optimal allocation strategies for the case that no inverter’s reactive power reaches the limits
when the reactive power demand is small and for the case that some inverters’ reactive powers reach their limits when the reactive power demand is large. We have
provided the analytical solution of the reactive power allocation strategies. Also, we
mathematically proved the strategies are optimal. In order to seek the inverters with
28
reactive power that hits its limits, an order has been used to sort all the DC-AC
inverters for the identification. Finally, we introduced an algorithm based on the optimal allocation strategy and compared such an algorithm with two other algorithms
to show its optimality in a case study. Based on the optimal reactive power allocation strategies given we proposed in this chapter, possible future research directions
include the development of distributed optimization algorithms, the control scheme
design, and the application of the control techniques in simulation or on real systems.
29
Chapter 3: Distributed Optimization of Reactive Power
Allocation in Large Scale Grid-connected Photovoltaic
Systems
3.1
Introduction
In Chapter 2, an optimal approach to allocate the reactive power on the DC-AC
inverters in large scale PV systems is proposed to maximize the “security” of the entire
system. In order to quantify the security of each DC-AC inverter, a power transfer
capability function is defined in Equation (2.1) for each DC-AC inverter in a system
with N ∈ N+ DC-AC inverters. Denoting Qi by xi , Equation (2.1) is expressed as
si (xi ) = Ci −
p
Pi2 + x2i
, i = 1, . . . , N
3|V |
(3.1)
where Ci ∈ R+ is the current limit, Pi ∈ R+ is the transferred real power, and xi ∈ R
is the generated reactive power, of the ith DC-AC inverter in the system, respectively.
It is assumed that Ci and Pi are known and the reactive power xi is the variable to
control. It is obvious that the ith power transfer capability function needs to be
non-negative, which gives the inequality constraints of the optimization problem in
Equation (2.5). The total reactive power generated by all the DC-AC inverters is
P
equal to the amount of desired reactive power QD ∈ R, i.e, N
i=1 xi = QD . Denote
30
the vector of reactive power by x = [x1 , . . . , xm ]⊤ , then the optimization problem of
maximizing the total power transfer capability of the system is defined in terms of x
in Equation (3.2).
min − sT = −
s.t. h(x) =
N X
i=1
N
X
i=1
p
Pi2 + x2i
Ci −
3|V |
xi − QD = 0
(3.2)
q
gi (xi ) = xi − 9|V |2 Ci2 − Pi2 , i = 1, . . . , N
q
gi+N (xi ) = −xi − 9|V |2 Ci2 − Pi2 , i = 1, . . . , N
The optimal solutions to Equation (3.2) are given by Theorems 2.3.1 and 2.3.3 in
Chapter 2 and provide an optimal approach to the reactive power allocation in the
large scale PV systems. However, the algorithm in Chapter 2 is a centralized algorithm which is not reliable under stress circumstances and not efficient for large scale
systems. Since some PV inverters in the market have the communication functions,
the large scale PV system consisting of multiple DC-AC inverters is possible to have
a communication network. Then, the centralized optimization algorithm can be developed to a distributed optimization algorithm based on the dual theory [13–15].
Distributed optimization algorithms have been widely applied to many engineering
problems such as flight formation [63], communication systems [77, 82], energy systems [41, 69, 84], and so on. By using a similar technique, we first analyze the convexity of the cost function of the problem in Chapter 2 and then develop the centralized
optimization algorithm into a distributed optimization algorithm by decomposing it
into sub-optimization problems.
31
3.2
Parallel Algorithm Based on the Decomposition of the
Optimization Problem
In order to develop a distributed algorithm for the optimal reactive power control approach proposed in Chapter 2, a necessary decomposition of the optimization
problem shown in Equation (3.2) is implemented based the dual theory [13, 14].
3.2.1
Convexity Analysis
The decomposition based on the dual theory requires convexity of the cost function
of the optimization problem. The cost function in the problem given by Equation (3.2)
is
f (x) = −
Then, we have the following lemma.
N X
i=1
p
Pi2 + x2i
Ci −
3|V |
(3.3)
Lemma 3.2.1. The cost function of the optimization problem in Equation (3.2),
f : RN 7→ R, given in Equation (3.3), is strictly convex.
Proof. To investigate the convexity of the cost function in Equation (3.3), it is convenient to determine the convexity of the individual function fi (xi ) : R 7→ R, i.e.,
p
Pi2 + x2i
fi (xi ) = − Ci −
3|V |
32
(3.4)
For all xi , yi ∈ R and for all α ∈ (0, 1), we have
=
=
=
=
αfi (xi ) + (1 − α)fi (yi ) − f (αxi + (1 + α)yi )
p
p
Pi2 + x2i
Pi2 + yi2
− (1 − α) Ci −
+ Ci −
− α Ci −
3|V |
3|V |
p
Pi2 + (αxi + (1 − α)yi )2
3|V |
p
p
p
(1 − α) Pi2 + yi2
α Pi2 + x2i
Pi2 + (αxi + (1 − α)yi )2
+
−
3|V |
3|V |
3|V |
2 p
p
2
2
2
2
2
2
α Pi + xi + (1 − α) Pi + yi − Pi + (αxi + (1 − α)yi )
p
p
p
2
2
2
2
2
2
3|V | α Pi + xi + (1 − α) Pi + yi + Pi + (αxi + (1 − α)yi )
p
α2 (Pi2 + x2i ) + (1 − α)2 (Pi2 + yi2 ) + 2α(1 − α) (Pi2 + x2i )(Pi2 + yi2 )
−
p
p
p
2
2
2
2
2
2
3|V | α Pi + xi + (1 − α) Pi + yi + Pi + (αxi + (1 − α)yi )
(3.5)
Pi2 + α2 x2i + (1 − α)2 yi2 + s2α(1 − α)xi yi
p
p
p
2
2
2
2
2
2
3|V | α Pi + xi + (1 − α) Pi + yi + Pi + (αxi + (1 − α)yi )
p
2α(1 − α) (Pi2 + x2i )(Pi2 + yi2 ) − 2α(1 − α)xi yi
=
p
p
p
2
2
2
2
2
2
3|V | α Pi + xi + (1 − α) Pi + yi + Pi + (αxi + (1 − α)yi )
It is obvious that
p
(Pi2 + x2i )(Pi2 + yi2 ) − xi yi > 0 for Pi > 0, hence for all α ∈ (0, 1),
the expression in Equation (3.5) is positive, i.e., αfi (xi ) + (1 − α)fi (yi ) − f (αxi + (1 +
α)yi ) > 0. Hence, the function fi (xi ) given in Equation (3.4) is strictly convex. In
addition, since the weighted sum of convex functions with positive weights is convex,
we conclude that the cost function given in Equation (3.3) is strictly convex.
Because of the desirable convexity property of the cost function, the decomposition
of the optimization problem is possible.
33
3.2.2
The Decomposition of the Optimization Problem
The optimization problem in Equation (3.2) can be decomposed into N independent subproblems based on the duality theory. In Equation (3.2), there are 2N
inequality constraints and one equality constraint, and x ∈ RN . The dual problem is
expressed as follows,
maximize q(µ, λ)
(3.6)
subject to µ ≥ 0, λ ∈ R
where the dual function is given by
q(µ, λ) =
=
inf
xi ∈R
i=1,...,N
N
X
i=1
X
N fi (xi ) + µi gi (xi ) + µi+N gi+N (xi ) + λh(x)
i=1
(3.7)
qi (µ, λ) − λQD
and
qi (µ, λ) = inf fi (xi ) + µi gi (xi ) + µi+N gi+N (xi ) + λxi , i = 1, . . . , N
xi ∈R
(3.8)
Because of the convexity of the cost function, according to Assumption 5.3.2 and
Proposition 5.3.2 in [13], there is no duality gap for the problem given in Equation (3.6) and there exists at least one Lagrange multiplier.
3.2.3
Communication Network
Assume that each DC-AC inverter is assigned a processor, and these processors are
able to communicate with each other. The entire communication network is described
by a directed graph G = (I, A), where I = {1, 2, . . . , N } represents the DC-AC
inverters in the network that we assume to be nodes, and A = {(i, j) : i, j ∈ I} is
a set of directed arcs that represents the communication links and A ⊂ I × I. For
each i ∈ A, there must exist (i, j) ∈ A such that each DC-AC inverter is guaranteed
34
to be connected to the network, and if (i, j) ∈ A then (j, i) ∈ A. We assume that
(i, i) ∈
/ A, as each inverter does not need to communicate with itself and does not
balance reactive power with itself.
For the ith node the rest of the N − 1 nodes connected to it through links formulate a spanning tree. Suppose that a packet is sent to the ith node from every
other node and we assume that packets can be “combined” for transmission on any
communication link, then if a sum consisting of one term from each node is required
by the ith node, the addition of scalars at a node is implemented by combining the
corresponding packets into a single packet. This is called single node accumulation.
Moreover, for the so-called multinode accumulation every node in the communication
network may require packets from all other nodes, then every node is specified as a
root node of a unique spanning tree and each root node is implementing the single
node accumulation [14].
3.2.4
Parallel Algorithms
Because of the strict convexity of the cost function given in Equation (3.3), the
dual function in Equation (3.7) is continuously differentiable (see the dual function
differentiability theorem in [14]), and for all (µ, λ), we have
∂q(µ, λ)
= x̄i − Qmax
, i = 1, . . . , N
i
∂µi
∂q(µ, λ)
= −x̄i + Qmin
i , i = 1, . . . , N
∂µi+N
N
∂q(µ, λ) X
=
x̄i − QD
∂λ
i=1
(3.9)
where x̄(µ, λ) = [x̄1 (µ, λ), . . . , x̄N (µ, λ)]⊤ is the unique vector that minimizes the Lagrangian function L(x, µ, λ). If we define a new vector p ∈ Γ = {z | z ∈ R2N +1 , zi ≥
35
0, i = 1, . . . , 2N }, p = [µ1 , . . . , µN , µN +1 , . . . , µ2N , λ]⊤ , the gradient of the dual function q(p) is
∇q(p) =
h
∂q(p)
∂q(p)
∂q(p) ∂q(p)
, . . . , ∂q(p)
, ∂µ
, . . . , ∂µ
, ∂λ
∂µ1
∂µN
N +1
2N
i⊤
(3.10)
Then, the dual problem in Equation (3.6) can be solved by using the gradient projection algorithm, i.e.,
p(t + 1) = p(t) + γ∇q(p(t))
+
(3.11)
where γ is a positive stepsize and [·]+ denotes the orthogonal projection of the vector
p onto the convex set Γ, i.e.,
[p]+ = arg min ||z − p||2
z∈Γ
(3.12)
Note that we have a plus sign in Equation (3.11) because the dual problem is a
maximization problem.
For the ith DC-AC inverter, it indicates, from Equation (3.8) and (3.9), that
∂q(µ, λ)
∂qi (µ, λ)
=
= x̄i − Qmax
, i = 1, . . . , N
i
∂µi
∂µi
∂qi (µ, λ)
∂q(µ, λ)
=
= −x̄i + Qmin
i , i = 1, . . . , N
∂µi+N
∂µi+N
(3.13)
Hence, the algorithm to calculate each µi and µi+N is implemented by each DC-AC
inverter, i.e.,
+ +
∂qi (µ, λ)
max
= µi (t) + γ x̄i (t) − Qi
µi (t + 1) = µi (t) + γ
∂µi
+ + (3.14)
∂qi (µ, λ)
min
µi+N (t + 1) = µi+N (t) + γ
= µi+N (t) + γ − x̄i (t) + Qi
∂µi
However, in order to find λ(t + 1), the sum of every x̄(t)i is needed for each DC-AC
inverter. Hence, a single node accumulation is required for each DC-AC inverter to
gather the information of every xi (t) and this is a multinode accumulation for the
36
entire communication network [14]. Then, we have, for each DC-AC inverter,
+ X
+
N
∂q(µ, λ)
λ(t + 1) = λ(t) + γ
= λ(t) + γ
x̄i (t) − QD
(3.15)
∂λ
i=1
where in Equation (3.14) and (3.15), x̄i (t) is calculated by using Equation (3.8), i.e.,
x̄i (t) = arg min qi (x(t))
(3.16)
where it also requires µi (t), µi+1 (t), and λ(t) to calculate the value of qi (x(t)). Then,
it is able to obtain the value of x̄i (t + 1) by using µi (t + 1), µi+N (t + 1), and λ(t + 1).
3.2.5
Simulation: A Case Study
Consider a 10 MW grid-connected PV system with two types of DC-AC inverters.
The required data of these inverters for the case study is shown in Table 2.1. In this
system, we have a total of N = 55 of DC-AC inverters. Assume that for i = 1, . . . , 30,
Ci = 301 A and Ci = 121 A for i = 31, . . . , 55. The nominal output voltage is 480 V
√
AC, line to line. Then, |V | = 480/ 3 = 277.1 V. For the system shown in Table
Table 2.1, if we assume the solar irradiation is 0.9 which means the output real power
of each inverter is 0.9Pimax for i = 1, . . . , N , then we obtain
Pi = 0.9Pimax = 0.9 × 250 = 225 kW, for i = 1, . . . , 30
Pi =
0.9Pimax
(3.17)
= 0.9 × 100 = 90 kW, for i = 31, . . . , 55
Based on the real power given in Equation (3.17) the limits of reactive power for type
1 inverters are
Qmin
i
Qmax
i
=−
=
q
9|V |2 Ci2 − Pi2 = −109.54 kVar, i = 1, . . . , 30
−Qmin
i
= 109.54 kVar, i = 1, . . . , 30
and the limits of reactive power for type 2 inverters are
q
min
Qi = − 9|V |2 Ci2 − Pi2 = −44.94 kVar, i = 31, . . . , 55
Qmax
i
=
−Qmin
i
(3.18)
= 44.94 kVar, i = 31, . . . , 55
37
(3.19)
Hence, we obtain the values of the left-hand side and right-hand side of Equation (2.8)
by applying Equations (3.17)-(3.19):
min X
N
Qi
max
Pi = −4.3816 MVar,
i=1,...,N
Pi i=1
min
i=1,...,N
N
X
Qmax
i
Pi = 4.3816 MVar
Pi i=1
(3.20)
We consider two cases:
• The desired total reactive power by the grid QD satisfies Equation (2.8). Here,
we assume QD = −3 MVar. By the theoretical results given in Chapter 2 we
know that the optimal reactive power is
Pi
Q∗i = PN
i=1
Q∗i
Pi
QD = 75 kVar, i = 1, . . . , 30
(3.21)
= 30 kVar, i = 31, . . . , 55
Now, by applying the distributed algorithm developed above we obtain the
simulation results of the optimal reactive power of type 1 and type 2 inverters
shown in Figure 3.1. It shows that the reactive power of all inverters does
not hit the lower limit which is expressed by the dashed line in Figure 3.6 and
the values of reactive power of both type 1 and type 2 inverters obtained from
the simulation of the distributed algorithm are the same as the ones obtained
from the theoretical results. Figure 3.2 shows that the total reactive power
of all the inverters meets the desired value QD = −3 MVar. The values of
Lagrange multipliers µi and µi+N for type 1 and type 2 inverters are all zeros
(not shown here), since all the inequality constraints are inactive. The Lagrange
multiplier λ is shown in Figure 3.3. Figures 3.4 and 3.5 show the individual
dual functions of type 1 and type 2 inverters, and the total dual function of the
system, respectively. It is shown that the value of the total dual function keeps
increasing until its optimal value but the values of individual dual functions are
38
4
0
x 10
R eacti ve Power of Typ e 1 I nverter W h en Q D = −3 MVar
4
0
Qi
in
Qm
i
x 10
R ea cti ve Power of Typ e 2 I nverter W h en Q D = −3 MVar
Qi
in
Qm
i
−0.5
Reactive p ower Q i (Var)
Reactive p ower Q i (Var)
−2
−4
−6
−8
−10
−1
−1.5
−2
−2.5
−3
−3.5
−4
−4.5
−12
0
10
20
30
40
50
60
70
80
90
100
0
Iteration steps
10
20
30
40
50
60
70
80
90
100
Iteration steps
Figure 3.1: Optimal reactive power obtained by the distributed reactive power allocation algorithm when the desired reactive power QD = −3 MVar. Left: reactive
power of all type 1 inverters; right: reactive power of all type 2 inverters.
decreasing. This is because that the dual problem is a maximization problem
of the total dual function but not individual ones. Note that there is a −λQD
difference between the total dual function and the sum of all individual dual
functions.
39
6
(Var)
0
Total Reactive Power of All Inverters When Q D = −3 MVar
i=1 Q i
−0.5
PM
Total reactive p ower
x 10
−1
−1.5
−2
−2.5
−3
0
10
20
30
40
50
60
70
80
90
100
Iteration steps
Figure 3.2: Total optimal reactive power of the system obtained by the distributed
reactive power allocation algorithm when the desired reactive power QD = −3 MVar.
Lagrange Multiplier λ
−4
Lagrange multiplier λ
x 10
3
2
1
0
0
20
40
60
80
100
Iteration steps
Figure 3.3: Lagrange multiplier λ when the desired reactive power QD = −3 MVar.
40
Du a l Fu n cti on q i of Typ e 1 I nverter, Q D = −3 MVar
−32
−34
−36
−38
−40
−42
−13
−14
−15
−16
−17
−18
−44
−46
Du a l Fu n cti on q i of Typ e 2 I nverter, Q D = −3 MVar
−12
Dual function, q i (µ, λ)
Dual function, q i (µ, λ)
−30
0
20
40
60
80
−19
100
0
20
Iteration steps
40
60
80
100
Iteration steps
Figure 3.4: Individual dual function when QD = −3 MVar. Left: dual function of
type 1 inverters; right: dual function of type 2 inverters.
Total Du al Fu n cti on q (µ , λ), Q D = −3 MVar
Dual function, q(µ, λ)
−600
−700
−800
−900
−1000
−1100
−1200
−1300
0
20
40
60
80
100
Iteration steps
Figure 3.5: Total dual function of the system when the desired reactive power QD =
−3 MVar.
41
• The desired total reactive power by the grid QD does not satisfy Equation (2.8).
Here, we assume QD = −4.4 MVar. By the theoretical results given in Chapter
2 we know that the reactive power of some inverters reaches its limit (lower
limit, since QD < 0). By applying Equation (3.22) for all the inverters, we
obtain
Qmin
Qmin
Qmin
Qmin
1
= . . . = 30 = −0.4868 ≥ 31 = 55 = −0.4993
P1
P30
P31
P55
(3.22)
Hence, the reactive power of all the type 1 inverters (i.e., i = 1,. . . ,30) hits
the lower limit which is given by Equation (3.18) and the reactive power of
the rest of the inverters (i.e., all the type 2 inverters) is calculated by using
Equation (2.15):
x∗i = Qmin
= −109.54 kVar, i = 1, . . . , 30
i
30
X
Pi
∗
min
= −44.55 kVar, i = 31, . . . , 55
QD −
xi = P55
Qi
i=31 Pi
i=1
(3.23)
Now, by applying the distributed algorithm developed above we obtain the
simulation results of the optimal reactive power of type 1 and type 2 inverters
shown in Figure 3.6. It shows that the reactive power of all type 1 inverters
hits the lower limit which is expressed by the dashed line in Figure 3.6 and the
values of reactive power of both type 1 and type 2 inverters obtained from the
simulation of the distributed algorithm are same as the ones obtained from the
theoretical results. Figure 3.7 shows that the total reactive power of all the
inverters meets the desired value QD = −4.4 MVar. The values of Lagrange
multipliers µi and µi+N for type 1 inverters are shown in Figure 3.8. We know
from the right plot of Figure 3.8 that µi+N 6= 0 for i = 1, . . . , 30. This implies
that the inequality constraints gi+N (xi ) = −xi + Qmin for i = 1, . . . , 30 are
42
4
0
x 10
R eacti ve Power of Typ e 1 I nverter W h en Q D = −4. 4 MVar
4
0
Qi
Q im in
x 10
R eacti ve Power of Typ e 2 I nverter W h en Q D = −4. 4 MVar
Qi
in
Qm
i
−0.5
Reactive p ower Q i (Var)
Reactive p ower Q i (Var)
−2
−4
−6
−8
−10
−1
−1.5
−2
−2.5
−3
−3.5
−4
X: 800
Y: −1.095e+05
X: 800
Y: −4.456e+04
−4.5
−12
0
100
200
300
400
500
600
700
800
0
Iteration steps
100
200
300
400
500
600
700
800
Iteration steps
Figure 3.6: Optimal reactive power obtained by the distributed reactive power allocation algorithm when the desired reactive power QD = −4.4 MVar. Left: reactive
power of all type 1 inverters; right: reactive power of all type 2 inverters.
active. The values of Lagrange multipliers µi and µi+N for type 2 inverters
are shown in Figure 3.9. The values of µi and µi+N for i = 31, . . . , 55 given
by Figure 3.9 are all zero, since the reactive power of all the type 2 inverters
does not hit the limits and all the inequality constraints for type 2 inverters
are inactive. The Lagrange multiplier λ is shown in Figure 3.10. Figures 3.11
and 3.12 show the individual dual function of type 1 and type 2 inverters, and
the total dual functions of the system, respectively. Similarly, the value of the
total dual function keeps increasing until its optimal value and the values of the
individual dual functions are decreasing.
3.3
Summary
In this chapter, a distributed optimization algorithm is developed based on the
dual theory for the optimal reactive power allocation in large scale grid-connected
43
Total reactive p ower
PM
i=1 Q i
(Var)
0
6
x 10 Total R eacti ve Power of A l l I nverters W h en Q D = −4. 4 MVar
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
−4
−4.5
X: 800
Y: −4.4e+06
0
100
200
300
400
500
600
700
800
Iteration steps
Figure 3.7: Total optimal reactive power of the system obtained by the distributed reactive power allocation algorithm when the desired reactive power QD = −4.4 MVar.
PV systems. The distributed optimization algorithm allows each DC-AC inverter to
autonomously optimize a cost function of itself which is obtained from the decomposition of the original optimization problem. A communication system is established for
the DC-AC inverters to share information and no centralized controller is required.
The optimal solution of each individual DC-AC inverter leads to the global optimal
solution of the entire system.
44
Lagrange Multiplier µi (i = 1, ..., 30) of type 1 Inverters
−6
8
1
0.8
x 10
Lagrange Multiplier µi+N (i = 1, ..., 30) of type 1 Inverters
7
Lagrange multiplier µi+N
Lagrange multiplier µi
0.6
0.4
0.2
0
−0.2
−0.4
6
5
4
3
2
−0.6
1
−0.8
−1
0
100
200
300
400
500
600
700
0
800
0
100
200
300
400
500
600
700
800
Iteration steps
Iteration steps
Figure 3.8: Lagrange multipliers µi and µi+N , i = 1, . . . , 30 (type 1 inverters) when
QD = −4.4 MVar. Left: µi ; right: µi+N
Lagrange Multiplier µi+N (i = 31, ..., 55) of type 2 Inverters
1
0.8
0.8
0.6
0.6
Lagrange multiplier µi+N
Lagrange multiplier µi
Lagrange Multiplier µi (i = 31, ..., 55) of type 2 Inverters
1
0.4
0.2
0
−0.2
−0.4
0.4
0.2
0
−0.2
−0.4
−0.6
−0.6
−0.8
−0.8
−1
0
100
200
300
400
500
600
700
−1
800
0
100
200
300
400
500
600
700
800
Iteration steps
Iteration steps
Figure 3.9: Lagrange multipliers µi and µi+N , i = 31, . . . , 55 (type 2 inverters) when
QD = −4.4 MVar. Left: µi ; right: µi+N
45
Lagrange Multiplier λ
−4
Lagrange multiplier λ
6
x 10
5
4
3
2
1
0
0
100
200
300
400
500
600
700
800
Iteration steps
Figure 3.10: Lagrange multiplier λ when the desired reactive power QD = −4.4 MVar.
Du al Fu n cti on q i of Typ e 1 I nverter, Q D = −4. 4 MVar
−35
−40
−45
−50
−55
−60
Du a l Fu n cti on q i of Typ e 2 I nverter, Q D = −4. 4 MVar
−12
Dual function, q i (µ, λ)
Dual function, q i (µ, λ)
−30
−14
−16
−18
−20
−22
−24
0
100
200
300
400
500
600
700
−26
800
Iteration steps
0
100
200
300
400
500
600
700
800
Iteration steps
Figure 3.11: Individual dual function when QD = −4.4 MVar. Left: dual function of
type 1 inverters; right: dual function of type 2 inverters.
46
Total Du a l Fu n cti on q (µ , λ), Q D = −4. 4 MVar
Dual function, q(µ, λ)
0
−200
−400
−600
−800
−1000
−1200
−1400
0
100
200
300
400
500
600
700
800
Iteration steps
Figure 3.12: Total dual function of the system when the desired reactive power QD =
−4.4 MVar.
47
Chapter 4: Reactive Power Control Based on Balancing
Strategies
4.1
Introduction
Large-scale grid-connected photovoltaic (PV) systems include a considerable number of PV panels which are connected in series and parallel to obtain higher power
supply capability, DC-DC converters which are responsible for regulating the voltages of PV panels to achieve maximum power point tracking (MPPT), and DC-AC
inverters which connected the entire system with the grid. Depending on the topology of the grid-connected PV system, one or more DC-AC inverters are applied. For
some cases, as the MPPT function is also able to be realized by DC-AC inverters, the
DC-DC converters are not necessary. Under partially shaded conditions, a large PV
system with one inverter (converter) may have multiple peaks on its power-voltage
(P-V) curve. Hence, in order to avoid such a problem multiple inverters are installed
to decentralize the MPPT control. In the future smart grid, distributed generation
(DG) such as the grid-connected PV systems will participate in not only real power
generation but also reactive power generation. The DC-AC inverters in the largescale PV systems have certain capability to cooperatively generate reactive power.
As a result, an allocation strategy is needed to distribute the reactive power on these
48
inverters. In Chapter 2, an optimal reactive power allocation strategy is proposed
for large scale PV systems with multiple DC-AC inverters. Such an optimal strategy
requires a centralized control scheme. Due the large size of the grid-connected PV
systems, the performance of a centralized controller is diminished by sensing noise,
communication constraints, and heavy computation load. Also, one single centralized
controller is not perfectly reliable. In Chapter 3, a distributed optimization algorithm
for the reactive power allocation in the large scale PV systems is developed based on
the separable principle and the decomposition of the optimization problem. However,
control schemes are needed to realize the optimal solutions that are derived so far. In
this chapter, a distributed reactive power control strategy is proposed for the large
scale grid-connected PV systems based on the balancing strategies [18, 31–33, 58].
4.2
The System Model
We first specify a system model for the large-scale grid-connected PV systems
with multiple DC-AC inverters. The system model is decentralized as those DC-AC
inverters are separate entities that have certain autonomy to regulate local real and
reactive power generation and all the DC-AC inverters are connected via a communication network. The entire model is in a discrete time framework and we assume
all the DC-AC inverters use the same global time reference. We also assume that
the dynamics and local control of individual DC-AC inverters are much faster than
the control for the entire system. By such an assumption, we consider the DC-AC
inverters as multiple nodes in the network and we focus on the balancing strategies
for the reactive power control of the overall system.
49
4.2.1
The Grid-connected PV Systems
The grid-connected PV systems discussed here are similar to the ones in Chapter
2. We consider a grid-connected PV system with N ∈ N+ DC-AC inverters. For each
DC-AC inverter, there are many PV panels that are connected together in series and
parallel then connected to the grid through the DC-AC inverter. If all the PV panels
associated with this DC-AC inverter are all connected in series, we call it a “PV
string”; otherwise, if there are also parallel connections, we call it a “PV sub-array”
to distinguish it from the entire system which sometimes is called a “PV array”. For
the following analysis, we use “PV sub-array” to refer to the PV unit that is connected
to one DC-AC inverter. Let the continuous variable xi ∈ R, i ∈ {1, . . . , N }, be the
amount of reactive power of the ith DC-AC inverter and Pi ∈ R+ be the amount
of real power generated by the ith PV sub-array and transferred by the ith DC-AC
P
inverter. Suppose that N
i=1 xi = QD , where QD is the reactive power demand from
the utility grid, which is a constant but it could be time-variant (i.e., the reactive
power demand of the grid varies for different times of the day). The ith DC-AC
inverter has limited capability to transfer real power and generate reactive power.
We still use a constant Ci > 0 to represent the current limit of the ith inverter. The
value of Ci > 0 is optimally designed based on the rating of the real power of the PV
sub-array attached to the ith inverter. This implies a trade-off between the inverter
cost and the power transfer capability of the ith inverter. As the current of the ith
DC-AC inverter is not allowed to exceed Ci , we have
p
Pi2 + Q2i
≥ 0 =⇒
Ci −
3|V |
q
q
− 9|V |2 Ci2 − Pi2 ≤ Qi ≤ 9|V |2 Ci2 − Pi2 , i = 1, . . . , N
50
(4.1)
where |V | is a positive constant that represents the magnitude of the grid voltage.
This is exactly the inequality constraints of the optimization problem in Chapter 2.
p
p
Then qimax = 9|V |2 Ci2 − Pi2 and qimin = − 9|V |2 Ci2 − Pi2 are the upper and lower
bounds of Qi . As we will present reactive power balancing strategies, we use a discrete
time formulation. Hence, we use xi (k) to denote the reactive power Qi at time k.
4.2.2
Communication Network
We adopt a communication network for the DC-AC inverters of the grid-connected
PV systems that is similar to the ones for the systems given by [33] and [58]. There are
different candidate topologies for the communication system of the DC-AC inverters
(i.e., line, ring, network, and so on). However, we assume that the communication
links and the topology are fixed. Also, we assume that the communication links have
sufficient capacity to transmit the required information, and the only deficiency is a
possible delay that occurs during the sensing process and the information transmission. As we assume that the local control of the DC-AC inverters dynamics is fast
enough such that the possible delays due to the local control operation are negligible.
The communication network of these DC-AC inverters I = {1, 2, . . . , N } is described
by a directed graph G = (I, A), where I represents the DC-AC inverters in the network that we assume to be nodes, and A = {(i, j) : i, j ∈ I} is a set of directed arcs
that represents the communication links and A ⊂ I × I. For each i ∈ A, there must
exist (i, j) ∈ A such that each DC-AC inverter is guaranteed to be connected to the
network, and if (i, j) ∈ A then (j, i) ∈ A. We assume that (i, i) ∈
/ A, as each inverter
does not need to communicate with itself and does not balance reactive power with
itself.
51
4.2.3
Operation Schemes
The grid-connected PV systems may operate in different ways to achieve different
objectives. Candidate operation schemes of large scale grid-connected PV systems
include:
• Night operation mode: During night, there is no solar irradiation. Hence,
the grid-connected PV system without energy storage units does not generate
real power but draws real power from the grid. However, the DC-AC inverters
are also able to supply reactive power to the grid provided that there are certain
demands.
• Optimal reactive power mode: We consider this operation mode during the
day based on the results given in Chapter 2.
• Microgrid islanding mode: If there is a grid-connected PV system in a microgrid, for the islanding operation mode of the microgrid, the real and reactive
power generated by the grid-connected PV system need to be both conservative,
P
PN
i.e., N
i=1 Pi = PD and
i=1 Qi = QD where, PD is the real power demand of
the microgrid.
• Other modes: The grid-connected system is also able to work in other operation modes. For instance, the objective of the PV system in [79] is to operate
all the DC-AC inverters such that
P2
P1
= max = . . . =
max
P1
P2
Q2
Q1
=
= ... =
Qmax
Qmax
1
2
PN
= rP0
max
PN
QN
0
= rQ
max
QN
(4.2)
where rp0 and rP0 are constants that represent the ratios of real and reactive
power of each inverter to their maximum values.
52
In this chapter, control strategies based on balancing the reactive power among all
DC-AC inverters are proposed for the first two operation schemes. The reactive power
is desired to be evenly allocated among all the inverters for the night operation mode
since there is no real power during the night. For the optimal reactive power mode,
the real power is transferred through each inverter at the maximum power point and
the reactive power is optimally allocated among all inverters to maximize the total
power transfer capability of the entire system. The reactive power balancing strategies
for other operation schemes are similar to the ones for the first two operation schemes
and not discussed here.
4.3
Stable Distributed Reactive Power Control Based on Balancing Strategies
According to the operation mode, the balancing strategy varies. Here, we propose
different balancing strategies via different operation modes and objectives. We will
prove the balancing strategies are stable with respect to an invariant set.
4.3.1
Uniformly Distributed Reactive Power
We consider the case where the reactive power is evenly balanced among the DCAC inverters. Such a balancing strategy is able to alleviate the stress of each DC-AC
inverter and can be applied in the night operation mode. We begin for this case
without the consideration of the bounds of the reactive power for each individual
DC-AC inverter.
53
Reactive Power Passing Strategies
Let xi (k) be the reactive power of inverter i at time k. For any (i, j) ∈ A, let
xij (k) be the amount of reactive power of inverter j that inverter i perceives at time
k. Define αii→j ≥ 0 to be the amount reactive power that inverter i passes to inverter
j. It is also the amount of reactive power removed from inverter i when i passes to
inverter j. Define αji→j ≥ 0 as the amount of reactive power received by inverter j
due to inverter i sending reactive power to j at time k. Let Ni = {j : (i, j) ∈ A}
be the subset of the neighboring nodes of inverter i. Then, the following conditions
define a class of reactive power passing strategies for inverter i at time k:
4.3.1-i) αii→j = 0, if xi (k) − xij (k) ≤ 0;
4.3.1-ii) xi (k) −
4.3.1-iii) If
αii→j
4.3.1-iv) If
αii→j
P
{j:j∈Ni }
αii→j ≥ xij (k) + αii→j , ∀j ∈ Ni such that xi (k) − xij (k) > 0;
> 0 for some j, then
> 0 for some j, then
j ′ ∈ Ni ;
xij (k)
∗
αii→j
+
αji→j
≤ max
′
j
xij ′ (k)
′
: j ∈ Ni ;
xij ′ (k) :
≥ γij ∗ xi (k)−xij ∗ (k) where j ∗ = arg min
′
j
In condition 4.3.1-iv), the term γij ∗ ∈ (0, 1) for j ∗ ∈ Ni is a coefficient that represents
the proportion of the reactive power imbalance that is sometimes guaranteed to be
reduced.
Remark 1. Condition 4.3.1-i) indicates that inverter i may only pass reactive power
to inverter j if its reactive power perception about inverter j is lower than its own
reactive power. It is worth pointing out that the reactive power of each inverter can be
negative. Negative reactive power means the inverters absorb reactive power from the
54
grid. Recall that we assume inverters can only pass positive reactive power to other
inverters i.e., αii→j ≥ 0. For the case of condition 4.3.1-i), if xi (k) is negative and
if xi (k) − xij (k) ≤ 0 for some j, then inverter i will not pass any positive amount
of reactive power to inverter j. This means inverter i will not take more capacitive
reactive power for this case. On the other hand, if xi (k) > 0 and xi (k) − xij (k) > 0,
condition 4.3.1-i) does not hold any longer. Then, inverter i may pass certain amount
of inductive (positive) reactive power to inverter j no matter whether the value of xij (k)
is positive or not.
Remark 2. Condition 4.3.1-ii) limits the amount of reactive power that inverter i
can pass to its neighboring nodes. It indicates that after the reactive power transfer
the reactive power of inverter i must not be lower than the reactive power perception
of any of its neighboring inverters. This condition excludes the oscillation of reactive
power between inverters. Condition 4.3.1-iii) guarantees that after taking the reactive
power passed by inverter i, the reactive power perception of inverter j is not higher
than the maximum reactive power perception of the neighboring nodes of inverter i.
Remark 3. Condition 4.3.1-iv) implies that if inverter i passes some amount of
reactive power to its neighboring nodes, then it must pass some nonnegligible amount
of reactive power to the neighboring inverter with minimum reactive power perception.
Achievement of Power Conservation
The total amount of reactive power provided by the grid-connected PV system and
the desired reactive power of the grid must be identical. In order to achieve the power
conservation, without loss of generality, we assume that there are M “leader nodes”
(DC-AC inverters) that are able to receive the information about the power demands
55
from the grid. Define L = {i : i ∈ L} to be the subset of the leader nodes, and
|L| = M . We assume that the leader nodes have memory units which can memorize
how much reactive power needs to pass to inverters. Also, we define M = {1, . . . , M }
to be the subset of the memory units and each r ∈ M corresponds to an i ∈ L. For
memory unit r ∈ M, we define xm
r (k) to be the total amount of reactive power that
the memory unit r of inverter i holds and needs to be allocated among inverters at
time k. Note that xm
r (k) is different from xi (k) which is the amount of reactive power
that inverter i is supplying at time k. We also define QD
r (k) to be the reactive power
demand information of the grid that inverter i (memory unit r) perceives at time k.
We assume that
M
X
QD
r (k) = QD (k)
(4.3)
r=1
where QD (k) is the total reactive power demand of the grid at time k. The reason to
have such an assumption is that the grid-connected PV system is geographically large
and reactive power demands could occur at different locations. The decentralized
reactive power demand commands are sent to nearby inverters that are leader nodes
in the communication network. When M = 1, it is the case that the total reactive
power demand is sent to the only leader node in the network.
For the inverters which are the leader nodes, we define αrr→i ≥ 0 to be the amount
of reactive power that the memory unit r of inverter i passes to inverter i and αii→r ≥ 0
to be the amount of reactive power that inverter i passes to its memory unit r. We
need to clarify that the memory unit is not able to supply reactive power, the value
of xm
r (k) is only the reactive power that needs to be allocated among inverters. The
term αrr→i is the amount of reactive power removed from the value of xm
r (k) which
is held by the memory unit of inverter i and also the amount of reactive power that
56
is added to the amount of reactive power that inverter i supplies and the term αii→r
is the amount of reactive power removed from inverter i and also the amount of
reactive power that is added to the value of xm
r (k). For memory unit r ∈ M, we have
additional conditions for the memory units:
• If xm
r (k) > 0:
m
4.3.1-a1) αii→r = 0, αrr→i ≥ γrm xm
r (k), where γr ∈ (0, 1) for r ∈ M is a coefficient
that represents minimum proportion of xm
r (k) that the memory unit passes
to inverter i;
r→i
4.3.1-a2) xm
≥ 0;
r (k) − αr
• If xm
r (k) < 0:
4.3.1-b1) αrr→i = 0, αii→r ≥ γrm |xm
r (k)|;
i→r
4.3.1-b2) xm
≤ 0;
r (k) + αi
Remark 4. Condition 4.3.1-a1) says if the value of xm
r (k) > 0, which means the
memory holds inductive reactive power that needs to be allocated, then the leader node
inverter i does not “pass” reactive power to its memory. Instead, the memory passes
a nonnegligible portion inductive reactive power to the inverter. Condition 4.3.1-b1)
says if the value of xm
r (k) < 0 which means the memory holds capacitive reactive power
that needs to be allocated, then inverter i needs to “pass” a nonnegligible amount of
reactive power to the memory. Actually, inverter i reduces αii→r reactive power and
this αii→r is added to the value of xm
r (k) to cancel some capacitive reactive power.
This means the memory unit allocates capacitive reactive power among the inverters.
57
i→r
Note that αii→r ≥ γrm |xm
≥ γrm |xi (k)|. This guarantees Condition b1)
r (k)| not αi
and Condition b2) do not conflict.
Remark 5. Condition 4.3.1-a2) guarantees that after the reactive power passing, the
m
value of xm
r (k) is still nonnegative provided that xr (k) > 0 before the reactive power
passing. Condition 4.3.1-b2) says if xm
r (k) < 0, after the reactive power passing, the
value of the reactive power in the memory unit should not be positive. These two
conditions prohibit alternating the sign of the reactive power value of the memory.
Note that Conditions 4.3.1-i)–4.3.1-iv) defined above are still effective for this case.
However, as inverter i passes αii→r to the associated memory, after the reactive power
P
passing the reactive power of inverter i, xi (k)− {j:j∈Ni } αii→j −αii→r , is not guaranteed
to not be less than xij (k) + αii→j : j ∈ Ni . This means we assume that xi (k) −
P
i→j
≥ min{xij (k) + αii→j , ∀j ∈ Ni for i ∈ L, but we do not assume that
{j:j∈Ni } αi
j
P
xi (k) − {j:j∈Ni } αii→j − αii→r ≥ xij (k) + αii→j , ∀j ∈ Ni . The reason to do this is
that inverter i ∈ L is assumed to pass a nonnegligible amount of reactive power to
P
its memory and assuming that xi (k) − {j:j∈Ni } αii→j − αii→r ≥ xij (k) + αii→j , ∀j ∈
Ni would prohibit the reactive power passing from inverter i to its memory when
P
xi (k) − {j:j∈Ni } αii→j = xij (k) + αii→j , for some j ∈ Ni .
Stability Analysis of Uniformly Balanced Reactive Power
Next, we will analyze the properties of the strategies based on Lyapunov approach.
With the reactive power balancing strategies presented above, the state equation is
expressed as follows:
xi (k + 1) = xi (k) −
X
αii→j +
{j:(i,j)∈A}
X
{j:(i,j)∈A}
58
αij→i , ∀ i ∈ I − L
(4.4)
where the term αij→i ≥ is the amount of reactive power that inverter j passes to i.
Equation Equation (4.4) says the reactive power of inverter i at time k + 1 equals the
reactive power of inverter i at previous time plus the total amount of reactive power
it receives and minus the total amount of reactive power it sends. Consider the leader
node inverter i ∈ L, the state equations are
X
X
xi (k + 1) = xi (k) −
αii→j +
{j:(i,j)∈A}
xm
r (k
+ 1) =
xm
r (k)
−
αrr→i
+
{j:(i,j)∈A}
αii→r
+
QD
r (k
αij→i − αii→m + αim→i , ∀ i ∈ L
+ 1) −
QD
r (k),
(4.5)
∀r∈M
D
where QD
r (k + 1) and Qr (k) are the reactive power demand from the grid at time
k + 1 and k. Equation (5) says for the leader nodes, the reactive power of the inverter
at time k + 1 should also add the amount of reactive power from the memory unit
and subtract the amount of reactive power it passes to the memory unit.
In order to show that the reactive power of all the inverters will be balanced finally
with the state equations given by Equation (4.4) and (4.5), we define x(k) ∈ RN ,
x(k) = [x1 (k), . . . , xN (k)]⊤ to be the state vector of the reactive power for inverters
m
⊤
and xm (k) ∈ RM , xm (k) = [xm
1 (k), . . . , xM (k)] to be the state vector of the reactive
power values in the memory units of leader nodes. We also create an augmented state
vector xA (k) ∈ RN +M , xA (k) = [x⊤ (k), xm⊤ (k)]⊤ . Next, we define an invariant set,
such that any state xA (k) in the invariant set exhibits the following properties: 1) the
reactive power between any two neighboring inverters is balanced; 2) all the memory
units of the leader nodes have zero reactive power values. Then, the invariant set is
chosen as
Xb =
A
N +M
x (k) ∈ R
m
: xi (k) = xj (k), ∀ (i, j) ∈ A; x (k) = 0
59
(4.6)
To study the ability of the system to automatically distribute the reactive power, we
use Lyapunov stability theoretic approach. We assume that the grid reactive power
demand is constant, i.e., QD (k) = QD . We have the following result:
Theorem 4.3.1. (No Inverter Constraints, Evenly Balanced Reactive Power,
Asymptotic Convergence) For the grid-connected PV systems with DC-AC inverters having the reactive power balancing strategies given by conditions 4.3.1-i)–4.3.1iv), 4.3.1-a1)–4.3.1-a2), and 4.3.1-b1)–4.3.1-b2) the invariant set Xb is asymptotically
stable in large.
Proof of Theorem 4.3.1. See Appendix A.
Theorem 4.3.1 shows the reactive power passing strategy guarantees asymptotic
stability to a balanced state. The rate at which the system converges to a balanced
state is also important. The reactive power passing strategies for inverters and memories of leader inverters are just conditions of reactive power passing behaviors but
do not indicate how often the reactive power passing occurs. In order to investigate
the convergence rate, without loss of generality, we assume that there exists B > 0,
such that there is the occurrence of reactive power passing (confined by the reactive
power passing conditions defined in subsection 4.3.1) for each inverter and memory
every B time steps.
Theorem 4.3.2. (No Inverter Constraints, Evenly Balanced Reactive Power,
Exponential Stability) Assume every B time steps, there is the occurrence of reactive power passing behaviors, that are defined by the reactive power balancing strategies
given in subsection 4.3.1, for each inverter and memory of the leader inverters in the
grid-connected PV systems, then the invariant set Xb is exponentially stable in large.
60
Proof of Theorem 4.3.2. See Appendix A.
Theorem 4.3.2 indicates that the convergence rate to the invariant set with the
assumption that the reactive power passing occurs every B time steps for each inverter. Notice that since the occurrence of the reactive power passing of each inverter
is asynchronous and random, without the assumption the convergence rate is not
determined and Theorem 4.3.2 is not guaranteed to be true.
4.3.2
Uniformly Distributed Reactive Power with Inverter
Constraints
We now modify the model of the system to allow for constraints on the inverter
capabilities of providing and absorbing reactive power. For instance, the amount of
reactive power that one inverter can generate under certain real power transferred
and certain power factor is bounded below (capacitive reactive power) and above
(inductive reactive power). Without loss of generality, we define qimin and qimax to be
the minimum and maximum reactive power that inverter i can generate, respectively,
and assume that q min < 0 and qimax > 0. Notice that qimax and qimin are not necessarily
time-invariant, i.e., their values can change when the environmental conditions of
inverter i change. Hence, we use qimax (k) and qimin (k) to denote the upper and lower
bounds of the reactive power of inverter i at time k. For simplicity, we ignore the
memory units of leader nodes in the network and the reactive passing for them.
P
We focus on a simplex ∆ = {x ∈ RN : N
i=1 xi = QD } in which the xi dynamics
evolve, where we assume that QD is constant and known. Additional analysis for
the case of considering leader nodes and their memory units is similar to the one in
subsection 4.3.1. Here, we let U(k) = {i ∈ I : qimin (k) < xi < qimax (k)} represent
61
the set of inverters with unsaturated reactive power, i.e., in which the reactive power
does not reach the bounds at time k, and let S(k) = I − U(k) represent the set
of inverters with saturated reactive power, i.e., in which the reactive power reaches
the bounds. We present a class of reactive power passing strategies considering the
reactive power bounds of inverters. Then, a distribution of reactive power is presented
by an invariant set and proved to be stable in the sense of Lyapunov under certain
conditions.
Reactive Power Passing Strategies
Due to the reactive power bounds of each inverter, we modify the model of reactive
power passing for the no bounds case. Here, we do not assume that the amount of reactive power being passed from one inverter to another is always inductive (positive).
Instead, for the case that the inverters are balancing a total amount of inductive
reactive power we assume that the reactive power being passed between inverters is
capacitive (negative), i.e., when QD > 0, we assume that αii→j ≤ 0. For the case that
the inverters are balancing a capacitive reactive power, we assume that the reactive
power being passed between inverters is inductive (positive), i.e, when QD < 0, we
assume that αii→j ≥ 0. Then, the following conditions define a class of reactive power
passing strategies for inverter i at time k with the considerations of inverter bounds:
• When QD > 0, we assume that inverters can only pass capacitive reactive power
between each other.
4.3.2-a1) αii→j = 0, if xi (k) − xij (k) ≥ 0 or if xi (k) = qimax (k);
4.3.2-a2) xi (k) −
P
{j:j∈Ni }
αii→j ≤ min{xij (k) + αii→j , qimax (k) + αii→j }, ∀ j ∈ Ni such
that xi (k) − xij (k) < 0;
62
i
′
xj ′ (k) : j ∈ Ni ;
4.3.2-a3) If
< 0 for some j, then
+
≥ min
j′
i→j
i→j ∗
i
4.3.2-a4) If αi < 0 for some j, then αi
≤ γij ∗ max xi (k) − xj ∗ (k) , xi (k) −
i
′
∗
max
xj ′ (k) : j ∈ Ni ;
where j = arg max
qi
′
αii→j
xij (k)
αji→j
j
• When QD < 0, we assume that inverters can only pass inductive reactive power
between each other.
4.3.2-b1) αii→j = 0, if xi (k) − xij (k) ≤ 0 or if xi (k) = qimin (k);
4.3.2-b2) xi (k) −
that
P
αii→j ≥ max{xij (k) + αii→j , qimin (k) + αii→j }, ∀j ∈ Ni such
{j:j∈Ni }
xi (k) − xij (k)
> 0;
i
′
xj ′ (k) : j ∈ Ni ;
4.3.2-b3) If
> 0 for some j, then
+
≤ max
j′
i→j ∗
i→j
i
4.3.2-b4) If αi > 0 for some j, then αi
≥ γij ∗ min xi (k) − xj ∗ (k) , xi (k) −
i
′
∗
min
xj ′ (k) : j ∈ Ni ;
where j = arg min
qi
′
αii→j
xij (k)
αji→j
j
Remark 6. Condition 4.3.2-a1) indicates that inverter i will not pass any capacitive
reactive power to its neighboring inverter j if its reactive power perception about inverter j is greater than its own reactive power, i.e., if the reactive power i is greater
than the reactive power perception of inverter j, inverter i will not increase the reactive power level of itself to decrease the reactive power level of inverter j. Also,
inverter i will not pass any capacitive reactive power to its neighboring inverters if
the reactive power of inverter i reaches the upper bound, i.e., inverter i cannot take
more inductive reactive power for this case.
Remark 7. Condition 4.3.2-a2) limits the amount of capacitive reactive power that
inverter i can pass to its neighbor nodes then limits the increase of the reactive power
63
level of inverter i. It indicates that after the reactive power transfer the reactive
power of inverter i must be not higher than the reactive power perception of any of
its neighbor inverters or its upper bound. This condition excludes the oscillation of
reactive power between inverters.
Remark 8. Condition 4.3.2-a4) implies that if inverter i passes some capacitive
reactive power to its neighboring nodes, then it must pass some nonnegligible amount
of capacitive reactive power to the neighboring inverter with maximum reactive power
level. Meanwhile, the reactive power of inverter i is guaranteed not to exceed the upper
bound.
The analysis for the case that QD < 0 is similar to the case that QD > 0, so we will
not present remarks for that case.
Distribution of Reactive Power
The state equation of xi with the reactive power passing strategies presented above
is
xi (k + 1) = xi (k) −
X
αii→j +
{j:(i,j)∈A}
X
{j:(i,j)∈A}
αij→i , ∀i ∈ I
(4.7)
Let X = ∆ be the set of states and x(k) = [x1 (k), . . . , xN (k)]⊤ ∈ X be the state
vector, with xi (k) the reactive power of inverter i at time k ≥ 0. Then, the set
Xc = x ∈ X : for all i ∈ I, either xi (k) = xj (k) for all (i, j) ∈ A such that
qjmin (k) < xj (k) < qjmax (k), xi (k) > xj (k) for all (i, j) ∈ A such that
(4.8)
xj (k) = qjmax (k), and xi (k) < xj (k) for all (i, j) ∈ A such that
min
max
min
xj (k) = qj (k); or xi (k) = qi (k) or qi (k)
represents a distribution of the reactive power on the inverter network. Any distribution x ∈ Xc is such that for any i ∈ I either xi = q max (k) when QD > 0, xi = q min (k)
64
when QD < 0; or if qimin (k) < xi < qimax (k) it must be the case that all neighboring
inverters j ∈ Ni such that qjmin (k) < xj < qjmax (k) have the same reactive power
levels as inverter i. In Xc , if xj = qjmax (k) when QD > 0 for j ∈ Ni , then xj ≤ xi ; if
xj = qjmin (k) when QD < 0 for j ∈ Ni , then xj ≥ xi . Notice that when x(k) ∈ Xc
there is only one reactive power passing strategy that satisfies conditions 4.3.2-a1)—
4.3.2-a4) or 4.3.2-b1)—4.3.2-b4), i.e., αji→j = 0 for all i ∈ I. Recall that for all x ∈ X
there exists a subset of inverters with unsaturated reactive power, denoted by U(k).
For any x ∈ Xc , the subset U(k) is not unique and the specific equalized reactive
power levels of inverters in this subset are not always known as a priori or at any
point before the set Xc is achieved. The set U(k) and the equalized reactive power
levels emerge while the reactive power is distributed over the inverters.
Emergence of Inverter Islands
According to the definition of Xc it is possible that inverters in the subset U(k) are
isolated (by the inverters with saturated reactive power) and have different reactive
power levels. This could occur, for instance, if two inverters with high reactive power
levels are separated by an inverter with saturated reactive power, i.e., xi−1 (k) 6= xi+1 ,
xi (k) = qimax (k), and xi (k) ≤ min{xi−1 (k), xi+1 (k)}. Hence, depending on the graph’s
topology, there could be isolated “islands” of inverters of which the reactive power
does not reach the bounds, where only inverters belong to the same island have the
same reactive power level. Moreover, notice that the formation of inverter islands
depends on the total reactive power, their initial distribution x(0), and the changes
of environmental conditions, i.e., qimax (k) and qimin (k).
65
Stability Analysis
Let us consider the reactive power distribution defined by Equation (4.8). As
discussed above, the invariant set consists of many elements which represent different
reactive power distributions and some distributions can lead to saturated reactive
power on certain inverters (i.e., reactive power of that inverter hits the bounds). The
next theorem shows that under certain situations there is no inverter with saturated
reactive power and the distribution represented by the invariant set is unique.
Theorem 4.3.3. (Uniform Distribution, Unsaturated Reactive Power, Uniqueness of Invariant Set) If qimax and qimin are consistent with time k and the total
amount reactive power satisfies N maxi {qimin } < QD < N mini {qimax }, then the invariant set Xc satisfies |Xc | = 1, and the invariant set Xc is simplified to Xc = {x ∈
X : for all i ∈ I, xi (k) = xj (k) for all (i, j) ∈ A}.
Proof of Theorem 4.3.3. See Appendix A.
Theorem 4.3.3 implies the conditions under which there is no inverter with saturated reactive power (i.e., no inverter’s reactive power hits the bounds) and the
uniqueness of the invariant set. All inverters will eventually have the same reactive
power level and the reactive power level only depends on the number of inverters N
and the total desired reactive power QD . Then, the following analysis considering
inverter reactive power bounds is restricted to the following scenario:
Assumption 4.3.1. (Complete Graph, Consistent Inverter Constraints, Saturated Reactive Power)
(i) The graph G = (I, A) is fully connected.
66
(ii) The environmental conditions of inverters are consistent, i.e., qimax and qimin are
time-invariant for all i ∈ I.
(iii) The total amount reactive power QD satisfies either N mini {qimax } < QD <
PN max
P
min
for QD > 0, or N
< QD < N maxi {qjmin } for QD < 0.
i=1 qi
i=1 qj
Assumption 4.3.1 (i) indicates a complete graph in which every node is connected
to other nodes; (ii) guarantees the bounds of reactive power for each inverter are fixed
and known; (iii) is the condition such that the reactive power of certain inverters in
the network will eventually reach either the lower bound or upper bound but QD is
less than the greatest total reactive power capability of the entire system.
Theorem 4.3.4. (Complete Graph, Uniqueness of Invariant Set) With conditions (i) and (ii) of Assumption 4.3.1 and any total amount of reactive power such
P
PN max
min
that N
q
<
Q
<
the invariant set Xc satisfies |Xc | = 1.
D
i
i=1
i=1 qi
Proof of Theorem 4.3.4. See Appendix A.
Theorem 4.3.4 implies that for a fully connected graph topology there is no isolated
inverters with different reactive power levels. The full connectivity of the inverters
leads to reactive power equalization across all inverters with unsaturated reactive
power and the emergence in some cases (i.e., the cases given by condition (iii) of
Assumption 4.3.1) of a set of inverters with saturated reactive power.
Theorems 4.3.3 and 4.3.4 studied the characteristics of the invariant set Xc that
represents the reactive power distribution for different total reactive power amount
and connectivity topologies of the network. We now focus on the analysis of inverters
approaching this set especially for the case that some inverters have saturated reactive
power.
67
Theorem 4.3.5. (Complete Graph, Emergence of Saturated Inverters, Asymptotic Stability in Large) With Assumption 4.3.1 and the reactive power passing
strategies 4.3.2-a1)–4.3.2-a4), 4.3.2-b1)–4.3.2-b4), the invariant set Xc is asymptotically stable in large.
Proof of Theorem 4.3.5. See Appendix A.
Theorem 4.3.5 considers the inverters with saturated reactive power and studies the stability properties of the invariant set. With the reactive power passing
conditions 4.3.2-a1)–4.3.2-a4) and 4.3.2-b1)–4.3.2-b4) Theorem 4.3.5 indicates on a
complete graph for the total reactive power QD that satisfies Assumption 4.3.1 the
reactive power distribution will eventually end in the invariant set Xc , i.e., the reactive power of some inverters is saturated at the bounds and other inverters equalize
the reactive power level.
We now assume more restrictive reactive power passing conditions in order to
study the rate of convergence to the desired distribution. In particular, we assume
Assumption 4.3.2. (Rate of Occurrence) Every B time steps, there is the occurrence of the reactive power passing behaviors that are defined by conditions 4.3.2a1)–4.3.2-a4) and 4.3.2-b1)–4.3.2-b4) for every inverter.
Then, the following theorem is derived.
Theorem 4.3.6. (Complete Graph, Emergence of Saturated Inverters, Exponential Stability) With Assumptions 4.3.1 and 4.3.2, and the reactive power
passing strategies defined by conditions 4.3.2-a1)–4.3.2-a4), 4.3.2-b1)–4.3.2-b4), the
invariant set Xc is exponentially stable in large.
Proof of Theorem 4.3.6. See Appendix A.
68
4.3.3
Optimally Distributed Reactive Power
Multiple capability-limited inverters in the network can cooperatively generate a
large amount of desired reactive power for the grid with the uniformly distributed
reactive power balancing conditions. Such conditions aim at achieving an equalized
reactive power level on all inverters in the system. Under some circumstances, the
inverter constraints confine the reactive power of certain inverters below the the equalized reactive power level of others when QD > 0 (above the equalized reactive power
level of others when QD < 0). We now modify the reactive power balancing conditions to consider the optimality of the allocation of reactive power on inverters. For
instance, as indicated in Chapter 2 an optimally allocated reactive power profile can
achieve a maximum total “safety margin” of the entire system. We now investigate
the reactive power balancing conditions for such optimal reactive power allocation
strategies. Consider a multiple DC-AC inverter system of which the communication
network is defined by a directed graph G = (I, A), then the optimal reactive power
allocation strategies given in Chapter 2 are represented as follows:
• If max
i∈I
√
−
9|V |2 Ci2 −Pi2
Pi
P
Pi
i∈I
≤ QD ≤ min
i∈I
i ∈ I, the optimal reactive power is
• If QD > min
i∈I
power is
√
9|V |2 Ci2 −Pi2
Pi
x∗i = P
P
i∈I
Pi
Pi
i∈I
Pi
√
9|V |2 Ci2 −Pi2
Pi
P
i∈I
Pi , then for all
QD , ∀ i ∈ I
(4.9)
> 0, then for all i ∈ I, the optimal reactive
x∗i = qimax , i = 1, . . . , r
r
X
Pi
∗
max
xi = P N
, i = r + 1, . . . , N
QD −
qi
i=r+1 Pi
i=1
69
(4.10)
where we assume that all inverters are sorted in a sequence such that
q2max
P2
≤ ... ≤
max
qN
,
PN
q1max
P1
≤
and the number r, which is the number of inverters with
saturated reactive power, is given by
Pi
r = arg min r : PN
i=r+1
• If QD < max
i∈I
power is
√
−
Pi
9|V |2 Ci2 −Pi2
Pi
QD −
P
i∈I
Pi
r
X
qimax
i=1
<
qimax ,
i = r + 1, . . . , N
< 0, then for all i ∈ I, the optimal reactive
x∗i = qimin , i = 1, . . . , t
t
X
Pi
∗
min
xi = P N
, i = t + 1, . . . , N
QD −
qi
i=t+1 Pi
i=1
(4.12)
where we assume that all inverters are sorted in a sequence such that
q2min
P2
≥ ... ≥
min
qN
,
PN
(4.11)
q1min
P1
≥
and the number t, which is the number of inverters with
saturated reactive power, is given by
Pi
t = arg min t : PN
i=t+1
Pi
QD −
t
X
qimin
i=1
We still focus on the same simplex ∆ = {x ∈ RN :
>
qimin ,
PN
i=1
i = t + 1, . . . , N
(4.13)
xi = QD } and assume QD is
constant and known. In order to develop a class of passing strategies for the optimally
allocated reactive power, we rewrite Equation (4.9) as
QD
x∗i
=P
Pi
i∈I Pi
(4.14)
Equation (4.14) implies that the ratio of optimally allocated reactive power to the real
power of all inverters with unsaturated reactive power is equal to the ratio of total
reactive power to the total real power. Hence, the reactive power passing conditions
are now modified based on
xi
Pi
instead of the reactive power xi .
70
Reactive Power Passing Strategies
In order to achieve a maximum “safety margin” of the system, the reactive power
passing strategies are based on the equalization of the ratio of reactive power to
the real power for each inverter. Also, due to the different capabilities of different
inverters, it is possible to have some inverters with saturated reactive power at the
bounds in the system. By taking these factors into account and assuming that the
reactive power being passed between inverter is capacitive (negative) when QD > 0
(inductive when QD < 0), the following conditions define a class of optimally allocated
reactive power passing strategies for inverter i at time k with the considerations of
inverter bounds:
• When QD > 0, we assume that inverters can only pass capacitive reactive power
between each other.
4.3.3-a1) αii→j = 0, if Pi1(k) xi (k) − Pj1(k) xij (k) ≥ 0 or if xi (k) = qimax (k);
P
i→j
i→j
1
1
1
i
4.3.3-a2) Pi (k) xi (k) −
, Pi (k) qimax (k) +
≤ min Pj (k) xj (k) + αi
αi
{j:j∈Ni }
αii→j , ∀ j ∈ Ni such that Pi1(k) xi (k) − Pj1(k) xij (k) < 0 ;
i
xj ′ (k)
i→j
i→j
1
4.3.3-a3) If αi < 0 for some j, then Pj (k) xij (k) + αj
≥ min
: j ′ ∈ Ni ;
Pj ′
j′
2[Pj ∗ (k)xi (k)−Pi (k)xij ∗ (k)]
i→j
i→j ∗
4.3.3-a4) If αi < 0 for some j, then αi
≤ γij ∗ max
, xi (k)−
Pi (k)+Pj ∗ (k)
i
xj ′ (k)
where j ∗ = arg max
: j ′ ∈ Ni ;
qimax
P ′ (k)
′
j
j
• When QD < 0, we assume that inverters can only pass inductive reactive power
between each other.
4.3.3-b1) αii→j = 0, if
1
x (k)
Pi (k) i
−
1
xi (k)
Pj (k) j
71
≤ 0 or if xi (k) = qimin (k);
1
Pi (k)
P
αii→j
1
Pj (k)
αii→j
1
Pi (k)
,
≥ max
xi (k) −
+
qimin (k) +
{j:j∈Ni }
αii→j , ∀ j ∈ Ni such that Pi1(k) xi (k) − Pj1(k) xij (k) > 0;
i
xj ′ (k)
i→j
i→j
1
≤ max
: j ′ ∈ Ni ;
4.3.3-b3) If αi > 0 for some j, then Pj (k) xij (k) + αj
Pj ′
j′
2[Pj ∗ (k)xi (k)−Pi (k)xij ∗ (k)]
i→j
i→j ∗
, xi (k)−
4.3.3-b4) If αi > 0 for some j, then αi
≥ γij ∗ min
Pi (k)+Pj ∗ (k)
i
xj ′ (k)
where j ∗ = arg min
: j ′ ∈ Ni ;
qimin
P ′ (k)
′
4.3.3-b2)
j
xij (k)
j
Remark 9. Condition 4.3.3-a1) indicates that inverter i will not pass any capacitive
reactive power to its neighboring inverter j if its reactive power perception about inverter j is optimally greater than its own reactive power, i.e., if the ratio of reactive
power to the real power of inverter i is greater than the corresponding ratio of inverter
j, inverter i will not increase the reactive power level of itself to decrease the reactive
power level of inverter j. Also, inverter i will not pass any capacitive reactive power
to its neighboring inverters if the reactive power of inverter i reaches the upper bound,
i.e., inverter i cannot take more inductive reactive power for this case.
Remark 10. Condition 4.3.3-a2) limits the amount of capacitive reactive power that
inverter i can pass to its neighbor nodes then limits the increase of the reactive power
level of inverter i. It indicates that after the reactive power transfer the ratio of
reactive power to real power of inverter i must not be higher than the corresponding
ratio of any of its neighbor inverters or the ratio of reactive power upper bound to
real power of itself. This condition excludes the oscillation of reactive power between
inverters.
Remark 11. Condition 4.3.3-a4) implies that if inverter i is not optimally balanced
with all of its neighbors, then it must pass some nonnegligible amount of capacitive
72
reactive power to the neighboring inverter with maximum optimal reactive power level.
Meanwhile, the reactive power of inverter i is guaranteed not to exceed the upper
bound. Condition 4.3.3-a4) is derived from
1
1
1
1
1
i→j ∗
i→j ∗
≤ γij ∗
α
xi (k) −
α
xj ∗ (k)
+
2 Pi (k) i
Pj ∗ (k) i
Pi (k)
Pj ∗ (k)
i
xj ′ (k)
′
∗
: j ∈ Ni . Equation (4.15) directly implies that
where j = arg min
P ′ (k)
′
j
(4.15)
j
∗
αii→j ≤ 2γij ∗
Pj ∗ (k)xi (k) − Pi (k)xij ∗ (k)
Pi (k) + Pj ∗ (k)
(4.16)
The analysis for the case that QD < 0 is similar to the case that QD > 0, so we will
not present remarks for that case.
Distribution of Optimal Reactive Power
The state equation of xi with the reactive power passing conditions 4.3.3-a1)–
4.3.3-a4) and 4.3.3-b1)–4.3.3-b4) is same as the one given by Equation (4.7). Let
X = ∆ be the set of states and x(k) = [x1 (k), . . . , xN (k)]⊤ ∈ X be the state vector,
with xi (k) the reactive power of inverter i at time k ≥ 0. Then, the set
1
1
xi (k) =
xj (k) for all (i, j) ∈ A such that
Xd = x ∈ X : for all i ∈ I, either
Pi (k)
Pj (k)
1
1
qjmin (k) < xj (k) < qjmax (k),
xi (k) >
xj (k) for all (i, j) ∈ A such that
Pi (k)
Pj (k)
1
1
xj (k) = qjmax (k), and
xi (k) <
xj (k) for all (i, j) ∈ A such that
Pi (k)
Pj (k)
min
max
min
xj (k) = qj (k); or xi (k) = qi (k) or qi (k)
(4.17)
represents a distribution of the reactive power on the inverter network. Any distribution x ∈ Xd is such that for any i ∈ I either xi = q max (k) when QD > 0, xi = q min (k)
when QD < 0; or if qimin (k) < xi < qimax (k) it must be the case that all neighboring
73
inverters j ∈ Ni such that qjmin (k) < xj < qjmax (k) have the same ratio of reactive
power to real power as inverter i. In Xc , if xj = qjmax (k) when QD > 0 for j ∈ Ni ,
then
1
x
Pj j
≤
1
x;
Pi i
if xj = qjmin (k) when QD < 0 for j ∈ Ni , then
1
x
Pj j
≥
1
x.
Pi i
Notice
that when x(k) ∈ Xd there is only one reactive power passing strategy that satisfies
conditions 4.3.3-a1)—4.3.3-a4) or 4.3.3-b1)—4.3.3-b4), i.e., αji→j = 0 for all i ∈ I.
Similar to the uniformly distributed reactive power case, for any x ∈ Xd , according
to the definition of Xd , it is possible that inverters in the subset U(k) are isolated (by
the inverters with saturated reactive power) and have different optimal reactive power
levels, i.e., the ratio of reactive power to real power. Hence, there could be isolated
“islands” of inverters in the network. The formation of inverter islands depends on
the total reactive power, their initial distribution, the real power of each inverter, and
the constraints on reactive power of each inverter, i.e., qimax and qimin .
Let us consider the reactive power distribution defined by Equation (4.17). The
invariant set consists of many elements which represent different optimal reactive
power distributions and some distributions can lead to saturated reactive power on
certain inverters (i.e., reactive power of that inverter hits the bounds). The next
lemma shows that under certain situations there is no inverter with saturated reactive
power and the distribution represented by the invariant set is unique.
Theorem 4.3.7. (Optimal Distribution, Unsaturated Reactive Power, Uniqueness of Invariant Set) If Pi , qimax , and qimin are consistent with time k for all i and
the total amount reactive power satisfies maxi {
qimin
}
Pi
<
Q
PN D
i=1
Pi
< mini {
qimax
},
Pi
then
the invariant set Xd satisfies |Xd | = 1, and the invariant set Xd is simplified to
Xd = {x ∈ X : for all i ∈ I,
1
x (k)
Pi i
=
1
x (k)
Pj j
Proof of Theorem 4.3.7. See Appendix A.
74
for all (i, j) ∈ A}.
Theorem 4.3.7 implies the conditions under which there is no inverter with saturated reactive power (i.e., no inverter’s reactive power hits the bounds) and the
uniqueness of the invariant set. All inverters will eventually have the same ratio of
reactive power to real power and the equalized ratio of reactive power to real power
P
only depends on the total real power N
i=1 Pi and the total desired reactive power
QD .
Next, let us assume a complete graph topology (i.e., every inverter connects to
every other inverter). By adding assumption, we can loose the assumption on QD
then we have the following theorem:
Theorem 4.3.8. (Optimal Distribution, Complete Graph, Uniqueness of
Invariant Set) For a fully connected graph (I, A) and any total amount of reactive
P
P
max
min
the invariant set Xd satisfies |Xd | = 1.
< QD < N
power such that N
i=1 qi
i=1 qi
Proof of Theorem 4.3.8. See Appendix A.
Theorem 4.3.8 implies that for a fully connected graph topology there is no isolated
inverters with different reactive power to real power ratios. The full connectivity of the
inverters leads to reactive power to real power ratio equalization across all inverters
with unsaturated reactive power and the emergence (in some cases) of a set of inverters
with saturated reactive power. Theorems 4.3.7 and 4.3.8 studied the characteristics
of the invariant set Xd that represents the optimal reactive power distribution for
different total reactive power amount and connectivity topologies of the network. We
now focus on the analysis of inverters approaching this set.
75
Stability Analysis
Let us now consider again a general graph topology (I, A) and assume that every
inverter is connected to the graph, but not every inverter connects to every other
inverter. Also, we assume that the environmental conditions of the system are consistent with time k, i.e., Pi , qimax , and qimin are time-invariant.
Theorem 4.3.9. (Optimal Distribution, Asymptotic Stability in Large) Given
(I, A) and the reactive power passing strategies 4.3.3-a1)–4.3.3-a4), 4.3.3-b1)–4.3.3b4), there exists a constant QD such that the total desired reactive power QD satisfies
maxi {
qimin
}
Pi
<
Q
PN D
i=1
Pi
< mini {
qimax
},
Pi
then the invariant set Xd is asymptotically stable
in large.
Proof of Theorem 4.3.9. See Appendix A.
Since Xd is asymptotically stable in large, there is only one equilibrium distribution
for each total amount reactive power QD which satisfies maxi {
mini {
qimax
}.
Pi
qimin
}
Pi
<
Q
PN D
i=1
Pi
<
Thus, for any initial reactive power distribution this equilibrium can be
achieved.
We now assume more restrictive reactive power passing conditions in order to
study the rate of convergence to the desired distribution. In particular, we assume
Assumption 4.3.3. (Rate of Occurrence) Every B time steps, there is the occurrence of the reactive power passing behaviors that are defined by conditions 4.3.3a1)–4.3.3-a4) and 4.3.3-b1)–4.3.3-b4) for every inverter.
Then, the following theorem is derived:
Theorem 4.3.10. (Optimal Distribution, Exponential Stability) Given (I, A)
and the reactive power passing strategies 4.3.3-a1)–4.3.3-a4), 4.3.3-b1)–4.3.3-b4),
76
there exists a constant QD such that the total desired reactive power QD satisfies
maxi {
qimin
}
Pi
<
Q
PN D
i=1 Pi
< mini {
qimax
},
Pi
then with Assumption 4.3.3 the invariant set Xd
is exponentially stable in large.
Proof of Theorem 4.3.10. See Appendix A.
It is shown in Theorem 4.3.9 and 4.3.10 the stability characteristics of the optimal
reactive power distribution Xd with assumptions on the total amount of reactive power
of the system. We now consider the stability of Xd for a more general QD but with
the assumption of a complete graph.
Theorem 4.3.11. (Optimal Distribution, Complete Graph, Emergence of
Saturated Inverters, Asymptotic Stability in Large) For a fully connected
P
min
graph (I, A), any total amount of reactive power that satisfies N
< QD <
i=1 qi
PN max
, and the reactive power passing strategies 4.3.3-a1)–4.3.3-a4), 4.3.3-b1)–
i=1 qi
4.3.3-b4), the invariant set Xd is asymptotically stable in large.
Proof of Theorem 4.3.11. See Appendix A.
Since we do not have the same restriction on QD as the one in Theorem 4.3.9,
there can be some inverters with saturated reactive power in the network. However,
Theorem 4.3.8 indicates the uniqueness of Xd for a fully connected graph. Then, any
initial reactive power distribution will eventually converge to the unique equilibrium
Xd . The rate of the convergence to Xd with Assumption 4.3.3 for this case is given
by the following theorem:
Theorem 4.3.12. (Optimal Distribution, Complete Graph, Emergence of
Saturated Inverters, Exponential Stability) For a fully connected graph (I, A),
77
any total amount of reactive power that satisfies
PN
min
i=1 qi
< QD <
PN
max
,
i=1 qi
and
the reactive power passing strategies 4.3.3-a1)–4.3.3-a4), 4.3.3-b1)–4.3.3-b4), with
Assumption 4.3.3 the invariant set Xd is exponentially stable in large.
Proof of Theorem 4.3.12. See Appendix A.
78
4.3.4
Simulation: A Case Study
Now let us aim at a sample 1.5 MW grid-connected PV system with a total number
of 8 inverters, two types of DC-AC inverters. The required data of these inverters
for the case study is shown in Table 2.1. In order to distinguish each inverter from
the others, we index these inverters from 1 to 8. Specifically, we index all 5 type 1
inverters to be inverter 1, 2, 4, 6, and 7; index all 3 type 2 inverters to be inverter 3,
5, and 8. Hence, Ci = 301 A for i = 1, 2, 4, 6, 7 and Ci = 121 A for i = 3, 5, 8. The
√
nominal output voltage is 480 V AC, line to line. Then, |V | = 480/ 3 = 277.1 V.
Next, we will test the reactive power balancing strategies for different scenarios and
graph topologies.
Uniformly Distributed Reactive Power
Let us consider the case that the reactive power is uniformly distributed among all
of the inverters. First, we investigate the ring topology of the communication system
which is shown in Figure 4.1, i.e., each inverter is connected to 2 other inverters. If
Inverter 1
Inverter 2
Inverter 3
Inverter 4
Inverter 8
Inverter 7
Inverter 6
Inverter 5
Figure 4.1: The ring topology of the communication system of the DC-AC inverter
network.
79
we assume a uniform 0.9 solar irradiation profile for all the photovoltaic panels, the
output real power of each inverter is 0.9Pimax for i = 1, . . . , N , then we have
Pi = 0.9Pimax = 0.9 × 250 = 225 kW, for i = 1, 2, 4, 6, 7
Pi =
0.9Pimax
(4.18)
= 0.9 × 100 = 90 kW, for i = 3, 5, 8
Based on the real power given in Equation (4.18) the limits of reactive power for type
1 inverters are
qimin
qimax
q
= − 9|V |2 Ci2 − Pi2 = −109.54 kVar
=
−qimin
= 109.54 kVar, i = 1, 2, 4, 6, 7
and the limits of reactive power for type 2 inverters are
q
min
qi = − 9|V |2 Ci2 − Pi2 = −44.94 kVar
qimax
=
(4.19)
−qimin
(4.20)
= 44.94 kVar, i = 3, 5, 8
Hence, the total maximum reactive power the system can provide is qtmax = 682.52
kVar. Now we fix the total desired reactive power (i.e., we assume that QD is time
invariant and do not consider the existence of leader inverters) and consider two cases:
• Inverters with unsaturated reactive power
According to Theorem 4.3.3, if all the inverters are not saturated, the total
desired reactive power needs to satisfy N maxi {qimin } < QD < N mini {qimax }.
In this case, N = 8, maxi {qimin } = −44.94 kVar, and mini {qimax } = 44.94 kVar.
Therefore, the total desired reactive power needs to satisfy −359.52 kVar <
QD < 359.52 kVar. In this simulation, we let QD = −200 kVar and this total
desired reactive power is randomly distributed among all 8 inverters initially.
By using the reactive power passing conditions 4.3.2-b1)–b4) and assuming that
the reactive power passing occurs once every 1s for each inverter, the reactive
power of the system reaches a uniform distribution which is shown in Figure 4.2.
80
Inver t er 1
Inver t er 2
0
−40
−40
−40
−40
−60
x 4 ( kVar )
−20
x 3 ( kVar )
−20
−60
−60
−60
−80
−80
−80
−80
−100
−100
−100
−100
0
200
−120
400
0
Time ( s )
−120
400
0
Inver t er 6
0
200
−120
400
Inver t er 7
0
−40
−40
−40
x 8 ( kVar )
−40
x 7 ( kVar )
−20
x 6 ( kVar )
−20
−60
−60
−80
−80
−80
−80
−100
−100
−100
−100
−120
0
200
Time ( s )
400
−120
0
200
400
−120
0
200
Time ( s )
Time ( s )
400
Inver t er 8
0
−20
−60
200
Time ( s )
−20
−60
0
Time ( s )
Time ( s )
Inver t er 5
0
200
Inver t er 4
0
−20
−120
x 5 ( kVar )
Inver t er 3
0
−20
x 2 ( kVar )
x 1 ( kVar )
0
400
−120
0
200
400
Time ( s )
Figure 4.2: Uniform reactive power distribution without saturated inverters. The
solid line of each subplot: reactive power of each inverter; the dashed line of each
subplot: lower bound of the reactive power.
In Figure 4.2, all the inverters will reach an equal reactive power level in less
than 200s (indicated by Theorem 4.3.5). It is worth pointing out that if the
simulation time is divided into multiple 1s time intervals, the time instant of
the occurrence of the reactive power passing in each time interval is random.
In order to show the convergence of the reactive power balancing strategies, we
investigate the total error which is given by
N X
xi (k) − x̄i E=
x̄i
(4.21)
i=1
Figure 4.3 shows that the error decreases to 0 in about 150s. Also, since we
assume the reactive power passing occurs every 1s for each inverter, the reactive
81
Tot al er r or of inver t er r eact ive p ower and t he unifor m r eact ive p ower level
6
5
3
PN
i= 1
| x i − x̄ i|
x̄ i
4
2
1
0
0
50
100
150
200
250
300
350
400
Ti me (s)
Figure 4.3: Total error of inverter reactive power and the uniform reactive power
level.
power distribution converges exponentially to the invariant set. The minimum
reactive power is always non-decreasing which is shown in Figure 4.4.
• Inverters with saturated reactive power
If the total desired reactive power is changed to −400 kVar, Theorem 4.3.3 is not
satisfied. Hence, there exist some inverters with the reactive power hitting the
lower bound. Similar to the previous case, we let QD = −400 kVar randomly
distribute among all 8 inverters initially and also assume the ring topology.
Figure 4.5 shows the simulation result of the reactive power balancing among all
8 inverters of a ring topology with total desired reactive power QD = −400 kVar.
It is observed that inverters 1 and 2 finally reached an equal reactive power level
of −50.77 kVar, inverters 6 and 7 reached a different equal reactive power level
of −59.34 kVar, inverter 4 reached a different reactive power level from all other
82
Mi n i mu m reacti ve p ower
−20
−30
−50
−60
−70
i
mi n{x i } (kVar)
−40
−80
−90
−100
−110
0
50
100
150
200
250
300
350
400
Ti me (s)
Figure 4.4: The minimum reactive power at each time step.
type 1 inverters at −44.94. The reactive power of all 3 type 2 inverters reached
the lower bound at −44.94 kVar. The reason for the emergence of inverter
islands is that unsaturated inverters are isolated by saturated inverters. For
instance, inverters 1 and 2 are isolated by inverters 3 and 8 from other type 1
inverters. In order to let all unsaturated inverters have the same reactive power
level while all saturated inverters have the reactive power at the lower bound, the
communication topology of the system is modified to a fully connected graph,
i.e., every inverter is connected to all other inverters in the network. Figure 4.6
shows the connection of inverter 1 in the complete graph. Figure 4.7 shows the
reactive power balancing of all 8 inverters for a fully connected topology. It is
shown that all unsaturated inverters reached one equal reactive power level and
all saturated inverters reached the reactive power lower bounds. The inverter
islands are avoided due to the fully connected graph of all the inverters.
83
−20
X: 199.2
Y: −50.77
−40
X: 199.2
Y: −50.77
−60
x 2 ( kVar )
−40
−60
−20
−20
−40
−40
−60
−80
−80
−100
−100
−100
−100
50
100
150
−120
200
0
50
Inver t er 5
0
100
150
−120
200
0
50
Time ( s )
Time ( s )
150
−120
200
Inver t er 7
0
−40
−40
−40
X: 199.2
Y: −59.34
−60
X: 199.2
Y: −59.34
x 8 ( kVar )
−40
x 7 ( kVar )
−20
x 6 ( kVar )
−20
−60
−80
−80
−80
−100
−100
−100
−100
50
100
150
200
−120
0
Time ( s )
50
100
150
200
Time ( s )
−120
0
50
100
150
200
−60
−80
0
100
150
Time ( s )
Inver t er 8
0
−20
−120
50
Time ( s )
−20
−60
0
Time ( s )
Inver t er 6
0
100
X: 199.2
Y: −44.94
−60
−80
0
Inver t er 4
0
−80
−120
x 5 ( kVar )
Inver t er 3
0
x 3 ( kVar )
−20
x 1 ( kVar )
Inver t er 2
0
x 4 ( kVar )
Inver t er 1
0
200
−120
0
50
100
150
200
Time ( s )
Figure 4.5: Uniform reactive power distribution: the emergence of inverter islands
due to saturated inverters and limited connection topology.
Inverter 1
Inverter 2
Inverter 3
Inverter 8
Inverter 4
Inverter 7
Inverter 6
Inverter 5
Figure 4.6: A complete graph of the DC-AC inverter network. It only shows the
connections of inverter 1.
84
Inver t er 1
Inver t er 2
0
−40
−40
−40
−40
−60
x 4 ( kVar )
−20
x 3 ( kVar )
−20
−60
−60
−60
−80
−80
−80
−80
−100
−100
−100
−100
0
20
−120
40
0
Time ( s )
Inver t er 5
0
20
−120
40
0
Time ( s )
−120
40
Inver t er 7
0
−40
−40
−40
x 8 ( kVar )
−40
x 7 ( kVar )
−20
x 6 ( kVar )
−20
−60
−60
−80
−80
−80
−80
−100
−100
−100
−100
−120
0
20
Time ( s )
40
−120
0
20
40
Time ( s )
−120
0
20
Time ( s )
40
Inver t er 8
0
−20
−60
20
Time ( s )
−20
−60
0
Time ( s )
Inver t er 6
0
20
Inver t er 4
0
−20
−120
x 5 ( kVar )
Inver t er 3
0
−20
x 2 ( kVar )
x 1 ( kVar )
0
40
−120
0
20
40
Time ( s )
Figure 4.7: Uniform reactive power distribution with saturated inverters for a fully
connected graph. The solid line of each subplot: reactive power of each inverter; the
dashed line of each subplot: lower bound of the reactive power.
85
Optimally Distributed Reactive Power
Now we consider the case that the reactive power is optimally distributed among
all of the inverters such that the entire system has a maximized “ safety margin”.
Similarly, we investigate the ring topology shown in Figure 4.1 first. We still consider
two cases:
• Inverters with unsaturated reactive power
We still assume a uniform 0.9 solar irradiation profile for all the PV panels and
the total desired reactive power is QD = −200 kVar for this case. Hence, the
ratio of total reactive power to total real power is
QD
PN
i=1 Pi
=
−200
200
=−
= −0.1434
225 × 5 + 90 × 3
1395
(4.22)
For type 1 inverters, the ratio of reactive power lower bound to real power and
the ratio of reactive power upper bound to real power are
q max
−109.54
q min
qimin
= −0.4868, i
=
= − i = 0.4868, for i = 1, 2, 4, 6, 7 (4.23)
Pi
225
Pi
Pi
For type 2 inverters, the ratio of reactive power lower bound to real power and
the ratio of reactive power upper bound to real power are
qimin
−44.94
q max
q min
=
= −0.4993, i
= − i = 0.4993, for i = 3, 5, 8
Pi
90
Pi
Pi
Hence, it is obvious that maxi {
qimin
}
Pi
<
Q
PN D
i=1
Pi
< mini {
qimax
}
Pi
(4.24)
and Theorem 4.3.7
is satisfied. By using the optimal reactive power passing conditions 4.3.3-b1)b4) and assuming that the reactive power passing occurs once every 1s for each
inverter, the reactive power balancing is shown in Figure 4.8 and Figure 4.9.
Figure 4.8 shows the reactive power and the lower bound of each inverter, Figure 4.9 shows the ratio of reactive power to real power of each inverter. For
86
the optimal reactive power distribution, inverters actually try to reach an equal
ratio of reactive power to real power rather than an equal reactive power level.
Figure 4.10 shows the current margin of the entire system. It is obvious from
−40
−40
−60
−80
−20
−30
−40
−60
−80
−40
−100
200
−120
400
0
−30
−40
200
400
−20
−20
−40
−40
−60
−80
−100
−100
0
200
400
−120
0
400
−120
Time ( s )
200
400
Time ( s )
Inver t er 8
0
−10
−60
−80
−120
200
Inver t er 7
0
x 7 ( kVar )
x 6 ( kVar )
−20
Time ( s )
0
Time ( s )
Inver t er 6
0
−10
0
−50
400
Time ( s )
Inver t er 5
0
200
−100
x 8 ( kVar )
0
Time ( s )
x 5 ( kVar )
−20
−10
−60
Inver t er 4
0
−80
−100
−50
Inver t er 3
0
x 3 ( kVar )
−20
x 2 ( kVar )
x 1 ( kVar )
−20
−120
Inver t er 2
0
x 4 ( kVar )
Inver t er 1
0
−20
−30
−40
0
200
Time ( s )
400
−50
0
200
400
Time ( s )
Figure 4.8: Optimal reactive power distribution with unsaturated inverters for a ring
connection topology. The solid line of each subplot: reactive power of each inverter;
the dashed line of each subplot: lower bound of the reactive power.
this figure that the optimal reactive power distribution increases the system
capability of taking more current (load).
• Inverters with saturated reactive power (partially shaded conditions)
For the optimal reactive power distribution, we do not assume an increased total
desired reactive power to have saturated inverters but consider a case that likely
87
−0.1
x 3 /p 3
−0.3
−0.3
−0.4
−0.4
−0.4
−0.5
−0.5
−0.5
−0.5
400
0
Inver t er 5
0
200
400
0
Time ( s )
Inver t er 6
0
−0.1
X: 400
Y: −0.1434
0
X: 400
Y: −0.1434
X: 400
Y: −0.1434
x 7 /p 7
−0.3
−0.3
−0.4
−0.4
−0.5
−0.5
−0.5
−0.5
200
400
0
200
400
Time ( s )
0
200
Time ( s )
400
X: 400
Y: −0.1434
−0.2
−0.4
Time ( s )
Inver t er 8
0
−0.4
0
400
−0.1
−0.2
−0.3
200
Time ( s )
−0.1
−0.2
x 6 /p 6
−0.3
400
Inver t er 7
0
−0.1
−0.2
200
Time ( s )
x 8 /p 8
200
Time ( s )
X: 400
Y: −0.1434
−0.2
−0.4
0
x 5 /p 5
−0.1
X: 400
Y: −0.1434
−0.2
−0.3
Inver t er 4
0
−0.1
X: 400
Y: −0.1434
−0.2
x 2 /p 2
−0.3
Inver t er 3
0
−0.1
X: 400
Y: −0.1434
−0.2
x 1 /p 1
Inver t er 2
0
x 4 /p 4
Inver t er 1
0
0
200
400
Time ( s )
Figure 4.9: The ratio of optimally distributed reactive power to real power with
unsaturated inverters for a ring connection topology. The solid line of each subplot:
the ratio of reactive power to real power; the dashed line of each subplot: the ratio
of reactive power lower bound to real power.
occurs for large scale PV systems: the partially shaded conditions. We assume
that the solar panels of all type 1 inverters are partially shaded by heavy clouds
such that they have 0.1 solar irradiation. Also, we assume that the solar panels
of inverter 5 (which is type 2 inverter) has a 0.1 solar irradiation profile as well.
Inverters 3 and 8 have the same 0.9 solar profile. The output real power of each
88
Tot al cur r ent mar gin of t he s ys t em
174
172
(A )
166
164
PN
i =1 C i
−
√
168
p 2i +x 2i
3| V |
170
162
160
158
156
154
0
50
100
150
200
250
300
350
400
Ti me (s)
Figure 4.10: The current margin of the system for the optimal reactive power distribution with unsaturated inverters.
inverter is
Pi = 0.1Pimax = 0.1 × 250 = 25 kW, for i = 1, 2, 4, 6, 7
Pi = 0.1Pimax = 0.1 × 100 = 10 kW, for i = 5
(4.25)
Pi = 0.9Pimax = 0.9 × 100 = 90 kW, for i = 3, 8
Based on the real power given in Equation (4.25) the limits of reactive power
of each inverter are
q
qimin = − 9|V |2 Ci2 − Pi2 = −248.99 kVar, qimax = −qimin = 248.99 kVar, i = 1, 2, 4, 6, 7
q
min
qi = − 9|V |2 Ci2 − Pi2 = −100.1 kVar, qimax = −qimin = 100.1 kVar, i = 5
q
min
qi = − 9|V |2 Ci2 − Pi2 = −44.94 kVar, qimax = −qimin = 44.94 kVar, i = 3, 8
(4.26)
89
Then, the ratio of reactive power lower bound to real power and the ratio of
reactive power upper bound to real power for each inverter are
q max
−248.99
q min
qimin
= −9.9596, i
=
= − i = 9.9596, for i = 1, 2, 4, 6, 7
Pi
25
Pi
Pi
min
max
min
qi
q
−100.1
q
(4.27)
= −10.01, i
=
= − i = 10.01, for i = 5
Pi
10
Pi
Pi
qimin
−44.94
qimax
qimin
=
= −0.4993,
=−
= 0.4993, for i = 3, 8
Pi
25
Pi
Pi
The total desired reactive power is still QD = −200 kVar for this case and the
ratio of total reactive power to total real power is
QD
PN
i=1 Pi
=
200
−200
=−
= −0.6349
25 × 5 + 90 × 2 + 10
315
(4.28)
Hence, Theorem 4.3.7 is not satisfied for this case and there are saturated
inverters in the system (which are inverters 3 and 8). Figure 4.11 and Figure 4.12
show the reactive power balancing for partially shaded conditions with a ring
topology of the system. Figure 4.12 shows that due to the saturated inverters
3 and 8, inverters 1 and 2 have an equal ratio of the reactive power to real
power while inverters 4-7 have a different equal ratio. This is because that the
saturated inverters 3 and 8 isolate them to form two islands. It is noticed that
even inverters 4-7 have the same ratio of reactive power to real power, inverter
5 has a different reactive power level from inverters 4, 6, and 7. This is due
to the different inverter types.
Figure 4.13 shows the current margin of the
entire system. It is obvious from this figure that the optimal reactive power
distribution increases the system capability of taking more current (load). In
order to avoid the emergence of inverter islands, a fully connected graph is
used. Figure 4.14 shows that the ratio of reactive power to real power of all
unsaturated inverters are equal for a fully connected graph. Also, it is observed
90
−50
−100
−150
−200
X: 400
Y: −33.18
−100
−150
200
300
400
0
100
Time ( s )
300
−50
400
−60
−80
−150
−200
−250
0
100
200
300
400
0
100
−100
−150
−200
300
400
Inver t er 8
0
X: 400
Y: −12.87
−50
200
Time ( s )
Inver t er 7
0
x 7 ( kVar )
−40
−100
Time ( s )
X: 400
Y: −12.87
−50
x 6 ( kVar )
x 5 ( kVar )
200
Inver t er 6
0
X: 400
Y: −5.148
−20
−30
Time ( s )
Inver t er 5
0
−20
−10
x 8 ( kVar )
100
X: 400
Y: −12.87
−50
−40
−250
0
Inver t er 4
0
−10
−200
−250
Inver t er 3
0
x 3 ( kVar )
X: 400
Y: −33.18
x 2 ( kVar )
x 1 ( kVar )
−50
Inver t er 2
0
x 4 ( kVar )
Inver t er 1
0
−100
−150
−200
−20
−30
−40
−100
−250
0
100
200
300
Time ( s )
400
−250
0
100
200
300
400
Time ( s )
0
100
200
300
Time ( s )
400
−50
0
100
200
300
400
Time ( s )
Figure 4.11: Optimal reactive power distribution with saturated inverters (partially
shaded conditions) for a ring connection topology. The solid line of each subplot:
reactive power of each inverter; the dashed line of each subplot: lower bound of the
reactive power.
from the comparison between Figures 4.13 and 4.15 that the fully connectedness
slightly increases the capability of the entire system.
91
−2
−2
X: 400
Y: −1.327
X: 400
Y: −0.5148
−2
−0.2
−6
Inver t er 4
0
−0.1
X: 400
Y: −1.327
x 3 /p 3
−6
Inver t er 3
0
−4
x 2 /p 2
−4
x 1 /p 1
Inver t er 2
0
−4
x 4 /p 4
Inver t er 1
0
−0.3
−6
−0.4
−8
−8
−8
−0.5
−10
−10
200
400
−10
0
Time ( s )
0
400
Inver t er 8
0
−0.1
−0.2
−4
−6
200
Time ( s )
X: 400
Y: −0.5148
−2
x 7 /p 7
x 6 /p 6
400
Inver t er 7
0
−4
−6
200
Time ( s )
X: 400
Y: −0.5148
−2
−4
x 5 /p 5
0
Inver t er 6
0
X: 400
Y: −0.5148
−2
400
Time ( s )
Inver t er 5
0
200
x 8 /p 8
0
−6
−0.3
−0.4
−8
−8
−8
−10
−10
−10
−0.5
0
200
Time ( s )
400
0
200
400
Time ( s )
0
200
Time ( s )
400
0
200
400
Time ( s )
Figure 4.12: The ratio of optimally distributed reactive power to real power with
saturated inverters for a ring connection topology. The solid line of each subplot: the
ratio of reactive power to real power; the dashed line of each subplot: the ratio of
reactive power lower bound to real power.
92
Tot al cur r ent mar gin of t he s ys t em
1420
1410
X: 400
Y: 1411
1390
1380
PN
i =1 C i
−
√
p 2i +x 2i
3| V |
(A )
1400
1370
1360
1350
1340
1330
0
50
100
150
200
250
300
350
400
Ti me (s)
Figure 4.13: The current margin of the system for the optimal reactive power distribution with saturated inverters (partially shaded conditions) for a ring connection.
93
X: 400
Y: −0.8157
−2
x 3 /p 3
x 2 /p 2
X: 400
Y: −0.8157
−2
−0.2
−6
Inver t er 4
0
−0.1
−4
−6
Inver t er 3
0
X: 400
Y: −0.8157
−2
−4
x 1 /p 1
Inver t er 2
0
−4
x 4 /p 4
Inver t er 1
0
−0.3
−6
−0.4
−8
−8
−8
−0.5
−10
−10
100
200
300
400
−10
0
100
Time ( s )
0
100
400
0
100
300
400
Inver t er 8
0
−0.1
−0.2
−4
−6
200
Time ( s )
X: 400
Y: −0.8157
−2
x 7 /p 7
x 6 /p 6
300
Inver t er 7
0
−4
−6
200
Time ( s )
X: 400
Y: −0.8157
−2
−4
x 5 /p 5
400
Inver t er 6
0
X: 400
Y: −0.8157
−2
300
Time ( s )
Inver t er 5
0
200
x 8 /p 8
0
−6
−0.3
−0.4
−8
−8
−8
−10
−10
−10
−0.5
0
100
200
300
Time ( s )
400
0
100
200
300
400
Time ( s )
0
100
200
300
Time ( s )
400
0
100
200
300
400
Time ( s )
Figure 4.14: The ratio of optimally distributed reactive power to real power with
saturated inverters for a fully connected graph. The solid line of each subplot: the
ratio of reactive power to real power; the dashed line of each subplot: the ratio of
reactive power lower bound to real power.
94
Tot al cur r ent mar gin of t he s ys t em
1420
X: 400
Y: 1416
1400
1390
PN
i =1 C i
−
√
p 2i +x 2i
3| V |
(A )
1410
1380
1370
1360
0
50
100
150
200
250
300
350
400
Ti me (s)
Figure 4.15: The current margin of the system for the optimal reactive power distribution with saturated inverters (partially shaded conditions) for a fully connected
graph.
95
Evaluation of the System Topology
In order to study the impact of the system topology on the reactive power
balancing of the 8 inverter system, we perform Monte Carlo runs to compare
the performance of different topologies. Specifically, we focus on the case where
there are only unsaturated inverters and we define the performance measure
as the time needed for the total difference between the inverter reactive power
level and the equalized reactive power level to reach and settle for a given range
N P
xi (k)−x̄i (2%), i.e., we measure the time ts that the total difference E =
x̄i i=1
settles into the 2% range. We will investigate different topologies in which each
inverter has a different number of neighbors. For instance, for the ring topology
each inverter has 2 neighbors and the number of cooperating inverters is 3;
for the fully connected graph each inverter has 7 neighbors and the number
of cooperating inverters is 8. We will investigate the cases that the number of
cooperating inverters is from 3 to 8. Here, we assume that cooperating inverters
are the ones being neighbors. Figure 4.16 shows the topologies in which the
number of cooperating inverters is from 4 to 7. Figures 4.1 and 4.6 show the
ring topology and a complete graph, respectively. Figure 4.17 shows that the
mean and the standard deviations of the settling time decrease as the number
of neighbors of inverters increases for the uniform reactive power distribution
on unsaturated inverters. It is obvious that the when the number of cooperative
inverters increases, each inverter is balancing reactive power with more other
inverters. Thus, it speeds up the convergence of the balancing process. Since the
system have more connections as the number of cooperative inverters increases,
different initial reactive power distributions have less impacts on the convergence
96
Inverter 1
Inverter 2
Inverter 8
Inverter 3
Inverter 1
Inverter 4
Inverter 7
Inverter 6
Inverter 8
Inverter 5
Inverter 2
Inverter 8
Inverter 6
Inverter 6
Inverter 5
(b) Number of cooperating inverters: 5
Inverter 3
Inverter 1
Inverter 4
Inverter 7
Inverter 3
Inverter 4
Inverter 7
(a) Number of cooperating inverters: 4
Inverter 1
Inverter 2
Inverter 2
Inverter 8
Inverter 5
Inverter 4
Inverter 7
(c) Number of cooperating inverters: 6
Inverter 3
Inverter 6
Inverter 5
(d) Number of cooperating inverters: 7
Figure 4.16: System topologies in which each inverter has a different number of
neighbors (only shows the connections of inverter 1).
time of the balancing process. This leads the decrease of the standard deviation
of the settling time.
For the optimal reactive power distribution case, the settling time is defined as
the time that the total difference between the ratio of reactive power to real
power of each inverter and the ratio of total desired reactive power to total real
power of the system reaches and settles for a given range (2%), i.e., we measure
the total difference
N X
E=
xi (k)
pi
i=1
−
Q
PN D
Q
PN D
i=1
i=1
pi
pi
(4.29)
We still perform Monte Carlo runs to compare the performance of different
topologies. Figure 4.18 shows that the mean and the standard deviations of the
97
Set t ling t ime of differ ent t op ologies for unifor mly dis t r ibut ed r eact ive p ower
200
180
S ettl i n g ti me t s (s)
160
140
120
100
80
60
40
20
0
2
3
4
5
6
7
8
9
N u mb er of co op era ti n g i nverters
Figure 4.17: Settling time of uniform reactive power distribution for different system topologies. Every data point represents 200 simulation runs with varying initial
conditions. The error bar are standard deviations for these runs.
settling time decrease as the number of neighbors of inverters increases for the
optimal reactive power distribution on unsaturated inverters.
4.4
Summary
In this chapter, distributed reactive power control based on balancing strategies
is proposed for the inverter network of large scale grid-connected PV systems. Uniform reactive power distribution and optimal reactive power distribution among all
inverters are considered. Reactive power balancing strategies are presented for both
desired distributions. Invariant sets are defined to denote the desired reactive power
distributions. Then, stability analysis is conducted for the invariant sets by using
98
Set t ling t ime of differ ent t op ologies for opt imally dis t r ibu t ed r eact ive p ower
300
S ettl i n g ti me t s (s)
250
200
150
100
50
0
2
3
4
5
6
7
8
9
N u mb er of co op era ti n g i nverters
Figure 4.18: Settling time of optimal reactive power distribution for different system topologies. Every data point represents 200 simulation runs with varying initial
conditions. The error bar are standard deviations for these runs.
Lyapunov stability theory. In order to validate the proposed reactive power balancing strategies, a case study is performed on a sample large scale grid-connected PV
system considering different conditions.
99
Chapter 5: Conclusions
5.1
Summary
In future smart grid, distributed generation especially grid-connected PV systems
will play an important role of generating both real and reactive power. This thesis
proposes a reactive power optimal allocation strategy for the DC-AC inverters in
large scale grid-connected PV systems to achieve a maximized total power transfer
capability. Optimal solutions are derived for both cases that the inverter constrains
are inactive and active. It is indicated by the optimal solutions that the reactive
power of each inverter in the system is proportional to the real power it transfers
when the inverter constraints are inactive. When the inverter constraints are active,
the DC-AC inverters generate reactive power at either the upper bound or lower
bound depending on the polarity of total desired reactive power.
Then, a distributed optimization algorithm is developed based on the optimal
solutions of the original optimization problem. The cost function of the optimization
problem is proved to be convex. Hence, the separable principle is applicable for the
decomposition of the problem. After decomposing the original optimization problem
into multiple sub-problems, the optimal algorithm for each inverter is derived and
100
each inverter only needs to optimize the cost of itself to achieve the optimization of
the cost function of the entire system.
Finally, control schemes based on balancing strategies are proposed for uniform
reactive power distribution and optimal reactive power distribution. The reactive
power balancing strategies only require inverters to have local communication with
neighboring inverters. Invariant sets are formulated to represent the desired uniform
distribution and optimal distribution of reactive power. It is proved that these invariant sets are unique under certain conditions, asymptotically stale in large, and
exponentially stable under certain assumptions. The simulation results of a case study
show that limited connection topologies of the communication system of the inverter
network can lead to the imbalance of reactive power among inverters and increasing
the connections of each inverter with the others can resolve this problem.
5.2
Contributions
The research work in this dissertation presents the reactive power allocation and
control problems in the large scale grid-connected PV systems which have not been
studied. The optimal strategies of the reactive power allocation for the DC-AC inverters in the large scale grid-connected PV systems provide analytical solutions of
maximizing the power transfer capability of the system. The strategies indicate how
the multiple DC-AC inverters in the system cooperatively generate the desired reactive power. The distributed optimization algorithm provides a robust method to
realize the optimization approaches. The reactive power balancing control proposes an
approach to realize the optimal reactive power allocation in large scale grid-connected
PV systems.
101
5.3
Future Work
Future research directions following the work in this thesis include both theoretical
and practical paths. Theoretically, many other reactive power optimization problems
for grid-connected PV systems can be studied and solved. For instance, the reactive
power allocation of PV inverters in a microgrid is a good candidate research problem,
since in a microgrid the real power of PV inverters without energy storage units
may not be controlled at the MPP due the real power conservation. Then, both
real and reactive power need to be optimally allocated in an proper way. Another
possible research direction is to study the reactive power allocation of PV inverters in a
distribution network. For this problem, the impedance of the connection lines and the
phase angles of distribution feeders need to be considered which makes the problem
quite different from this thesis. Practically, the optimal reactive power allocation
strategies, the distributed optimization algorithm, and the reactive power balancing
control need to be tested in a real PV system. A lot of issues that are ignored by the
theoretical solutions need to be considered.
102
Appendix A: Proofs of Theorems
A.1
Optimal Reactive Power Allocation Strategies
Proof of Theorem 2.3.1. Here, we use Lagrange multiplier method [13] to solve the
T
problem given in Equation (2.6). Let Q = Q1 , . . . , QN , the Lagrangian function is
constructed as follows,
p
X
N
Pi2 + Q2i
+λ
Qi − QD
L(Q, λ, µ) = −
Ci −
3|V
|
i=1
i=1
N
q
X
+
µi Qi − 9|V |2 Ci2 − Pi2
N X
+
i=1
N
X
i=1
µi+N
(A.1)
q
− 9|V |2 Ci2 − Pi2 − Qi
where λ, µj , j = 1, . . . , 2N are Lagrange multipliers. As for this case we assume
the reactive power of the ith inverter Qi satisfies Equation (2.5) with strictly inequalities, the inequality constraints are inactive. Hence, the Lagrangian function in
Equation (A.1) becomes
L(Q, λ) = −
N X
i=1
p
X
N
Pi2 + Q2i
+λ
Qi − QD
Ci −
3|V |
i=1
(A.2)
Let the gradient of the Lagrangian function Equation (A.2) ∇Q L(Q, λ) = 0, we have
Q
p i
= −λ, i = 1, . . . , N
3|V | Pi2 + Q2i
103
(A.3)
From Equation (A.3) we know that Qi and λ have opposite signs, and |V | and Pi are
both positive, so we obtain
Qi = − p
If we substitute Equation (A.4) into
as the only variable,
−p
3|V |Pi λ
PN
i=1
3|V |Pi λ
1 − 9|V |2 λ2
By solving Equation (A.5), we have
(A.4)
1 − 9|V |2 λ2
Qi = QD , we obtain one equation with λ
N
X
i=1
Pi − QD = 0
(A.5)
Q2D
λ2 =
9|V |2
PN
i=1
Pi
2
+ Q2D
As Qi and λ have opposite signs, it is obvious that QD and λ also have opposite signs.
Then, λ is expressed as follows,
λ=−
3|V |
s
QD
PN
i=1
Pi
2
(A.6)
+ Q2D
Substituting Equation (A.6) into Equation (A.3) we have the reactive power Q∗i ,
i = 1, . . . , N ,
Pi
Q∗i = PN
i=1
Pi
QD , i = 1, . . . , N
(A.7)
To guarantee Q∗i in Equation (A.7) is the optimal reactive power for the ith inverter,
we need to the Hessian of the Lagrangian function to be positive definite [13]. The
Hessian of the Lagrangian function is

P2
 3|V


∇QQ L(Q∗ , λ∗ ) = 


0
1
|(P12 +Q21 )3/2
0
..
.
0
3|V
104
P22
2
|(P2 +Q22 )3/2
..
.
0
···
0
···
...
···
0
..
.
3|V
2
PN
2
|(PN +Q2N )3/2







where, λ∗ is the one given in Equation (A.6). For all y 6= 0 such that ∇(
QD )T y = 0, we have
∗
T
∗
y ∇QQ L(Q , λ )y =
N
X
i=1
PN
i=1
Qi −
Pi2
y2 > 0
2 3/2 i
2
3|V |(Pi + Qi )
Hence, the Hessian of the Lagrangian function is positive definite. So Q∗i given by
Equation (A.7) is the optimal reactive power profile. To let the inactive inequalities
assumption hold, we need Q∗i to satisfy Equation (2.5). Then, we have
−
q
9|V
|2 Ci2
−
Pi2
Pi
≤ PN
i=1 Pi
QD ≤
q
9|V |2 Ci2 − Pi2 , i = 1, . . . , N
(A.8)
As Equation (A.8) needs to hold for all inverters, we obtain
p
p
N
N
− 9|V |2 Ci2 − Pi2 X
9|V |2 Ci2 − Pi2 X
max
Pi ≤ QD ≤ min
Pi
i=1,...,N
i=1,...,N
Pi
Pi
i=1
i=1
which proves Equation (2.8).
Proof of Theorem 2.3.2. Let us consider the case where the reactive power demand
QD > 0. Suppose that all the inverters are already sorted in the order given by
Equation (2.10) and in such an order the reactive power of the first r − 1 inverters
already hit their upper bounds. Now consider the assumption that the reactive power
of the rth inverter does not reach its upper bound, i.e., Qr < Qmax
, and the reactive
r
power of the (r + 1)th inverter hits its upper bound, i.e., Qr+1 = Qmax
r+1 . As indicated
by the assumption, the reactive powers Qr and Qi , i = r + 2, . . . , m, do not hit their
upper bounds, according to Theorem 2.3.1 for these N − r reactive powers, we have
Pr−1 max
QD − i=1
Qi − Qmax
r+1
Pi , i = r + 1, r + 2, . . . , N
(A.9)
Qi = P N
Pr−1
i=1 Pi −
i=1 Pi − Pr+1
, then we have
For the rth inverter, we substitute Equation (A.9) into Qr < Qmax
r
Pr−1 max
QD − i=1
Qi − Qmax
r+1
Pr < Qmax
(A.10)
Pm
Pr−1
r
P
−
P
−
P
r+1
i=1 i
i=1 i
105
The (r + 1)th inverter’s reactive power Qr+1 , by the assumption, hits the upper
bounds. If we apply Theorem 2.3.1 and calculate Qr+1 by using a manner similar to
Equation (A.9), the reactive power Qr+1 will exceed the upper bound Qmax
r+1 . Based
on this, we have such inequality
Pr−1 max
QD − i=1
Qi
PN
Pr−1 Pr ≥ Qmax
r+1
i=1 Pi −
i=1 Pi
(A.11)
As Pi > 0, i = 1, . . . , m, from Equation (A.10) and Equation (A.11) we obtain
Pr−1 max
QD − i=1
Qi − Qmax
Qmax
r+1
< r
(A.12)
PN
Pr−1
Pr
i=1 Pi −
i=1 Pi − Pr+1
and
Pr−1 max
Qmax
QD − i=1
Qi
≥ r+1
PN
Pr−1
Pr+1
i=1 Pi −
i=1 Pi
(A.13)
Subtract the left-hand side of Equation (A.12) by the left hand side of Equation (A.13),
we obtain the following inequality
Pr−1 max
Pr−1 max
QD − i=1
Qi − Qmax
QD − i=1
Qi
r+1
− PN
PN
Pr−1
Pr−1
i=1 Pi −
i=1 Pi − Pr+1
i=1 Pi −
i=1 Pi
Pr−1 max
PN
Pr−1
(QD −
Q )Pr+1 − ( i=1 Pi − i=1 Pi )Qmax
r+1
≥0
= PN i=1 Pi r−1
P
Pr−1
( i=1 Pi − i=1 Pi − Pr+1 )( N
P
−
P
i=1 i
i=1 i )
(A.14)
The reason that Equation (A.14) holds is that from Equation (A.11) we know
X
N
r−1
r−1
X
X
max
Pr+1 ≥
QD −
Pi −
Qi
Pi Qmax
r+1
i=1
i=1
i=1
and the denominator of the second line of Equation (A.14) is obviously positive.
Hence,
Pr−1 max
Pr−1 max
QD − i=1
Qi − Qmax
QD − i=1
Qi
r+1
≥ PN
PN
Pr−1
Pr−1
i=1 Pi −
i=1 Pi − Pr+1
i=1 Pi −
i=1 Pi
(A.15)
From Equation (A.11), Equation (A.12), and Equation (A.15), we have the following
inequality,
Pr−1 max
Pr−1 max
QD − i=1
Qi − Qmax
Qmax
QD − i=1
Qi
Qmax
r+1
r+1
r
> PN
≥ PN
Pr−1
Pr−1 ≥
Pr
P
P
−
P
−
P
P
−
P
r+1
r+1
i=1 i
i=1 i
i=1 i
i=1 i
106
(A.16)
The inequality in Equation (A.16) shows
Qmax
r
Pr
>
Qmax
r+1
Pr+1
which contradicts the order
in Equation (2.10). Hence, the assumption that Qr < Qmax
while Qr+1 = Qmax
r
r+1 is
invalid. Then we conclude that when QD > 0 the first r inverters’ reactive power Qi ,
i = 1, . . . , r in the order given in Equation (2.10) hit their upper bounds. For the rest
N − r inverters, the following inequality holds,
Pi
PN
i=r+1 Pi
QD −
r
X
Qmax
j
i=1
< Qmax
, i = r + 1, . . . , N
i
(A.17)
Thus, r is the minimum number that makes Equation (A.17) hold. Similarly, we can
prove the case that QD < 0.
Proof of Theorem 2.3.3. We use Lagrange multiplier method [13] to prove this theorem. The Lagrangian function is the one given in Equation (A.1). We have two
cases.
• The reactive power demand QD > 0. For this case, all the inverters are in
the order given in Equation (2.10). By Theorem 2.3.2, we know that those r
inverters with reactive power that hits the upper bound are the first r inverters
in that order. Hence, the inequality constraints
gi (Qi ) = Qi −
q
9|V |2 Ci2 − Pi2 ≤ 0, i = 1, . . . , r
are active. For i = 1, . . . , r, we have
Qi = Qmax
, i = 1, . . . , r
i
(A.18)
and by taking the gradient of Equation (A.1) we have
µi = −
Q
p i
− λ, i = 1, . . . , r
3|V | Pi2 + Q2i
107
(A.19)
For those N − r inverters with inactive inequality constraints, we have
Q
p i
+ λ = 0, i = r + 1, . . . , N
3|V | Pi2 + Q2i
(A.20)
Also, we have the equality constraints which we are
r
X
Qi +
i=1
N
X
i=r+1
Qi − QD = 0
(A.21)
From Equation (A.20) we obtain,
3|V |Pi λ
Qi = − p
, for i = r + 1, . . . , N
1 − 9|V |2 λ2
(A.22)
Substitute Equation (A.18) and Equation (A.22) into Equation (A.21), and we
obtain
r
X
N
X
3|V |λ
Qi − p
Pi − QD = 0
2 λ2
1
−
9|V
|
i=1
i=r+1
P
QD − ri=1 Qi
3|V |λ
p
= − PN
1 − 9|V |2 λ2
i=r+1 Pi
X
2
N
r
2
X
2 2
9|V | λ
Pi = QD −
Qi (1 − 9|V |2 λ2 )
i=r+1
2 2
9|V | λ
X
N
(A.23)
i=1
Pi
i=r+1
2
+ QD −
r
X
i=1
Qi
2 = QD −
r
X
i=1
Qi
2
From the second line of Equation (A.23) we know that λ has the opposite sign
P
of QD − ri=1 Qi . In this case, λ is negative. As Qi = Qmax
, i = 1, . . . , r. Hence,
i
P
QD − ri=1 Qmax
i
s
λ=−
2 2
PN
Pr
max
+ QD − i=1 Qi
3|V |
i=r+1 Pi
(A.24)
Substitute Equation (A.18) and Equation (A.24) into Equation (A.22), and we
obtain Equation (2.14).
108
• The reactive power demand QD < 0. For this case, all the inverters are in
the order given in Equation (2.12). By Theorem 2.3.2, we know that those t
inverters with reactive power that hits the upper bound are the first t inverters
in that order. Hence, the inequality constraints
gi+N (Qi ) = Qi −
q
9|V |2 Ci2 − Pi2 ≤ 0, i = 1, . . . , t
are active. Similarly, we can prove Equation (2.15).
For both cases,
µj > 0, ∀j ∈ A(Q)
(A.25)
where A(Q) = {j | gj (Q) = 0} is the index set that the inequality constraints are
active. Now we show the reason why Equation (A.25) holds. For the case that QD > 0,
as we assume some inverters’ reactive powers hit their upper bounds, Equation (2.8)
does not hold. Then, consider the inverters in the order given by Equation (2.10).
For i = 1, . . . , r, Qi = Qmax
, then
i
Pi
Qmax
< PN
i
i=1
Pi
QD , i = 1, . . . , r
The reactive power Qi , i = 1, . . . , r, reaches its upper bound, so the amount of reactive
power
P
PN i
i=1
Pi
QD − Qmax
, i = 1, . . . , r, will be allocate on other inverters. Hence,
i
Pi
Qj ≥ PN
i=1 Pi
Then we obtain
QD , j = r + 1, . . . , N
Qj
Qmax
i
<
, i = 1, . . . , r, j = r + 1, . . . , N
Pi
Pj
109
(A.26)
(A.27)
From Equation (A.19) and Equation (A.24), µ∗i is expressed as
P
QD − ri=1 Qmax
i
∗
s
µi =
2
2 Pr
PN
max
3|V |
+ QD − i=1 Qi
i=r+1 Pi
(A.28)
Qmax
p i
, i = 1, . . . , r
)2
3|V | Pi2 + (Qmax
i
P
For the case that QD > 0, we have QD − ri=1 Qmax
> 0 and Qmax
> 0. We can turn
i
i
−
Equation (A.28) into the following form,
1
µ∗i =
3|V |
s
PN
i=r+1 Pi
P
QD − ri=1 Qmax
i
2
1
−
3|V |
+1
s
Pi
Qmax
i
2
, i = 1, . . . , r
(A.29)
+1
Now consider the denominators of those two terms in Equation (A.29). From Equation (2.14), we know that
PN
Pj
i=r+1 Pi
P
, j = r + 1, . . . , N
=
Qj
QD − ri=1 Qmax
i
From Equation (A.27) we know that
Pi
Pj
< max
Qj
Qi
Hence, in Equation (A.29) the denominator of the first term is smaller than the
denominator of the second term. Similarly, we can show it for the case that QD < 0.
Then, we conclude that µj > 0, ∀j ∈ A(Q). For all y 6= 0 such that ∇h(Q)T y = 0,
and ∇gj (Q)T y = 0, ∀j ∈ A(Q), we have
∇QQ L(Q∗ , λ∗ , µ∗ ) =

P2
 3|V





1
|(Q21 +P12 )3/2
0
..
.
0
0
P22
3|V |(Q22 +P22 )3/2
..
.
0
···
···
...
···
110
0
0
..
.
3|V
2
PN
2
2 )3/2
|(QN +PN







(A.30)
and
∗
T
∗
∗
y ∇QQ L(Q , λ , µ )y =
N
X
i=1
Pi2
y2 > 0
3|V |(Q2i + Pi2 )3/2 i
(A.31)
The Hessian of the Lagrangian function is positive definite. Hence, the reactive
power profile given by Equation (2.14) and Equation (2.15) are the optimal allocation
reactive power profile when Equation (2.8) is not satisfied.
A.2
Reactive Power Control Based on Balancing Strategies
Proof of Theorem 4.3.1. Let x̄A = [x̄1 , . . . , x̄N , 0, . . . , 0]⊤ . Choose
| {z }
M 0
A
ρ(x (k), Xb ) = inf
max{|x1 (k) − x̄1 |, . . . , |xN (k) −
m
x̄N |, |xm
1 |, . . . , |xM |}
A
: x̄ ∈ Xb
(A.32)
and
A
V (x (k)) = max
i
X
N
M
1 X
xj (k) − xi (k) +
|xm
r (k)|
N j=1
r=1
(A.33)
Note that the last term in Equation (A.33) represents the sum of the absolute values
of the elements of xm (k). It is obvious that
1
A
max{xi (k)} − min{xi (k)}
ρ(x (k), Xb ) ≥
i
i
2
M
X
A
ρ(x (k), Xb ) ≤ max{xi (k)} − min{xi (k)} +
|xm
r (k)|
i
i
≤ max{xi (k)} − min{xi (k)} + N
i
i
(A.34)
r=1
M
X
r=1
|xm
r (k)|
We also know that
N
M
X
1 X
V (x (k)) =
xj (k) − min{xi (k)} +
|xm
r (k)|
i
N j=1
r=1
A
≤ max{xi (k)} − min{xi (k)} +
i
i
111
M
X
r=1
|xm
r (k)|
(A.35)
From Equation (A.34) we know that
2ρ(xA (k), Xb ) ≥ max{xi (k)} − min{xi (k)}
i
2ρ(xA (k), Xb ) +
M
X
r=1
i
|xm
r (k)| ≥ max{xi (k)} − min{xi (k)} +
i
i
M
X
r=1
|xm
r (k)|
(A.36)
m
Recall that ρ(xA (k), Xb ) = inf{max{|x1 (k) − x̄1 |, . . . , |xN (k) − x̄N |, |xm
1 |, . . . , |xM |} :
x̄A ∈ Xb }. Then, it is obvious that
ρ(xA (k), Xb ) ≥ max{|xm
i (k)|}
i
A
M ρ(x (k), Xb ) ≥ M
max{|xm
i (k)|}
i
≥
M
X
r=1
|xm
r (k)|
(A.37)
Hence, from Equation (A.35)-(A.37) we obtain
V (xA (k)) ≤ 2ρ(xA (k), Xb ) +
M
X
r=1
A
|xm
r (k)| ≤ (M + 2)ρ(x (k), Xb )
(A.38)
Notice that
N
1
1 X
max{xi (k)} + (N − 1) min{xi (k)}
xj (k) ≥
i
i
N j=1
N
(A.39)
Combining Equation (A.35) and (A.39), we have
M
X
1
max{xi (k)} + (N − 1) min{xi (k)} − min{xi (k)} +
|xm
V (x (k)) ≥
r (k)|
i
i
i
N
r=1
X
M
1
max{xi (k)} − min{xi (k)} +
=
|xm
r (k)|
i
i
N
r=1
M
X
1
1
m
max{xi (k)} − min{xi (k)} + N
|xr (k)| ≥ ρ(xA (k), Xb )
=
i
i
N
N
r=1
(A.40)
A
Hence, from Equation (A.38) and (A.40) we have
1
ρ(xA (k), Xb ) ≤ V (xA (k)) ≤ (M + 2)ρ(xA (k), Xb )
N
Thus,
112
(A.41)
• For c1 =
max{xi (k)} − min{xi (k)} > 0 it is possible to find a c2 =
i
i
1
max{xi (k)}−min{xi (k)} > 0 such that V (xA (k)) ≥ c2 and ρ(xA , Xb ) ≥ c1 ;
2N
1
2
i
i
M
P
• For c3 = max{xi (k)} − min{xi (k)} + N
|xm
r (k)| > 0, it is possible to find a
i
i
r=1
M
M
P
P m
c4 = 2 max{xi (k)} − min{xi (k)} + N
|xm
|xr (k)| > 0 such that
r (k)| +
i
i
r=1
A
r=1
A
when ρ(x (k), Xb ) ≤ c3 we have V (x (k)) ≤ c4 ;
• The function V (xA (k)) is nonincreasing with the reactive power passing strategy
given in subsection 4.3.1. The reason is as follows: these inverters try to balance
the reactive power between each other, so the first term of V (xA (k)) will not
increase. The second term of V (xA (k)) represents the total reactive power
value of all the memory units. The memory units try to allocate reactive power
to leader node inverters. Hence, the value of second term of V (xA (k)) will
decrease. However, when the memory units pass reactive power to the leader
node inverters, the reactive power of the leader node inverters may increase.
But note that the decrease of reactive power of the memory units is eqaul to
the increase in reactive power of the leader node inverters. Hence, V (xA (k)) is
a nonincreasing function.
Then, we conclude that for the system with the reactive power balancing strat
egy given in subsection 4.3.1, the invariant set Xb = xA (k) ∈ RN +M : xi (k) =
xj (k), ∀ (i, j) ∈ A; xm (k) = 0 is asymptotically stable in large.
Proof of Theorem 4.3.2. Choose the Lyapunov function V (xA (k)) to be given as follows
A
V (x (k)) = max
i
N
M
X
1 X
xj (k) − xi (k) + 2
|xm
r (k)|
N j=1
r=1
113
(A.42)
It is obvious that
M
1 X m
|x (k)|
ρ(x (k), Xb ) ≥
N r=1 r
A
(A.43)
Similar to Equation (A.38) we know that
A
A
V (x (k)) ≤ 2ρ(x (k), Xb ) + 2
Then, there exist c1 =
1
N
M
X
r=1
A
|xm
r (k)| = (2N + 2)ρ(x (k), Xb )
(A.44)
and c2 = (2N + 2) such that
c1 ρ(xA (k), Xb ) ≤ V (xA (k)) ≤ c2 ρ(xA (k), Xb )
(A.45)
Let γ1 = min{γij }, γ2 = min{γrm }, and γ = min{γ1 , γ2 }. For any i ∈ I − L and
r
i,j
k ≥ 0, we know from condition 4.3.1-ii) in subsection 4.3.1 that if the reactive power
i→j
i→j
i
passing occurs for inverter i, and if αi > 0, then, αi > γ xi (k) − xj (k) . We
have xi (k + 1) ≥ xij (k) + γ xi (k) − xij (k) for j ∈ Ni . If the reactive power passing
does not occur or αii→j = 0, then xi (k + 1) = xi (k). It follows that in any case,
xi (k + 1) ≥ min{xi (k)} + γ xi (k) − min{xi (k)} , ∀i ∈ I − L
i
i
(A.46)
For any i ∈ L, when the reactive power passing occurs, inverter i passes a nonnegligible amount of reactive power to its memory. Hence, Equation (A.46) does not hold
m
any more for this case when xm
r (k) < 0 ( Equation (A.46) still holds if xr (k) > 0)
due to αii→r . Hence, we need to modify Equation (A.46) considering αii→m . For any
i→r
r ∈ M and k ≥ 0, we know that αrr→i ≥ γrm |xm
≥ γrm |xm
r (k)| and αi
r (k)|. Let
αr = αii→r if the rth memory unit receives reactive power at time k and let αr = αrr→i
if the rth memory unit passes reactive power at time k. For either case, it is obvious
that
M
X
r=1
m
m
|xm
r (k)| − |xr (k + 1)| = αr ≥ γ|xr (k)|
|xm
r (k)|
−
M
X
r=1
|xm
r (k
+ 1)| =
M
X
r=1
114
αr ≥ γ
M
X
r=1
|xm
r (k)|
(A.47)
Hence, based on Equation (A.47) by taking consideration of αii→r for the case that
r→i
xm
for the case that xm
r (k) < 0 and αr
r (k) > 0, Equation (A.46) is changed to what
follows,
xi (k + 1) −
M
X
r=1
|xm
r (k
+ 1)| ≥ min{xi (k)} −
i
M
X
r=1
|xm
r (k)|+
γ xi (k) − min{xi (k)} +
i
M
X
r=1
(A.48)
|xm
r (k)| , ∀i ∈ I
P
m
It is worth pointing out that min{xi (k)} − M
r=1 |xr (k)| is a nondecreasing function
i
P
m
of k. Because the term M
i=1 |xr (k)| is decreasing as k increases, and the possible
P
m
decrease on the term min{xi (k)} is not greater than the decrease of M
i=1 |xr (k)|.
i
We now show via induction that
xi (k + t) −
M
X
r=1
|xm
r (k
+ t)| ≥ min{xi (k)} −
i
min{xi (k)} +
i
M
X
r=1
M
X
r=1
|xm
r (k)|
t
+ γ xi (k)−
(A.49)
|xm
r (k)| , ∀i ∈ I
for all t ≥ 0. When t = 1, Equation (A.49) is turned to be Equation (A.48). Now
we assume that Equation (A.49) is true for an arbitrary t, and we show that Equation (A.49) also holds for the case of t + 1. According to Equation (A.48) for any
115
i ∈ I at time k + t + 1 we have
xi (k + t + 1) −
M
X
r=1
≥ min{xi (k + t)} −
i
M
X
r=1
|xm
r (k
+ t)|
≥ min{xi (k + t)} −
i
M
X
r=1
|xm
r (k
M
X
|xm
r (k
M
X
|xm
r (k
r=1
r=1
t
i
r=1
M
X
r=1
|xm
r (k
t
+ t)| + γ min{xi (k)} −
i
+ t)|
r=1
|xm
r (k
i
r=1
i
= min{xi (k)} −
i
min{xi (k)} +
i
= min{xi (k)} −
M
X
r=1
M
X
r=1
M
X
r=1
|xm
r (k)|
|xm
r (k)|
r=1
|xm
r (k)|
r=1
M
X
r=1
M
X
r=1
|xm
r (k)|
|xm
r (k)|
+
i
− min{xi (k)} +
+γ
i
t+1
−
M
X
i
+ γ min{xi (k)} −
|xm
r (k)|+
(A.50)
i
|xm
r (k)| + γ min{xi (k)} −
|xm
r (k)|
M
X
+ t)| + γ min{xi (k)}−
i
M
X
M
X
+ γ xi (k) − min{xi (k)} +
γ t (xi (k) − min{xi (k)} +
i
M
X
≥ (1 − γ) min{xi (k)} −
i
i
i
|xm
r (k)|
+ t)| + γ xi (k + t) − min{xi (k + t)}+
= (1 − γ) min{xi (k + t)} −
M
X
+ t)| + γ xi (k) − min{xi (k)} +
min{xi (k + t)} +
|xm
r (k + t + 1)|
M
X
r=1
M
X
r=1
r=1
M
X
r=1
|xm
r (k)|
|xm
r (k)|
Thus, Equation (A.49) must be valid for all t ≥ 0.
116
+ t)|
|xm
r (k)|+
t
+ γ xi (k)−
xi (k) − min{xi (k)} +
i
|xm
r (k
M
X
r=1
|xm
r (k)|
Fix i ∈ I and k ≥ 0. We now show that the reactive power of all neighbors of i
are bounded from below for all k ′ , k ′ ≥ k + N B. Specifically, we will show that
M
M
X
X
′
m ′
m
k′ −k
xj (k ) −
|xr (k )| ≥ min{xi (k)} −
xi (k)−
|xr (k)| + γ
i
r=1
r=1
min{xi (k)} +
i
M
X
r=1
(A.51)
′
|xm
r (k)| , ∀ k ≥ k + N B, j ∈ Ni
There are times kp ≥ k, p ∈ {1, 2, . . .} such that the reactive power passing occurs,
and the reactive power passing does not occur for k ′ 6= kp . We know from the
assumption we made for this theorem that k ≤ k1 < k+B, kp−1 < kp < kp−1 +B, ∀ p ∈
{2, 3, . . .}.
• Let us consider time kp , p ∈ {1, 2, . . .}, and j ∈ Ni , such that xj (kp ) < xi (kp )
and xj (kp ) ≤ xj ′ (kp ) for all j ′ ∈ Ni . According to condition 4.3.1-iv), we have
i→j
αi ≥ γ xi (kp )−xj (kp ) . Now consider node j, where j ∈ Ni , xj (kp ) < xi (kp ),
and xj (kp ) ≤ xj ′ (kp ), denote the set of the neighboring nodes of node j as Nj .
– For certain node q ∈ Nj at time kp , if xq (kp ) < xj (kp ), we have
X j→q
xj (kp ) −
αj ≥xq (kp ) + αjj→q , ∀ q ∈ Nj ,
q
(A.52)
such that xj (kp ) > xq (kp )
Then,
X
xj (kp + 1) = xj (kp ) −
X
αjj→q +
q
q′
αqq′ →j , ∀ q, q ′ ∈ Nj
′
(A.53)
and if this node j ∈ L, Equation (A.53) is modified to
xj (kp + 1) = xj (kp ) −
X
αjj→q +
q
X
q′
αqq′ →j − αjj→r , ∀ q, q ′ ∈ Nj (A.54)
′
if node j passes reactive power to the memory unit, or
xj (kp + 1) = xj (kp ) −
X
αjj→q +
q
X
q′
117
αqq′ →j + αrr→j , ∀q, q ′ ∈ Nj
′
(A.55)
if the memory unit passes reactive power to node j. No matter which
M
M
P
P
j→r
case is true, since
|xm
(k
)|
−
|xm
(or αrr→j ) > 0,
p
r
r (kp + 1)| ≥ αj
r=1
r=1
Equations (A.53)-(A.55) imply that
xj (kp + 1) −
M
X
r=1
|xm
r (kp
+ 1)| ≥xj (kp ) −
X
M
X
r=1
q ′ →j
αq ′
q′
|xm
r (kp )| −
, ∀ q, q ′ ∈ Nj
X
αjj→q +
q
(A.56)
As we know that node i is one of the neighbors of node j, i.e., j ∈ Ni =⇒
i ∈ Nj , Equation (A.56) is written as
xj (kp + 1) −
M
X
r=1
|xm
r (kp
+ 1)| ≥xj (kp ) −
X
M
X
r=1
′
αqq′ →j
|xm
r (kp )| −
αii→j ,
+
q ′ ,q ′ 6=i
X
αjj→q +
q
∀ q, q ′ ∈ Nj
(A.57)
From condition 4.3.1-ii), we know that for node j at time kp , xj (kp ) −
P j→q
∗
≥ xq∗ (kp ) + αjj→q , where xq∗ (kp ) ≤ xq (kp ), ∀ q ∈ Nj . Therefore,
q αj
we derive from Equation (A.57) that
xj (kp + 1) −
M
X
r=1
∗
j→q
|xm
+ αii→j −
r (kp + 1)| ≥xq ∗ (kp ) + αj
X
q ′ ,q ′ 6=i
′
αqq′ →j ,
′
∀ q ∈ Nj
M
X
r=1
|xm
r (kp )|+
From condition 4.3.1-iv), we know that
j→q ∗
i→j
αj
≥ γ xj (kp ) − xq∗ (kp ) , αi ≥ γ xi (kp ) − xj (kp )
(A.58)
(A.59)
Equation (A.59) holds because that node j has the minimum reactive
power among all neighbors of node i and node q ∗ has the minimum reactive power among all neighbors of node j at time kp . Equations (A.58)
118
and (A.59) imply that
xj (kp + 1) −
M
X
r=1
|xm
r (kp + 1)|
≥xq∗ (kp ) + γ xi (kp ) − xq∗ (kp ) −
≥xq∗ (kp ) + γ xi (kp ) − xq∗ (kp ) −
≥ min{xi (kp )} −
i
M
X
r=1
|xm
r (kp )|
M
X
r=1
M
X
r=1
|xm
r (kp )| +
X
αqq′ →j
′
q ′ ,q ′ 6=i
(A.60)
|xm
r (kp )|
+ γ xi (kp ) − min{xi (kp )}
i
– If xj (kp ) ≤ xq (kp ), ∀ q ∈ Nj , i.e., node j has the minimum reactive power
among all its neighbors, we have
xj (kp + 1) −
≥xj (kp ) −
≥xj (kp ) −
M
X
r=1
M
X
r=1
M
X
r=1
i→j
|xm
r (kp )| + αi
|xm
r (kp )| + γ xi (kp ) − xj (kp )
≥ min{xi (kp )} −
i
|xm
r (kp + 1)|
M
X
r=1
|xm
r (kp )|
(A.61)
+ γ xi (kp ) − min{xi (kp )}
i
Hence, no matter what the reactive power level of node j is, we have
xj (kp + 1) −
M
X
r=1
≥ min{xi (kp )} −
i
|xm
r (kp + 1)|
M
X
r=1
|xm
r (kp )|
+ γ xi (kp ) − min{xi (kp )}
=(1 − γ) min{xi (kp )} + γxi (kp ) − (1 − γ)
i
119
i
M
X
r=1
|xm
r (kp )|
−γ
M
X
r=1
|xm
r (kp )|
(A.62)
Recall that mini {xi } −
xj (kp + 1) −
M
X
r=1
PM
r=1
|xm
r | is nondecreasing, Equation (A.62) implies
|xm
r (kp + 1)|
≥(1 − γ) min{xi (k)} + γxi (kp ) − (1 − γ)
i
M
X
r=1
|xm
r (k)|
−γ
M
X
r=1
|xm
r (kp )|
M
M
X
X
= min{xi (k)} + γ xi (kp ) − min{xi (k)} − (1 − γ)
|xm
(k)|
−
γ
|xm
r
r (kp )|
i
i
r=1
r=1
(A.63)
If we apply Equation (A.49) to Equation (A.63) with t = kp − k, we have
xj (kp + 1) −
M
X
r=1
|xm
r (kp + 1)|
≥ min{xi (k)} + γ min{xi (k)} + γ
i
M
X
r=1
|xm
r (k)|
(1 − γ)
M
X
r=1
+
i
M
X
r=1
|xm
r (k)|
= min{xi (k)} + γ
|xm
r (kp )|
−γ
kp −k+1
i
+γ
M
X
r=1
|xm
r (kp )|
= min{xi (k)} −
i
M
X
r=1
|xm
r (k)|
−γ
M
X
r=1
kp −k
M
X
r=1
−
M
X
r=1
xi (k) − min{xi (k)}+
i
|xm
r (k)|
− min{xi (k)} −
i
|xm
r (kp )|
min{xi (k)} − min{xi (k)} +
M
X
r=1
i
i
|xm
r (k)|
− (1 − γ)
M
X
r=1
M
X
r=1
|xm
r (k)|
|xm
r (k)|
−γ
M
X
r=1
|xm
r (kp )|
kp −k+1
min{xi (k)} − min{xi (k)}+
|xm
r (k)| + γ
i
i
(A.64)
120
If we apply Equation (A.49) to xj with k = kp + 1, t = k ′ − kp − 1, we have
′
xj (k ) −
M
X
r=1
′
|xm
r (k )|
≥ min{xi (kp + 1)} −
i
min{xi (kp + 1)} +
i
M
X
r=1
M
X
r=1
|xm
r (kp
|xm
r (kp
+ 1)| + γ
+ 1)|
k′ −kp −1
=(1 − γ k −kp −1 ) min{xi (kp + 1)} − (1 − γ k −kp −1 )
′
′
i
γ
k′ −kp −1
M
X
r=1
|xm
r (kp
≥ min{xi (k)} + γ
+ 1)| + γ
k′ −kp −1
i
(1 − γ
k′ −kp −1
)
M
X
r=1
≥ min{xi (k)} + γ
k′ −kp −1
r=1
|xm
r (kp
−γ
M
X
i
i
+ 1)| − (1 − γ
≥ min{xi (k)} −
i
r=1
|xm
r (k)|
k′ −kp −1
+γ
)
M
X
r=1
k′ −k
|xm
r (kp
r=1
M
X
r=1
+ 1)|
|xm
r (kp
+ 1)| −
|xm
r (kp + 1)|
r=1
M
X
min{xi (k)} +
i
M
X
r=1
M
X
i
k′ −kp −1
|xm
r (kp + 1)|−
xj (kp + 1) +
|xm
r (kp
r=1
M
X
+ γ kp −k+1 min{xi (k)} − min{xi (k)} +
M
X
M
X
xj (kp + 1) − min{xi (k)} +
|xm
r (k)|
i
k′ −kp −1
xj (kp + 1)−
r=1
+ 1)| −
M
X
r=1
|xm
r (k)|
|xm
r (k)| − min{xi (k)}+
|xm
r (k)|
−γ
i
k′ −kp −1
M
X
r=1
xi (k) − min{xi (k)} +
i
M
X
r=1
|xm
r (kp + 1)|
|xm
r (k)|
(A.65)
Hence, we have
′
xj (k ) −
M
X
r=1
′
|xm
r (k )|
≥ min{xi (k)} −
i
min{xi (k)} +
i
121
M
X
r=1
M
X
r=1
|xm
r (k)|
+γ
k′ −k
xi (k)−
′
|xm
r (k)| , ∀k ≥ kp + 1
(A.66)
• Now let us consider time kp , p ∈ {1, 2 . . .}, and j ′ ∈ Ni such that at some time
′
ks , 1 ≤ s < p, αii→j ≥ γ(xi (ks ) − xj ′ (ks )), i.e., at time ks , node i passes at least
γ(xi (ks ) − xj ′ (ks )) to node j ′ . Consider any j ∈ Ni such that xj (kp ) ≥ xj ′ (kp ).
Applying Equation (A.49) to xj with k = kp , t = k ′ − kp yields
xj (k ′ ) −
M
X
r=1
′
|xm
r (k )|
≥ min{xi (kp )} −
i
M
X
r=1
|xm
r (kp )|
+γ
k′ −kp
xj (kp ) − min{xi (kp )} +
i
≥(1 − γ k −kp ) min{xi (kp )} − (1 − γ k −kp )
′
′
i
γ k −kp
′
M
X
r=1
γ
r=1
|xm
r (kp )|−
′
i
M
X
r=1
r=1
|xm
r (kp )|
k −kp
xj (kp )
|xm
r (kp )| + γ
≥ min{xi (k)} −
k′ −kp
M
X
M
X
M
X
r=1
|xm
r (k)|
+γ
k′ −kp
xj ′ (kp ) −
M
X
r=1
|xm
r (kp )|
− min{xi (k)} +
i
|xm
r (k)|
(A.67)
Equation (A.67) holds for all k ′ ≥ kp . It is clear that Equation (A.66) applies
to node j ′ for all k ′ , k ′ ≥ ks + 1. Because kp ≥ ks + 1, we can substitute in
Equation (A.67) for xj ′ (kp ) from Equation (A.66) to arrive at
′
xj (k ) −
M
X
r=1
′
|xm
r (k )|
≥ min{xi (k)} −
i
γ
kp −k
M
X
r=1
|xm
r (k)|
+γ
xi (k) − min{xi (k)} +
i
k′ −kp
M
X
r=1
min{xi (k)} −
i
|xm
r (k)|
122
M
X
r=1
|xm
r (k)|+
M
X
′
− min{xi (k)} + γ k −kp
|xm
r (k)|
i
r=1
(A.68)
Hence, we have
′
xj (k ) −
M
X
r=1
′
|xm
r (k )|
≥ min{xi (k)} −
i
γ
k′ −k
M
X
r=1
|xm
r (k)|+
xi (k) − min{xi (k)} +
i
M
X
r=1
|xm
r (k)|
, ∀ k ′ ≥ kp
(A.69)
Note that Equation (A.69) is only valid for node j, such that xj (kp ) ≥ xj ′ (kp ),
i→j ′
′
where j is the node such that at time ks , 1 ≤ s < p, αi
≥ γ xi (ks )−xj ′ (ks ) .
In particular, at time kp , if xj (ks ) ≥ xj ′ (kp ), ∀j ∈ Ni , j 6= j ′ , Equation (A.69)
is valid for all neighbors of node i.
Now, let us consider the neighboring node of inverter i with the minimum reactive
power at every kp , p ∈ {1, 2, . . . , }. We have three cases:
1. At every time kp , αii→j ≥ γ(xi (kp )−xj (kp )) for j ∈ Ni such that xj (kp ) ≤ xj ′ (kp ),
i→j
′
∀j ∈ Ni ; and at every time ks , s < p, αi < γ xi (ks ) − xj (ks ) (i.e., node i
passes a nonnegligible amount of reactive power at time kp to its neighbor node
with minimum reactive power to which it has not passed a nonnegligile amount
of reactive power since before kp ).
2. At every time kp , αii→j ≥ γ(xi (kp )−xj (kp )) for j ∈ Ni such that xj (kp ) ≤ xj ′ (kp ),
i→j
′
∀j ∈ Ni ; and at every time ks , s < p, αi ≥ γ xi (ks ) − xj (ks ) (i.e., node i
passes a nonnegligible amount of reactive power at time kp to its neighbor node
with minimum reactive power to which it has passed a nonnegligile amount of
reactive power before kp ).
If case 1 is true, then we have Equation (A.66) is valid ∀ j for k ′ ≥ kp + 1; if
case 2 is true, then we have Equation (A.69) is valid ∀ j ∈ Ni for k ′ ≥ kp . As
123
we assume that |Ni | < N , case 1 occurs at every kp , p ∈ {1, 2, . . .} until at most
time kp′ , where p′ = |Ni | and at time kp′ case 1 must have occurred for each
j ∈ Ni , otherwise case 2 must occur before kp′ . Both these two possibilities
indicate that
′
xj (k ) −
M
X
r=1
′
|xm
r (k )|
≥ min{xi (k)} −
i
M
X
r=1
|xm
r (k)|
M
X
r=1
|xm
r (k)|
+γ
k′ −k
xi (k) − min{xi (k)}+
i
, ∀ k ′ ≥ k + N B, j ∈ Ni
(A.70)
Otherwise, if neither of these two possibilities is true, then we can only have
the third case to be true before kN , i.e.,
3. There is time kp , p ∈ {1, 2, . . .}, such that xi (kp ) ≤ xj (kp ), ∀ j ∈ Ni (i.e., node
i has the minimum reactive power among its all neighbors). In this case, we
have, for any j ∈ Ni , from Equation (A.49) with k = kp and t = k ′ − kp that
xj (k ′ ) −
M
X
r=1
′
|xm
r (k )| ≥ min{xi (kp )} −
i
γ
k′ −kp
=(1 − γ
i
) min{xi (kp )}−
(1 − γ k −kp )
′
i
M
X
r=1
≥ min{xi (k)} −
i
γ
k′ −kp
r=1
|xm
r (kp )|+
xj (kp ) − min{xi (kp )} +
k′ −kp
M
X
r=1
|xm
r (kp )|
′
M
X
r=1
|xm
r (k)|+
i
M
X
r=1
|xm
r (k)|
(A.71)
k −kp
xj (kp )
|xm
r (kp )| + γ
xi (kp ) − min{xi (k)} +
124
M
X
Applying Equation (A.49) for node i with t = kp − k we obtain
xi (kp ) −
M
X
r=1
|xm
r (kp )|
≥ min{xi (k)} −
i
γ
kp −k
i
i
γ
kp −k
r=1
|xm
r (k)|+
xi (k) − min{xi (k)} +
xi (kp ) ≥ min{xi (k)} −
M
X
M
X
r=1
M
X
|xm
r (k)|
M
X
|xm
r (k)|
r=1
(A.72)
|xm
r (k)|+
xi (k) − min{xi (k)} +
i
r=1
Equation (A.71) and (A.72) imply that
′
xj (k ) −
M
X
r=1
′
|xm
r (k )|
≥ min{xi (k)} −
i
min{xi (k)} +
i
M
X
r=1
M
X
r=1
|xm
r (k)|
+γ
k′ −k
xi (k)−
(A.73)
′
|xm
r (k)| , ∀ k ≥ kp , j ∈ Ni
Thus, considering all three cases we conclude that
′
xj (k ) −
M
X
r=1
′
|xm
r (k )|
≥ min{xi (k)} −
i
min{xi (k)} +
i
M
X
r=1
M
X
r=1
|xm
r (k)|
|xm
r (k)|
+γ
k′ −k
xi (k)−
(A.74)
′
, ∀k ≥ k + N B, j ∈ Ni
Now, we extend Equation (A.74) to
′
xj (k ) −
M
X
r=1
′
|xm
r (k )|
≥ min{xi (k)} −
i
min{xi (k)} +
i
M
X
r=1
M
X
r=1
|xm
r (k)|
|xm
r (k)|
+ (γ
k′ −k l
) xi (k)−
(A.75)
′
, ∀k ≥ k + lN B, j ∈ Ni
where node j is any node that is reachable from i by spanning l node connections.
Equation (A.74) is the case that Equation (A.75) with l = 1. We assume that
Equation (A.75) is valid for any node j at a distance l from i, and we prove that
Equation (A.75) is valid for any node j ′ at a distance l + 1 from i, for j ′ ∈ Nj . Since
125
node j ′ is a neighbor of node j, we can apply Equation (A.75) for node j ′ and j with
time k + lN B, then we have
′
xj ′ (k ) −
M
X
r=1
′
|xm
r (k )|
≥ min{xi (k + lN B)} −
i
min{xi (k + lN B)} +
i
= 1−γ
γ
k′ −(k+lN B)
k′ −(k+lN B)
|xm
r (k
r=1
M
X
|xm
r (k
r=1
+ lN B)| + γ
+ lN B)|
k′ −(k+lN B)
min{xi (k + lN B)} − 1 − γ
i
i
min{xi (k)} +
M
X
r=1
M
X
r=1
|xm
r (k)|
|xm
r (k)|
+γ
k′ −k
xj (k + lN B)−
k′ −(k+lN B)
xj (k + lN B)
≥ min{xi (k)} −
i
M
X
X
M
r=1
|xm
r (k + lN B)|+
xj (k + lN B)−
, ∀k ′ ≥ k + lN B + N B, j ′ ∈ Nj
126
(A.76)
Substituting Equation (A.75), based on our hypothesis, into Equation (A.76), we
have
′
xj ′ (k ) −
M
X
r=1
′
|xm
r (k )|
≥ min{xi (k)} −
i
M
X
r=1
|xm
r (k)|
+
min{xi (k)} +
i
≥ min{xi (k)} −
i
min{xi (k)} +
i
= min{xi (k)} −
i
min{xi (k)} +
i
M
X
r=1
M
X
r=1
M
X
r=1
M
X
r=1
M
X
r=1
M
X
r=1
M
X
r=1
|xm
r (k)|
+γ
′
|xm
r (k )|
|xm
r (k)|
k′ −k
+ (γ
min{xi (k)}−
k′ −k l
i
) xi (k)−
− min{xi (k)} +
i
M
X
r=1
|xm
r (k)|
k −k
(γ k −k )l xi (k)−
|xm
r (k)| + γ
′
|xm
r (k)|
|xm
r (k)|
|xm
r (k)|
′
+ (γ
k′ −k l+1
)
xi (k)−
(A.77)
where Equation (A.77) is valid for all k ′ ≥ k + (l + 1)N B. Hence, Equation (A.75)
is valid for l ≥ 1. As every node in the network can be reached from i by spanning
fewer than N arcs, Equation (A.75) implies
′
xj (k ) −
M
X
r=1
′
|xm
r (k )|
≥ min{xi (k)} −
i
min{xi (k)} +
i
M
X
r=1
M
X
r=1
|xm
r (k)|
|xm
r (k)|
+ (γ
k′ −k N
)
xi (k)−
(A.78)
′
2
, ∀ k ≥k+N B
Since Equation (A.78) is valid for all i ∈ I. Hence, Equation (A.78) is written as
M
M
X
X
′
m ′
m
k′ −k N
xj (k ) −
|xr (k )| ≥ min{xi (k)} −
|xr (k)| + (γ
) max{xi (k)}−
r=1
i
min{xi (k)} +
i
r=1
M
X
r=1
127
i
|xm
r (k)|
, ∀ k ′ ≥ k + N 2 B, ∀ j ∈ I
(A.79)
Equation (A.79) is valid for all j ∈ I. Hence, it is valid for the node with minimum
reactive power at time k ′ , i.e.,
′
min{xi (k )} −
i
M
X
r=1
′
|xm
r (k )|
≥ min{xi (k)} −
i
min{xi (k)} +
i
M
X
r=1
M
X
r=1
|xm
r (k)|
|xm
r (k)|
+ (γ
k′ −k N
)
max{xi (k)}−
i
, ∀ k′ ≥ k + N 2B
(A.80)
It is obvious that
N
N
M
M
X
X
1 X
1 X
2
m
2
xj (k) +
xj (k + N B) +
|xr (k)| ≥
|xm
r (k + N B)|
N j=1
N
r=1
r=1
j=1
(A.81)
Choose k ′ = k + N 2 B, for each k ≥ 0, xA (k) ∈
/ Xb , Equation (A.42), (A.80), and
(A.81) imply that
V (xA (k)) − V (xA (k + N 2 B))
X
N
M
X
1
xj (k) − xi (k) + 2
|xm
= max
r (k)|−
i
N j=1
r=1
X
N
M
X
1
2
2
2
max
xj (k + N B) − xi (k + N B) − 2
|xm
r (k + N B)|
i
N j=1
r=1
N
N
M
M
X
X
1 X
1 X
2
m
2
xj (k) +
xj (k + N B) −
=
|xr (k)| −
|xm
r (k + N B)|
N j=1
N
r=1
r=1
j=1
M
M
X
X
2
m
2
m
+ min{xi (k + N B)} −
|xr (k + N B)| − min{xi (k)} −
|xr (k)|
i
≥γ N
3B
i
r=1
max{xi (k)} − min{xi (k)} +
i
i
M
X
r=1
|xm
r (k)|
r=1
(A.82)
We know from Equation (A.34) that
ρ(xA (k), Xb ) ≤ max{xi (k)} − min{xi (k)} +
i
i
Hence, Equation (A.82) and (A.83) imply
V (xA (k)) − V (xA (k + N 2 B)) ≥ γ N
128
3B
M
X
r=1
|xm
r (k)|
ρ(xA (k), Xb )
(A.83)
(A.84)
Then, we conclude that the invariant set Xb is exponentially stable if the reactive
power passing occurs every B time steps for each inverter.
Proof of Theorem 4.3.3. If qimax and qimin are consistent with time k and N maxi {qimin } <
QD < N mini {qimax }, for any x ∈ Xc , we have xi = QD /N which implies that
qimin < xi < qimax for all i ∈ I, i.e., the reactive power of all inverters will not
saturate at the bounds. Since we assume QD is constant, there is only one reactive
power level for all inverters, i.e., QD /N . Hence, we conclude that |Xc | = 1.
Proof of Theorem 4.3.4. With fixed qimax and qimin for all i ∈ I (as indicated by condition (ii) of Assumption 4.3.1) if N maxi {qimin } < QD < N mini {qimax }, this leads
P
max
to the case of Theorem 4.3.3; if N mini {qimax } < QD < N
for QD > 0, or
i=1 qi
PN min
< QD < N maxi {qjmin } for QD < 0, the reactive power of some inverters
i=1 qj
saturate at the bounds. If in addition we assume a complete graph, i.e., the case given
by condition (i) of Assumption 4.3.1, there are no “isolated” inverters due to some
saturated inverters. The unsaturated inverters are connected together and have the
same reactive power level. Moreover, if we assume there are r < N number of saturated inverters (this number depends on QD , qimax , and qimax ), then we know that the
inverters with saturated reactive power are the first r inverters in the sequence such
max
min
that q1max ≤ q2max ≤ . . . ≤ qN
for QD > 0 and q1min ≥ q2min ≥ . . . ≥ qN
for QD < 0.
P
The unsaturated inverters have the same reactive power level (QD − ri=1 xi )/(N −r).
129
Proof of Theorem 4.3.5. Recall that U is the subset of inverters with unsaturated
reactive power and S is the subset of inverters with saturated reactive power. The
terms |U| and |S| denote the numbers of elements in U and S, i.e., the number of
inverters with unsaturated reactive power and the number of inverters with saturated
reactive power, respectively.
First, let us consider the case that QD > 0. With Assumption 4.3.1 the invariant
set becomes
Xc+
=
x ∈ X : for all i ∈ U, xi (k) = xj (k), for all j ∈ U;
max
xi (k) = qi (k) for all i ∈ S
(A.85)
Consider the state x̄ ∈ Xc+ , define Sc to be the subset of inverters such that for all
i ∈ Sc , x̄i = qimax and define Uc to be the subset of inverters such that for all i ∈ Uc ,
x̄i < qimax . As discussed above, we know that for any x̄ ∈ Xc+ ,
x̄i = qimax , for all i ∈ Sc
X 1
x̄i =
QD −
xj , for all i ∈ Uc
|Uc |
j∈S
(A.86)
c
and
qimax
X 1
QD −
≤
xj , for all i ∈ Sc
|Uc |
j∈S
(A.87)
c
Choose
ρ(x(k), Xc+ )
= inf
max{|xi (k) − x̄i |} : x̄ ∈
i∈I
Xc+
(A.88)
and
V (x(k)) = max
i∈Uc
X
1 X
max xj (k) − xi (k) +
xi (k) − qi |Uc | j∈U
i∈S
c
(A.89)
c
From Equation (A.86) we know that
ρ(x(k), Xc+ )
1
≥
max{xi (k)} − min{xi (k)}
i∈Uc
2 i∈Uc
130
(A.90)
and
ρ(x(k), Xc+ )
X
max ≤ max{xi (k)} − min{xi (k)} +
x
(k)
−
q
i
i
i∈Uc
i∈Uc
(A.91)
i∈Sc
The reason that Equation (A.91) holds is as follow:
• At time k, if min{xi (k)} ≤ x̄i for i ∈ Uc , then max{xi (k)} − min{xi (k)} ≥
i∈Uc
i∈Uc
i∈Uc
max{|xi (k) − x̄i |}. It is obvious that Equation (A.91) holds;
i∈Uc
• At time k, if min{xi (k)} ≥ x̄i for i ∈ Uc , then max{xi (k)} − min{xi (k)} ≤
i∈Uc
i∈Uc
i∈Uc
P
|xi (k) − qimax | ≥
max{|xi (k) − x̄i |}. However, max{xi (k)} − min{xi (k)} +
i∈Uc
i∈Uc
i∈Uc
i∈Sc
P
|xi (k) − qimax |
max{|xi (k) − x̄i |}, because for this case max{|xi (k) − x̄i |} ≤
i∈Uc
i∈Uc
i∈Sc
for i ∈ Uc , i.e., the maximum difference between xi (k) and the final equalized
reactive power level of inverter i ∈ Uc is less than the total difference between
current reactive power levels and the final saturated reactive power levels of
inverters in the subset Sc . In other words, since min{xi (k)} ≥ x̄i for i ∈ Uc , all
i∈Uc
inverters in the subset of Uc will decrease their reactive power levels to the final
equalized level by passing reactive power to inverters in the subset of Sc with
the reactive power passing strategies 4.3.2-a1)–4.3.2-a4). Hence, max{xi (k)} −
i∈Uc
P
max
|xi (k) − qi | ≥ max{|xi (k) − x̄i |} implies Equation (A.91).
min{xi (k)} +
i∈Uc
i∈Sc
i∈Uc
Equation (A.89) implies that
X 1 X
max V (x(k)) =
xj (k) − min{xi (k)} +
x
(k)
−
q
i
i
i∈Uc
|Uc | j∈U
i∈Sc
c
X max
xi (k) − qi ≤ max{xi (k)} − min{xi (k)} +
i∈Uc
i∈Uc
i∈Sc
131
(A.92)
Equation (A.90) implies that
2ρ(x(k), Xc+ ) ≥ max{xi (k)} − min{xi (k)}
i∈Uc
i∈Uc
X X +
max
max
xi (k) − qi ≥ max{xi (k)} − min{xi (k)} +
xi (k) − qi 2ρ(x(k), Xc ) +
i∈Uc
i∈Uc
i∈Sc
i∈Sc
(A.93)
We also know that
ρ(x(k), Xc+ ) ≥ max{|xi (k) − qimax |}
i∈Sc
|Sc |ρ(x(k), Xc ) ≥ |Sc | max{|xi (k) −
i∈Sc
qimax |}
X max ≥
xi (k) − qi (A.94)
i∈Sc
Hence, we obtain from Equation (A.92)–(A.94) that
X max +
V (x(k)) ≤ max{xi (k)} − min{xi (k)} +
x
(k)
−
q
i
i
≤ (2 + |Sc |)ρ(x(k), Xc )
i∈Uc
i∈Uc
i∈Sc
(A.95)
Notice that
1
1 X
max{xi (k)} + (|Uc | − 1) min{xi (k)}
xj (k) ≥
i∈Uc
|Uc | j∈U
|Uc | i∈Uc
(A.96)
c
Combining Equation (A.92) and (A.96), we obtain
1
max{xi (k)} + (|Uc | − 1) min{xi (k)} −
V (x(k)) ≥
i∈Uc
|Uc | i∈Uc
X
max x
(k)
−
q
min{xi (k)} +
i
i
i∈Uc
i∈S
c
X
1
max
xi (k) − qi =
max{xi (k)} − min{xi (k)} +
i∈Uc
|Uc | i∈Uc
i∈Sc
X 1
xi (k) − qimax max{xi (k)} − min{xi (k)} + |Uc |
=
i∈Uc
|Uc | i∈Uc
i∈S
(A.97)
c
1
≥
ρ(x(k), Xc+ )
|Uc |
Hence, Equation (A.92) and (A.97) imply that
1
ρ(x(k), Xc+ ) ≤ V (x(k)) ≤ (2 + |Sc |)ρ(x(k), Xc+ )
|Uc |
Thus,
132
(A.98)
• For c1 =
max{xi (k)} − min{xi (k)} > 0 it is possible to find a c2 =
i∈Uc
i∈Uc
1
max{xi (k)} − min{xi (k)} > 0 such that V (x(k)) ≥ c2 and ρ(x(k), Xc+ ) ≥
|Uc |
1
2
i∈Uc
i∈Uc
c1 ;
P max x
(k)
−
q
• For c3 = max{xi (k)} − min{xi (k)} +
i
i
> 0, it is possible to
i∈Uc
i∈Uc
i∈Sc
find a c4 = (2 + |Sc |)c3 such that when ρ(x(k), Xc+ ) ≤ c3 we have V (x(k)) ≤ c4 ;
• The function V (x(k)) is non-increasing with the reactive power passing strategies a1)–a4). The reasons are as follow:
– At time k, if min{xi (k)} ≤ x̄i for i ∈ Uc , then the average reactive
i∈Uc
P
power level |U1c |
xj (k) tends to decrease to x̄i due to the capacitive rej∈Uc
active power passed from the inverters in the subset of Sc . Also, since
min{xi (k)} ≤ x̄i , the inverter with the reactive power of min{xi (k)}
i∈Uc
i∈Uc
tends to pass capacitive reactive power to others to increase its reactive
power level which makes min{xi (k)} increase and − min{xi (k)} decrease,
i∈Uc
i∈Uc
or min{xi (k)} can decrease due to some capacitive reactive power it rei∈Uc
ceives from inverters in the subset of Sc . However, the reactive power
increase of some inverters in the subset of Sc cancels the decrease of
P max x
(k)−q
min{xi (k)}. Now consider the term
i
i
. Since we assume the
i∈Ic
i∈Sc
graph is complete, i.e., each inverter (node) in the graph is fully connected
P max x
(k)−q
to other inverters, and x̄i ≥ x̄j for i ∈ Ic and j ∈ Sc , then
i
i
i∈Sc
tends to decrease due to the capacitive reactive power that the inverters
in the subset Sc passes to inverters in the subset of Uc , i.e., inverters in the
subset Sc tends to increase their reactive power levels.
133
– At time k, if min{xi (k)} ≥ x̄i for i ∈ Uc , it is possible that min{xi (k)}
i∈Uc
i∈Uc
decreases and − min{xi (k)} increases. However, for this case all inverters
i∈Uc
in the subset of Uc receives capacitive power from inverters in the subset
of Sc , then the decrease of min{xi (k)} is not greater than the decrease of
i∈Uc
P max xi (k) − qi . Hence, the function V (x(k)) is non-increasing.
i∈Sc
• Furthermore, with the reactive power passing strategies defined by 4.3.2-a1)–
4.3.2-a4) V (x(k)) → 0 as k → 0 for all x(k) ∈ X .
Then, we conclude that with the reactive power passing strategies defined by con
+
ditions 4.3.2-a1)–4.3.2-a4) the invariant set Xc = x ∈ X : for all i ∈ U, xi (k) =
max
xj (k), for all j ∈ U; xi (k) = qi (k) for all i ∈ S is asymptotically stable in large.
−
Similarly, we can also prove that the invariant set Xc = x ∈ X : for all i ∈
min
U, xi (k) = xj (k), for all j ∈ U; xi (k) = qi (k) for all i ∈ S is asymptotically stable
in large for the case that QD < 0. Hence, the invariant set Xc is asymptotically stable
in large.
Proof of Theorem 4.3.6. First, let us consider the case that QD > 0. With Assumption 4.3.1 the invariant set becomes Xc+ that is given in Equation (A.85). We choose
ρ(x(k), Xc+ ) the same as the one in Equation (A.88) and the Lyapunov function
1 X
1 X max (A.99)
x
(k)
−
q
V (x(k)) = max xi (k) −
xj (k) +
i
i
i∈Uc
|Uc | j∈U
|Uc | i∈S c
c
Equation (A.91) is rewritten as
max
+
ρ(x(k), Xc ) ≤ max{xi (k)} − min{xi (k)} + max xi (k) − qi i∈I
i∈Uc
Equation (A.100) holds because
134
i∈Sc
(A.100)
max • It is obvious that max xi (k) − qi is identical with max{|xj (k) − x̄j |}.
i∈Sc
j∈Sc
• Also, it is obvious that min{xi (k)} <=
i∈Uc
1
|Uc |
P
xj (k),
j∈Uc
so max{xi (k)} − min{xi (k)} > max{|xj (k) − x̄j |}.
i∈I
j∈Uc
i∈Uc
Hence, we have
ρ(x(k), Xc+ )
= inf
max{|xi (k) − x̄i |} : x̄ ∈
i∈I
Xc+
Equation (A.99) implies that
≤
max{xi (k)} − min{xi (k)} .
i∈I
i∈I
1 X
1 X max V (x(k)) = max{xi (k)} −
xj (k) +
xi (k) − qi i∈Uc
|Uc | j∈U
|Uc | i∈S
c
c
X max ≤ max{xi (k)} − min{xi (k)} +
x
(k)
−
q
i
i
i∈Uc
i∈Uc
(A.101)
i∈Sc
Hence, from Equations (A.93), (A.94), and (A.101) we arrive at the same result as
Equation (A.95), i.e., V (x(k)) ≤ (2 + |Sc |)ρ(x(k), Xc+ ). Similar to Equation (A.97)
we obtain
1
(|Uc | − 1) max{xi (k)} + min{xi (k)} +
V (x(k)) ≥ max{xi (k)} −
i∈Uc
i∈Uc
i∈Uc
|Uc |
X
1
xi (k) − qimax |Uc | i∈S
c
1
1 X max xi (k) − qi =
max{xi (k)} − min{xi (k)} +
i∈Uc
|Uc | i∈Uc
|Uc | i∈S
c
X
1
xi (k) − qimax =
max{xi (k)} − min{xi (k)} +
i∈Uc
|Uc | i∈Uc
i∈S
(A.102)
c
1
ρ(x(k), Xc+ )
≥
|Uc |
Hence, we have
1
ρ(x(k), Xc+ ) ≤ V (x(k)) ≤ (2 + |Sc |)ρ(x(k), Xc+ )
|Uc |
(A.103)
Let γ = min{γij }. For any i ∈ I and k ≥ 0, we know from condition 4.3.2i,j∈I
a2) that if the reactive power passing occurs for inverter i, and if αii→j < 0, then
135
αii→j
≤ γ xi (k) −
xij (k)
. We have xi (k + 1) ≤
xij (k)
+ γ xi (k) −
xij (k)
for j ∈ Ni .
If the reactive power passing does not occur or αii→j = 0, then xi (k + 1) = xi (k). It
follows that in any case,
xi (k + 1) ≤ max{xi (k)} + γ xi (k) − max{xi (k)} , ∀ i ∈ I
i∈I
i∈I
(A.104)
Since x̄i > max{qjmax } for i ∈ Uc , max{xi (k)} > max{qjmax } always holds. Then,
j∈Sc
i∈Uc
j∈Sc
max{xi (k)} is a nonincreasing function of k. We now show via induction that
i∈I
xi (k + t) ≤ max{xi (k)} + γ xi (k) − max{xi (k)} , ∀ i ∈ I
t
i∈I
i∈I
(A.105)
for all t ≥ 0. When t = 1, Equation (A.105) is turned to be Equation (A.104). Now we
assume that Equation (A.105) holds for an arbitrary t and show that Equation (A.105)
also holds for the case of t + 1. According to Equation (A.104) for any i ∈ I at time
k + t + 1 we have
xi (k + t + 1) ≤ max{xi (k + t)} + γ xi (k + t) − max{xi (k + t)}
i∈I
i∈I
≤ max{xi (k)} + γ xi (k + t) − max{xi (k)}
i∈I
i∈I
≤ max{xi (k)} + γ max{xi (k)}+
i∈I
i∈I
t
γ xi (k) − max{xi (k)} − max{xi (k)}
i∈I
i∈I
t+1
xi (k) − max{xi (k)}
≤ max{xi (k)} + γ
(A.106)
i∈I
i∈I
Thus, Equation (A.105) must be valid for all t ≥ 0.
• Fix i ∈ Uc and k ≥ 0, we now show that reactive power of all neighbors of i are
bounded from above for all k ′ , k ′ ≥ k + B. Specifically, we will show that
′
k′ −k
xi (k) − max{xi (k)} , ∀ k ′ ≥ k + B, j ∈ Ni
xj (k ) ≤ max{xi (k)} + γ
i∈I
i∈I
(A.107)
136
Since we assume a fully connected graph, Equation (A.107) is turned into
′
max{xi (k )} ≤ max{xi (k)} + γ
i∈I
k′ −k
i∈I
xi (k) − max{xi (k)} , ∀ k ′ ≥ k + B, i ∈ Uc
i∈I
(A.108)
There are times kp ≥ k, p ∈ {1, 2, . . .} such that the reactive power passing
occurs for inverter i, and the reactive power passing does not occur for k ′ 6= kp .
We know from Assumption 4.3.2 that k ≤ k1 < k + B, kp−1 < kp < kp−1 +
B, ∀ p ∈ {2, 3, . . .}. Now let us consider two cases:
– Let us consider a time kp , p ∈ {1, 2, . . .}, and j ∈ Ni such that xj (kp ) >
xi (kp ), i.e., at time kp inverter i passes a nonnegligible amount of capacitive
reactive power to inverter j . According to condition 4.3.2-a2), we have
xj (kp ) −
X
r
αrj→r ≤ xr (kp ) + αrj→r , ∀ r ∈ Nj such that xj (kp ) < xr (kp )
(A.109)
Equation (A.109) implies that
xj (kp )−
X
r
∗
αrj→r ≤ xr∗ (kp )+αrj→r
, for some r ∈ {r : xr ≥ xr′ , ∀ r′ ∈ Nj }
∗
(A.110)
From time kp to kp + 1 we have
X
X r′ →j
αr ′ ,
xj (kp + 1) = xj (kp ) −
αrj→r +
r
r′
′
∀ r, r ∈ Nj such that xj (kp ) < xr (kp ) and xj (kp ) > xr′ (kp )
(A.111)
As i ∈ Nj , i.e., inverter i is one of the neighboring inverter of inverter j,
and xj (kp ) > xi (kp ), Equation (A.111) becomes
xj (kp + 1) = xj (kp ) −
X
αrj→r +
r
137
X
r ′ ,r ′ 6=i
αrr′ →j + αji→j
′
(A.112)
Equations (A.110)–(A.112) imply that
∗
xj (kp + 1) ≤ xr∗ (kp ) + αrj→r
+
∗
X
αrr′ →j + αji→j
′
(A.113)
r ′ ,r ′ 6=i
αji→j
≤ γ xi (kp ) − xj (kp ) for i ∈
j→r ∗
Uc ; since we assume a fully connected graph, αr∗ ≤ γ xj (kp ) − xr∗ (kp )
From condition 4.3.2-a4) we know that
for j ∈ Uc . Thus, by applying these two equations for Equation (A.112)
P r′ →j
and using the fact that
αr′ ≤ 0 we obtain
r ′ ,r ′ 6=i
∗
xj (kp + 1) ≤ xr∗ (kp ) + αrj→r
+
∗
X
αrr′ →j + αji→j
′
r ′ ,r ′ 6=i
≤ xr∗ (kp ) + γ xi (kp ) − xj (kp ) + γ xj (kp ) − xr∗ (kp )
= xr∗ (kp ) + γ xi (kp ) − xr∗ (kp )
≤ max{xi (k)} + γ xi (kp ) − max{xi (k)}
i∈I
i∈I
(A.114)
By applying Equation (A.105) to xi (kp ) in Equation (A.114) we have
kp −k
xi (k) − max{xi (k)} −
xj (kp + 1) ≤ max{xi (k)} + γ max{xi (k)} + γ
i∈I
i∈I
i∈I
max{xi (k)}
i∈I
kp −k+1
xi (k) − max{xi (k)}
= max{xi (k)} + γ
i∈I
i∈I
(A.115)
138
If we apply Equation (A.105) to xj with k = kp + 1 and t = k ′ − kp − 1,
we obtain
′
xj (k ) ≤ max{xi (kp + 1)} + γ
k′ −kp −1
i∈I
xj (kp + 1) − max{xi (kp + 1)}
i∈I
≤ max{xi (k)} + γ k −kp −1 max{xi (k)}+
i∈I
i∈I
kp −k+1
xi (k) − max{xi (k)} − max{xi (k)}
γ
i∈I
i∈I
k′ −k
xi (k) − max{xi (k)} , ∀ k ′ ≥ kp + 1
= max{xi (k)} + γ
′
i∈I
i∈I
(A.116)
– let us consider time kp , p ∈ {1, 2 . . .}, and j ′ ∈ Ni such that xj ′ (kp ) ≤
xi (kp ), i.e., inverter i does not pass a nonnegligible amount of capacitive reactive power to inverter j ′ . In this case, it is obvious from Equation (A.105)
with k = kp and t = k ′ − kp that
xj ′ (k ) ≤ max{xi (kp )} + γ
xj (kp ) − max{xi (kp )}
i∈I
i∈I
k′ −kp
xi (kp ) − max{xi (k)}
≤ max{xi (k)} + γ
′
k′ −kp
(A.117)
i∈I
i∈I
for all k ′ ≥ kp . From Equation (A.105) with t = kp − k, it is also clear that
xi (kp ) ≤ max{xi (k)} + γ
kp −k
i∈I
xi (k) − max{xi (kp )}
i∈I
(A.118)
It follows from Equations (A.117) and (A.118) that
′
k′ −kp
xj ′ (k ) ≤ max{xi (k)} + γ
max{xi (k)}+
i∈I
i∈I
kp −k
xi (k) − max{xi (kp )} − max{xi (k)}
(A.119)
γ
i∈I
i∈I
k′ −k
xi (k) − max{xi (k)} , ∀ k ′ ≥ kp
= max{xi (k)} + γ
i∈I
i∈I
Notice that at each time kp , p ∈ {1, 2, . . .}, for any i ∈ Uc and any j ∈ Uc
one of the two cases shown above must be valid for a fully connected graph.
139
Also, for certain i ∈ Uc and certain j ∈ Uc one of the two cases must occur
every B steps. Hence, if we choose kp = k1 and k ′ > kp , Equations (A.117)
and (A.119) indicate that Equation (A.107) is valid for all k ′ ≥ k + B, j ∈ Ni .
Also, since we assume a fully connected graph and max{xi (k ′ )} = max{xi (k ′ )},
i∈Uc
i∈I
Equation (A.107) is turned into Equation (A.108). As we made no assumptions
to the contrary, Equation (A.108) is valid for any i ∈ Uc . Hence, we can replace
xi (k) with min{xi (k)} and Equation (A.108) becomes
i∈Uc
′
max{xi (k )} ≤ max{xi (k)} + γ
i∈I
k′ −k
i∈I
min{xi (k)} − max{xi (k)} , ∀ k ′ ≥ k + B
i∈I
i∈Uc
(A.120)
• Next, fix i ∈ Sc and k ≥ 0, similar to the analysis for Equation (A.105) we
obtain from condition 4.3.2-2a) that
xi (k + t) ≤
qimax
t
+ γ xi (k) −
qimax
, ∀ i ∈ Sc
(A.121)
– Similar to the analysis for inverter i ∈ Uc , we consider a time kp , p ∈
{1, 2, . . .}, and j ∈ Ni such that xj (kp ) > xi (kp ), i.e., at time kp inverter
i passes a nonnegligible amount of capacitive reactive power to inverter
j. Equations (A.109)–(A.113) also apply to j ∈ Ni . Since i ∈ Sc , ac
i→j
cording to condition 4.3.2-a4) we know that αj
≤ γ max xi (kp ) −
max
xj (kp ) , xi (kp ) − qi
max
, i.e.,
∗ Let us consider the case that xi (kp ) − xj (kp ) ≥ xi (kp ) − qi
i→j
max
xj (kp ) ≤ qi . Then, αj ≤ γ xi (kp ) − xj (kp ) . Consider j ∈ Uc ,
j→r ∗
then αr∗ ≤ γ xj (kp ) − xr∗ (kp ) , the following analysis is the same
as the one for the case that i ∈ Uc and we directly obtain the same
140
result as Equation (A.116). As max{xi (k)} > qimax for any i ∈ Sc ,
i∈I
Equation (A.116) is turned into
′
xj (k ) ≤ max{xi (k)} + γ
k′ −k
i∈I
xi (k) −
qimax
, ∀ k ′ ≥ kp + 1 (A.122)
xi (kp ) − qimax
, i.e.,
∗ Let us consider the case that xi (kp ) − xj (kp ) <
xj (kp ) > qimax . Then, αji→j ≤ γ xi (kp ) − qimax . Consider j ∈ Uc ,
P r′ →j
∗
using the fact that αrj→r
≤ 0 and
αr′ ≤ 0 Equation (A.114) is
∗
r ′ ,r ′ 6=i
rewritten as
xj (kp + 1) ≤ xr∗ (kp ) + αji→j
≤ max{xi (k)} + γ xi (kp ) −
i∈I
qimax
(A.123)
By applying Equation (A.121) to xi (kp ) in Equation (A.123) with t =
kp − k we arrive at
kp −k
qimax
xi (k) −
+γ
max
kp −k+1
xi (k) − qi
≤ max{xi (k)} + γ
xj (kp + 1) ≤ max{xi (k)} + γ
i∈I
qimax
−
qimax
i∈I
(A.124)
If we apply Equation (A.105) to xj with k = kp + 1 and t = k ′ − kp − 1,
we obtain
xj (kp + 1) − max{xi (kp + 1)}
xj (k ) ≤ max{xi (kp + 1)} + γ
i∈I
i∈I
max
max
kp −k+1
k′ −kp −1
−
xi (k) − qi
qi + γ
≤ max{xi (k)} + γ
i∈I
max
qi
max
k′ −k
, ∀ k ′ ≥ kp + 1
xi (k) − qi
= max{xi (k)} + γ
′
k′ −kp −1
i∈I
(A.125)
141
– Let us consider time kp , p ∈ {1, 2, . . . , } and j ′ ∈ Ni such that inverter
i ∈ Sc does not pass a nonnegligible amount of capacitive reactive power
to inverter j ′ . It is either the case that xj ′ kp < xi (kp ) or xi (kp ) = qimax .
If it is the case that xj ′ kp < xi (kp ), by conducting a similar analysis to
the case that i ∈ Uc and xj ′ kp < xi (kp ), we can derive the same result as
Equation (A.125). If it is the case that xi (kp ) = qimax , it is obvious that
Equation (A.125) still holds.
As Equation (A.125) is valid for all j ∈ Uc , we have
max{xi (k)} ≤ max{xi (k)} + γ
k′ −k
i∈I
i∈Uc
xi (k) −
qimax
, ∀ k ′ ≥ kp + 1
(A.126)
Since every B time steps at least one of the two cases discussed above must
occur, Equation (A.126) must be valid for all k ′ ≥ k+B. Also, since we made no
contrary to xi for any i ∈ Sc , Equation (A.125) is modified to Equation (A.127):
max{xi (k)} ≤ max{xi (k)} + γ
i∈Uc
k′ −k
i∈I
max
, ∀ k ′ ≥ k + B (A.127)
min xi (k) − qi
i∈Sc
By adding Equation (A.127) to Equation (A.120), we obtain
′
γ k −k
′
max{xi (k )} ≤ max{xi (k)} +
min{xi (k)}−
i∈I
i∈I
i∈Uc
2
max
, ∀ k′ ≥ k + B
max{xi (k)} + min xi (k) − qi
i∈I
(A.128)
i∈Sc
Using the fact that xi (k) ≤ qimax for all i ∈ Sc , Equation (A.128) implies that
max{xi (k)} − max{xi (k ′ )}
i∈I
i∈I
k′ −k
γ
max
max{xi (k)} − min{xi (k)} − min xi (k) − qi
≥
i∈Sc
i∈I
i∈Uc
2
k′ −k
γ
max
max{xi (k)} − min{xi (k)} + max xi (k) − qi , ∀ k ′ ≥ k + B
=
i∈Sc
i∈I
i∈Uc
2
142
(A.129)
Notice that at any moment max{xi (k)} = max{xi (k)}. By applying Equation (A.99)
i∈I
i∈Uc
to V (x(k)) and V (x(k ′ )) we have
V (x(k)) − V (x(k ′ ))
1 X
1 X max xi (k) − qi +
= max{xi (k)} − max{xi (k )} −
xj (k) +
i∈I
i∈I
|Uc | j∈U
|Uc | i∈S
c
c
X
X
1
1
xi (k ′ ) − qimax xj (k ′ ) −
|Uc | j∈U
|Uc | i∈S c
c
1 X
1 X max
1 X
xj (k) −
xi (k) +
q +
= max{xi (k)} − max{xi (k ′ )} −
i∈I
i∈I
|Uc | j∈U
|Uc | i∈S
|Uc | i∈S i
c
c
c
X
X
1 X
1
1
xj (k ′ ) +
xi (k ′ ) −
q max
|Uc | j∈U
|Uc | i∈S
|Uc | i∈S i
′
c
c
c
(A.130)
It is obvious that
1
|Uc |
P
j∈Uc
xj (k) +
1
|Uc |
P
xi (k) is consistent with time. Hence, by
i∈Sc
applying Equation (A.129) and Equation (A.100), Equation (A.130) is turned into
V (x(k)) − V (x(k ′ )) = max{xi (k)} − max{xi (k ′ )}
i∈I
i∈I
k′ −k
γ
max
≥
max{xi (k)} − min{xi (k)} + max xi (k) − qi i∈Sc
i∈I
i∈Uc
2
′
γ k −k
ρ(x(k), Xc+ ), ∀ k ′ ≥ k + B
≥
2
(A.131)
Equation (A.103) indicates that c1 ρ(x(k), Xc+ ) ≤ V (x(k)) ≤ c2 ρ(x(k), Xc+ ), where
c1 =
1
|Uc |
and c2 = (2+|Sc |), and Equation (A.130) indicates that V (x(k))−V (x(k ′ )) ≥
c3 ρ(x(k), Xc+ ) for all k ′ ≥ k + B, where c3 =
γk
′ −k
2
. It is obvious that
c3
c2
∈ (0, 1).
Hence, the invariant set Xc+ is exponentially stable in large. Similarly, we can prove
that the invariant set Xc− is exponentially stable in large for the case when QD < 0.
Hence, with Assumptions 4.3.1 and 4.3.2, and the reactive power passing conditions
4.3.2-a1)–4.3.2-a4) and 4.3.2-b1)–4.3.2-b4), the invariant set Xc is exponentially stable
in large.
143
Proof of Theorems 4.3.7 – 4.3.12. The proofs are similar to the ones of the uniform
reactive power distribution case which are not presented here.
144
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