A Distributed Feedback Control Approach
to the Optimal Reactive Power Flow Problem
Saverio Bolognani, Guido Cavraro, and Sandro Zampieri
Department of Information Engineering, University of Padova, Italy
{saverio.bolognani,guido.cavraro,zampi}@dei.unipd.it
Abstract. We consider the problem of exploiting the microgenerators
connected to the low voltage or medium voltage grid in order to provide distributed reactive power compensation in the power distribution
network, solving the optimal reactive power flow problem for the minimization of power distribution losses subject to voltage constraints. The
proposed strategy requires that all the intelligent agents, located at the
generator buses, measure their voltage and share these data with the
other agents via a communication infrastructure. The agents then adjust
the amount of reactive power injected into the grid according to a policy
which is a specialization of duality-based methods for constrained convex optimization. Convergence of the algorithm to the configuration of
minimum losses and feasible voltages is proved analytically. Simulations
are provided in order to demonstrate the algorithm behavior, and the
innovative feedback nature of such strategy is discussed.
Keywords: cyber-physical systems, networked control, power distribution grid, distributed control, reactive power compensation.
1
Introduction
Recent technological advances, together with environmental and economic reasons, have been motivating the deployment of small power generators in the low
voltage and medium voltage power distribution grid. The availability of a large
number of these generators in the distribution grid can yield relevant benefits
for the network operation, which go beyond the availability of clean, inexpensive
electrical power. They can be used to provide a number of ancillary services that
are of great interest for the management of the grid [1,2].
We focus in particular on the problem of optimal reactive power compensation
for power losses minimization and voltage support. In order to properly command
the operation of these devices, the distribution network operator is required to
solve an optimal reactive power flow (ORPF) problem. This problem is one
among the typical tasks that transmission grid operators need to solve for the
efficient and safe operation of the high voltage grid. Indeed, powerful solvers have
been designed for the ORPF problem, and advanced optimization techniques
D.C. Tarraf (ed.), Control of Cyber-Physical Systems,
Lecture Notes in Control and Information Sciences 449,
c Springer International Publishing Switzerland 2013
DOI: 10.1007/978-3-319-01159-2_14, 259
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S. Bolognani, G. Cavraro, and S. Zampieri
have been recently specialized for this task [3,4]. However, these solvers generally
assume that an accurate model of the grid is available, that all the grid buses
are monitored, that loads announce their demand profiles in advance, and that
generators and actuators can be dispatched on a day-ahead, hour-ahead, and
real-time basis. For this reason, these solvers are in general offline and centralized,
and they collect all the necessary field data, compute the optimal configuration,
and dispatch the reactive power production at the generators.
These tools cannot be applied directly to the ORPF problem faced in the
low voltage or medium voltage power distribution network. The main reasons
are that not all the buses of the grid are monitored, individual loads are unlikely to announce they demand profile in advance, the availability of small size
generators is hard to predict (being often correlated with the availability of renewable energy sources). Moreover, the grid parameters, and sometimes even
the topology of the grid, are partially unknown, and generators are expected
to connect and disconnect, requiring an automatic reconfiguration of the grid
control infrastructure (the so called plug and play approach).
Different strategies have been recently proposed in order to address these
issues, ranging from purely local algorithms, in which each generator is operated
according to its own measurements [2], to distributed approaches that do not
require any central controller, but still require measurement at all the buses
of the distribution grid [5]. Only recently, algorithms that are truly scalable in
the number of generators and do not require the monitoring of all the buses
of the grid, have been proposed for the problem of power loss minimization
(with no voltage constraints) [6,7]. While these algorithms have been designed
by specializing classical nonlinear optimization algorithms to the ORPF problem,
they can also be considered as feedback control strategies. Indeed, the key feature
of these algorithms is that they require the alternation of measurement and
actuation, and therefore they are inherently online algorithms. In particular, the
reactive power injection of the generators is adjusted by these algorithms based
on the phasorial voltage measurements that are performed at the buses where
the generators are connected. The resulting closed loop system features a tight
dynamic interconnection of the physical layer (the grid, the generators, the loads)
with the cyber layer (where communication, computation, and decision happen).
In this paper, we design a distributed feedback algorithm for the ORPF problem
with voltage constraints, in which the goal is minimizing reactive power flows
while ensuring that the voltage magnitude across the network is larger than a
given threshold.
In Section 2, a model for the cyber-physical system of a smart power distribution grid is provided. In Section 3, the optimal reactive power flow problem
is stated. An algorithm for its solution is proposed in Section 4, and its convergence is studied in Section 5. Some simulations are provided in Section 6, while
Section 7 concludes the paper discussing some relevant features of the feedback
nature of the proposed strategy.
A Distributed Feedback Control Approach to the ORPF Problem
communication
Cyber layer
Physical layer
261
h∈
/C
1
h∈C
Fig. 1. A schematic representation of the the physical layer (the electric network)
and the physical layer (the communication and control resources) in a smart power
distribution grid. Circled nodes in the lower panel are the buses of the grid where a
microgenerator is connected. Node 1 is the point of connection to the transmission grid
(PCC). The other nodes are buses where loads are connected. The upper panel shows
how intelligent agents in the cyber level correspond to the nodes where some sensing
and actuation capabilities have been deployed, i.e. the PCC and the generator buses.
2
A Smart Power Distribution Grid
In this work, we envision a smart power distribution network as a cyber-physical
system, in which
– the physical layer consists of the power distribution infrastructure, including the power lines, the loads, the microgenerators, and the point of
connection to the transmission grid, while
– the cyber layer consists of intelligent agents, dispersed in the grid, and
provided with sensing, communication, and computational capabilities.
2.1
Model for the Physical Layer
We consider a portion of the power distribution network which is populated by
a number of small-size generators, together with regular loads. We model this
grid via a radial directed graph G (i.e. a tree) in which edges represent the power
lines, and the n nodes (whose set is denoted by V) represent the buses and also
the point of common coupling (PCC), i.e. the point where the distribution grid
that we are considering is connected to the transmission grid (see Figure 1).
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Given an edge e, we denote by σ(e) its source node and by τ (e) its terminal
node. We can therefore introduce the incidence matrix of G, defined via its
elements
⎧
⎨ −1 if v = σ(e)
Aev =
1 if v = τ (e)
⎩
0 otherwise.
We introduce the following assumption on the power line impedances.
Assumption 1. Let all the grid power line impedances have the same inductance/resistance ratio, i.e. for any edge e of the graph G, its impedance ze satisfies
ze = |ze | exp(jθ).
The grid electric topology and the grid power line parameters are therefore fully
described by the parameter θ and by the weighted Laplacian L ∈ IRn×n , defined
as
L = AT Z −1 A,
(1)
where
⎡
⎤
..
⎢ .
Z=⎢
⎣ |ze |
..
⎥
⎥
⎦
.
is the diagonal matrix of the absolute values of power line impedances.
We limit our study to the electric steady state behavior of the system, in
which all voltages and currents are sinusoidal signals at the same frequency ω.
We therefore assume that each signal (voltages and currents) can be represented
via a complex number whose absolute value corresponds to the signal rootmean-square value, and whose phase corresponds to the phase of the signal with
respect to an arbitrary common reference. In this notation, the state of a grid is
therefore described by the bus voltages uv ∈ C, v ∈ V.
In the following, we present a static model for the nodes of G. We model node
1 (corresponding to the PCC) as a slack bus, i.e. a constant voltage generator
at the nominal voltage UN ∈ IR and zero angle
u 1 = UN .
We assume instead that every other node h is a PQ-bus, i.e. the complex power
injected at the bus is independent from the bus voltage uh and is equal to
ph + jqh ,
where ph and qh are the injected active and reactive powers, respectively. Microgenerators fit in the PQ model (or constant power model) once they are
A Distributed Feedback Control Approach to the ORPF Problem
263
commanded via a complex power reference as in [8,9]. It is also a reasonable
approximation for many residential and industrial loads1 .
2.2
Model for the Cyber Layer
We assume that every generator bus, and also the PCC, correspond to an agent
in the cyber layer (see the upper panel of Figure 1). We denote by C (with
|C| = m) this subset of the nodes of G.
Each agent is provided with some computational capability, and with some
sensing capability, in the form of a phasor measurement unit (i.e. a sensor that
can measure voltage amplitude and angle [11]). Agents can communicate, via
some communication channel that could possibly be the same power lines (via
power line communication – PLC – technologies).
3
Optimal Reactive Power Flow Problem Formulation
Given the models and the definitions introduced above, we formulate the ORPF
problem in the following form,
min
qh ,h∈C\{1}
subject to
Jlosses
|uh | ≥ Umin ,
∀h ∈ C,
|uh | ≤ Umax ,
∀h ∈ C,
(2)
where Jlosses are the power distribution losses on the lines, and Umin is a given
lower bound for bus voltage magnitudes.
Notice that the decision variables in the optimization problem (2) are the
reactive power injections at the microgenerators, which are the only physical
devices that we aim to control via the proposed feedback algorithm. The reactive
power flowing from the transmission grid into the distribution grid via the PCC
(which we modeled as a slack bus) is not part of the set of decision variables,
because it automatically adjusts in order to ensure that power balance is satisfied
at any time in this portion of the distribution grid.
Notice moreover that the voltage constraints are defined only on the nodes
where we deployed sensing and actuation devices. Given the fact that the remaining nodes (corresponding to loads) are unmonitored, we cannot aim to design
any algorithm that can guarantee the satisfaction of operational constraints at
such nodes. However, in the case in which the voltage magnitude of some nodes
is of particular interest, the network operator has the flexibility of deploying
agents (i.e. PMU sensing units and the corresponding communication devices)
also in those nodes, and include them in the algorithm in order to guarantee voltage feasibility also in those buses. Alternatively, given a priori bounds on the
1
More general models could be considered, namely the exponential model and the
ZIP model, similarly to what has been done in [10]. The following analysis would
remain exactly the same, at the cost of a slightly more complex notation.
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S. Bolognani, G. Cavraro, and S. Zampieri
maximum power demand of the loads, the network operator can infer worst-casetype bounds on the maximum voltage drop that can occur between unmonitored
nodes and the closest agent, and therefore increase the voltage bound Umin accordingly (possibly also agent-wise). Both these solutions (ad-hoc deployment
of the agents, and worst-case guarantees for the voltage on unmonitored nodes)
are implementation issues that we are not addressing in this paper, but that can
be easily included on top of the approach that we describe here.
The ultimate goal of ORPF strategies is therefore the minimization of reactive
power flows across the power grid (by injecting reactive power as close at possible
to the buses that need it) while at the same time ensuring that the bus voltages
(which are a function also of reactive power flows) is kept inside a given range,
in order to guarantee more reliable and robust operation of the grid (see voltage
stability issues in [12]).
4
Proposed ORPF Algorithm
In order to formally describe the algorithm, we need the following definitions.
Definition 1 (Path). Let h, k ∈ V be two nodes of the graph G. The path
Phk = (v1 , . . . , v ) is the sequence of nodes, without repetitions, that satisfies
– v1 = h
– v = k
– for each i = 1, . . . , − 1, the nodes vi and vi+1 are connected by an edge.
Notice that, as the distribution grid topology is radial, there is only one path
connecting a pair of nodes h, k ∈ V.
Definition 2 (Neighbors in the cyber layer). Let h ∈ C. The set of nodes
that are neighbors of h in the cyber layer, denoted as N (h), is the subset of C
defined as
N (h) = {k ∈ C | Phk ∩ C = {h, k}} .
Figure 2 gives an example of such set. We assume that every agent h ∈ C knows
its set of neighbors N (h), and can communicate with them. Notice that this
architecture can be constructed by each agent in a distributed way, for example
by exploiting the PLC channel (as suggested for example in [13]). This allows
also a plug-and-play reconfiguration of such architecture when new agents are
connected to the grid.
It is also assumed that each agent h ∈ C has a local knowledge of the grid
electric parameters, and in particular of the following parameter.
Definition 3 (G-parameters). For each pair h, k ∈ C, let us define the parameter
.
(3)
Ghk = |ih | u = 1
k
u = 0, ∈ C\{k}
i = 0, ∈
/C
A Distributed Feedback Control Approach to the ORPF Problem
265
k ∈ N (h)
h
k ∈
/ N (h)
Fig. 2. An example of neighbor nodes in the cyber layer. Circled nodes (both gray and
black) are nodes in C. Nodes circled in black belong to the set N (h) ⊂ C. Node circled
in gray are agents which do not belong to the set of neighbors of h. For each agent
k ∈ N (h), the path that connects h to k does not include any other agent besides h
and k themselves.
i.e. the current that would be injected at node h if
– node k was replaced with a unitary voltage generator;
– all other nodes in C (the other agents) were replaced by short circuits;
– all nodes not in C (load buses) were replaced by open circuits.
Notice that the parameters Ghk depend only on the grid electric topology, and
that
Ghk = 0 if and only if k ∈ N (h).
(4)
Figure 3 gives a representation of this definition. Notice that, in the special
case in which the paths from h to its neighbors are all disjoint paths, then
Ghk = 1/ |Zhk |, where Zhk is the impedance of the electric path connecting
h to k. The G-parameters could also be defined from algebraic operations on
the Laplacian of the graph G (Schur complement), corresponding to a node
elimination procedure on the electrical network, as discussed in detail in [14].
As suggested in [13], these parameters can be estimated in an initialization
phase via some ranging technologies over the PLC channel. Alternatively, this
limited amount of knowledge of the grid topology can be stored in the agents at
the deployment time. Finally, the same kind of information can be inferred by
specializing the procedures that use the extended capabilities of the generator
power inverters for online grid sensing and impedance estimation [15,16].
Given the above definitions, we can state the proposed algorithm to solve
the optimization problem (2). We will show in Section 5 how the algorithm is
inspired by a dual decomposition approach [17] to (2). While problem (2) might
not be convex in general, we rely on the results presented in [18] which show that
zero duality gap holds for the ORPF problems, under some conditions that are
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S. Bolognani, G. Cavraro, and S. Zampieri
u = 0
∀ ∈ C\{h}
k ∈ N (h)
u = 0
∀ ∈ C\{h}
h
k ∈ N (h)
uh = 1
h
uh = 1
i = 0
∀ ∈
/C
i = 0
∀ ∈
/C
Fig. 3. A representation of how the elements Gkh are defined. Notice that in the
configuration of the left panel, as the paths from h to its neighbors k ∈ N (h) do not
share any edge, the gains Gkh corresponds to the absolute value of the path admittances
1/|Zkh |.
commonly verified in practice and in particular in radial networks like the ones
that we are considering. We will also show that, introducing an approximate
model for the grid, convergence of the algorithm can be studied analytically.
Finally, in Section 6, we will validate the proposed algorithm via simulations,
introducing all the non-idealities that have been neglected in the analytic study
of the algorithm convergence.
ORPF Algorithm. Let all agents store two auxiliary scalar variables λ+
h and
.
Let
γ
be
a
positive
scalar
parameter,
and
let
θ
be
the
impedance
angle
defined
λ−
h
in Assumption 1. At every synchronous iteration of the algorithm, each agent
h ∈ C executes the following operations:
– measures the voltage uh = |uh | exp(j∠uh );
– gathers the measurements {uk , k ∈ N (h)} from its neighbors;
−
– updates the auxiliary variables λ+
h and λh as
+
2
2
,
λ+
h ← λh + γ Umin − |uh |
+
−
2
2
λ−
,
h ← λh − γ Umax − |uh |
+
(5)
(6)
where [·]+ stands for the operation of projection on the positive orthant;
– updates the injected reactive power qh as
−
qh ← qh + 2 sin θ(λ+
Ghk |uh ||uk | sin(∠uk − ∠uh − θ). (7)
h − λh ) +
k∈N (h)
Notice that the agent located at the PCC does not need to perform the updates
(5) and (7), because the PCC is a constant-voltage slack bus, and therefore the
reactive power injected by the PCC automatically results from the power balance
in the grid.
A Distributed Feedback Control Approach to the ORPF Problem
5
267
Convergence Analysis
Before presenting the main result about the conditions that guarantee convergence of the proposed method, we show how this algorithm derives from a dual
decomposition of the original problem and from the specialization of the dual
ascent methods [17] to this specific system.
Let u and u be the vectors obtained by stacking the voltages uh , h ∈ C and
uk , k ∈ V\C, respectively. In the same way, let us define i, i , p, p , q, q as the
vectors of injected currents, active power, and reactive power, respectively.
Given Assumption 1, and given the weighted Laplacian L defined in (1), the
following grid equation is satisfied
i
u
exp(−jθ)L = .
i
u
It is also possible to construct a positive semi-definite symmetric pseudoinverse
X ∈ IRn×n (see [10]) such that
u
i
T
I − 1le1
= exp(jθ)X ,
(8)
i
u
where 1l is a vector all ones, and eh is a vector which is valued 1 in position h,
and 0 everywhere (i.e., is the h-th canonical basis vector).
The matrix X has some notable properties, including the fact that
(eh − ek )T X(eh − ek ) = |Zhk | ,
Xe1 = 0,
h, k ∈ V
(9)
(10)
and the fact that
Xhh ≥ Xhk ≥ 0
h, k ∈ V.
(11)
With the same partitioning as before, we can also partition X into blocks, as
M N
X=
.
(12)
NT Q
This notation allows us to introduce the approximate model for the power flows
in the grid proposed in [10], where a rigorous derivation and analysis of the
approximation error is also provided. According to this model
– the bus voltages of nodes in C can be expressed as
p − jq
1
exp(jθ) MN
u = UN 1l +
+
o
;
2
p − jq UN
UN
– reactive power injections satisfy, at every time, the balance
1
T q
1l
=o
;
q
UN
(13)
(14)
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S. Bolognani, G. Cavraro, and S. Zampieri
– the problem of optimal reactive power flow for losses minimization is equivalent to the problem of minimizing
1
M
J = qT q + qT N q + o
.
(15)
2
UN
This also allows us to express the squared voltage magnitudes |uh |2 via (13),
obtaining
⎡
⎤
..
.
⎢
⎥
p − jq
1
2
⎢|uh |2 ⎥ = UN
M
N
1
l
+
2
Re
exp(jθ)
+
o
.
(16)
⎣
⎦
p − jq UN
..
.
The proposed approximation is based on the fact that the grid operating point,
in its regular regime, is characterized by a relatively high nominal voltage compared to the voltage drops across the power lines, and by relatively small power
distribution losses, compared to the power delivered to the loads. Notice that
similar approximations have been used before in the literature for the problem of
estimating power flows on the power lines (see among the others [19,20] and references therein). It also shares some similarities with the DC power flow model
[21, Chapter 3], extending it to the case in which lines are not purely inductive
but also resistive (which is crucial in power distribution networks). In the analysis of the algorithm convergence, we will neglect the infinitesimal terms in both
(15) and (16).
In the following, we will consider only the lower bound constraint in the
optimization problem (2). We will therefore need only one of the two set of
auxiliary variables, i.e. λ+
h (which we will rename λh in order to simplify the
notation). The upper voltage constraint can be included following exactly the
same lines that will be presented hereafter for the lower voltage constraint, with
no additional complication to the design. Notice that the two constraints |uh | ≥
Umin and |uh | ≤ Umax at any node h will never be violated at the same time,
and therefore only one of the two additional variables (Lagrange multipliers) λ+
h
and λ−
h will be different from zero in practice.
We then consider the following problem in the decision variables q
min
1lT q=c
subject to
M
q + qT N q
2
(17a)
2
Umin
1l − v(q) ≤ 0,
(17b)
qT
where c = −1lT q and where v(q) is defined as
p − jq
2
1l + 2 Re exp(jθ) M N
.
v(q) = UN
p − jq (18)
Based on the proposed approximated model, and in particular by plugging (13)
into (7), we can also rewrite (via some algebraic manipulations) the optimization
A Distributed Feedback Control Approach to the ORPF Problem
269
algorithm presented in the previous section as a m-dimensional discrete time
system, with the following update equations.
q(t)
(19a)
q(t + 1) = q(t) − G M N
+ 2 sin θλ(t) + k(q(t), λ(t))e1
q
2
(19b)
λ(t + 1) = λ(t) + γ Umin 1l − v(q(t + 1)) +
where G is the matrix whose elements Ghk have been defined in Definition 3,
and k is a scalar that guarantees that constraint (14) is met at every iteration:
q(t)
T
k(q(t), λ(t)) = 1l G M N
− 2 sin θ1lT λ(t).
q
Notice that the term ke1 models the fact that, when the system is actuated at
the end of every algorithm iteration, the reactive power injected by the slack
bus 1 (the PCC) automatically balances the variations in the reactive power
injection that have been commanded to the generators.
In the following, we show how the algorithm (19) is a specialization of the
dual ascent algorithms for the solution of the optimization problem (17). The
Lagrangian of the problem (17) is
L(q, λ) = q T
2
M
q + q T N q + λT Umin
1l − v(q)
2
(20)
where λ is the Lagrangian multiplier (dual variables).
A dual ascent algorithm consists in the iterative execution of the following
alternated steps
1. minimization of the Lagrangian with respect to the primal variables q
q(t + 1) = arg min L(q(t), λ(t)),
1lT q=c
2. dual gradient ascent step on the dual variables
∂L(q(t + 1), λ(t)
λ(t + 1) = λ(t) + γ
.
∂λ
+
(21)
(22)
The partial derivative of the Lagrangian with respect to q results to be, by
inspecting (20) and (18),
∂L
∂v(q)
= M q + N q −
λ = M q + N q − 2 sin θM λ.
∂q
∂q
In order to show that (19a) is indeed the primal step that minimizes the Lagrangian with respect to the primal variable q, we need the following technical
lemma.
Lemma 1. Let G be defined element-wise as in Definition 3, and let M be
defined as in (12). Then
M G = I − 1leT1 .
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S. Bolognani, G. Cavraro, and S. Zampieri
Proof. From the definition of G, and by adopting the vector notation presented
in this section, we have that, given Assumption 1 and when i = 0,
i = exp(−jθ)Gu.
Then, by using (8), we have that
u
M N exp(−jθ)Gu
T
I − 1le1
,
= exp(jθ)
0
NT Q
u
and thus the conclusion.
By evaluating ∂L
∂q in q(t + 1) defined as is (19a), and by using the result of
Lemma 1, we have
∂L(q(t + 1), λ(t))
= M q(t + 1) + N q − 2 sin θM λ(t)
∂q
q(t)
= M q(t) − M G M N
+ N q
q
= 1leT1 N q ,
which is orthogonal to the feasible set 1lT q = c, and therefore proves that q(t+ 1)
solves (21).
In order to prove that (19b) is the dual ascent step described in (22), it is
enough to evaluate ∂L
∂λ to see that
∂L(q(t + 1), λ(t))
2
= Umin
1l − v(q(t + 1)),
∂λ
in accordance to the well known result in dual decomposition that the dual
ascent direction is given by the constraint violation.
We can then state the following convergence result.
Theorem 1. Consider the algorithm described in (19) for the optimization problem (17), which is an approximated description of the algorithm presented in
Section 4 for the optimization problem described in Section 3. The algorithm
converges if
1
γ≤
,
2
4 sin θDm
where m is the cardinality of C (the number of generator buses plus one) and
D = maxh |Z1h | is the maximum electric distance of a generator bus from the
PCC.
The proof is provided in the Appendix.
A Distributed Feedback Control Approach to the ORPF Problem
6
271
Simulations
The algorithm has been tested on the testbed IEEE 37 [22], which is an actual
portion of power distribution network located in California. The load buses are a
blend of constant-power, constant-current, and constant-impedance loads, with
a total power demand of almost 2 MW of active power and 1 MVAR of reactive
power (see [22] for the testbed data). The length of the power lines range from
a minimum of 25 meters to a maximum of almost 600 meters. The impedance
of the power lines differs from edge to edge (for example, resistance ranges from
0.182 Ω/km to 1.305 Ω/km). However, the inductance/resistance ratio exhibits
a smaller variation, ranging from ∠ze = 0.47 to ∠ze = 0.59. This justifies Assumption 1, in which we claimed that ∠ze can be considered constant across the
network.
The lower and upper bounds for voltage magnitudes has been set to 96% and
104% of the nominal voltage UN = 4800, respectively. The algorithm presented
in Section 4 has been simulated on a nonlinear exact solver of the grid. None of
the assumptions that have been considered during the design of the algorithm
(constant power loads, constant line impedance angle ∠ze , linearized power flow
equations) has been used in these simulations, being only a tool for the design
of the algorithm and for the study of the algorithm’s convergence.
When the grid is operated according to the testbed data, all voltages are
above the threshold. In this configuration, the proposed algorithm is capable of
reducing the power distribution losses practically to the minimum. Indeed we
have that
Power distribution losses
With no optimization
With the proposed algorithm
With numerical nonlinear optimizer
47.164 KW
38.309 KW
37.931 KW
and therefore the proposed optimization algorithm can achieve more than 95%
of the potential loss reduction.
In order to evaluate the performance of the algorithm when voltage constraints are active, the active power demand of two nodes of the grid has been
increased step-wise (see Figure 4). While this variation in the active power demand does not have effect on the optimal reactive power configuration (up to
second-order effects), the increased load causes a drop in the voltage magnitudes
and an increase in power distribution losses up to 57.5 kW. At this point voltage
constraints are not satisfied for one of the agents. The algorithm then drives the
system to a new optimal configuration that guarantees satisfaction of all voltage
constraints, at the cost of slightly larger losses (from 57.5 kW to 57.7 kW).
Additional simulations, including the case of time varying generation profiles
and intermittent loads, are available in [23].
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S. Bolognani, G. Cavraro, and S. Zampieri
60
·104
voltage magnitudes |uh (t)| [V]
power losses Jlosses [W]
4,800
50
4,700
4,650
4,600
40
10
4,750
12
14
16
18
20
10
t
12
14
16
18
20
t
Fig. 4. The left panel represents total power losses, while the right panel represents
the voltage magnitudes at the buses where generators are connected. The solid line
represents the behavior of the proposed algorithm, with the parameter γ equal to 0.1
(one half of the bound provided by Theorem 1). The dashed line, on the other hand,
represent the behavior of the loss minimization algorithm if no voltage constraints are
enforced. The red thick line in the right panel represents the desired voltage bound.
7
Conclusion
In this paper we proposed a distributed algorithm for the problem of optimal reactive power flow in a smart power distribution grid, that is based on a feedback
strategy, in the sense that it requires the interleaving of actuation and measurement, and that the control action is a function of real time data collected from
the agents. Figure 5 provides a block diagram representation of the proposed algorithm. The two feedback functions K1 and K2 (both functions of the measured
voltages) are defined element-wise as
[K1 (u)]h =
Ghk |uh ||uk | sin(∠uk − ∠uh − θ)
(23)
k∈N (h)
and
2
[K2 (u)]h = Umin
− |uh |2 .
(24)
This interpretation of a dual-ascent optimization algorithm as a control feedback
loop with memory, resembles what has been recently done in [24], and allows to
state some final remarks.
A Distributed Feedback Control Approach to the ORPF Problem
273
By adopting a feedback strategy on the measured voltages, the active power
injections in the grid (ph , h ∈ V) and the reactive power injection of the loads
(qh , h ∈ V\C) can be considered as disturbances for the control system. This
means that these quantities do not need to be known to the agents: in some
sense, the agents are implicitly inferring this information from the voltage measurements performed on the grid. This feature differentiates the proposed algorithm from basically all the ORPF algorithms available in the power system
literature, with the exception of some works, like [20], where however the feedback is only local, with no communication between the agents, and of [6] and [7].
Moreover, because of this feedback strategy, the controller (or optimizer) does
not need to solve any model of the grid in order to find the optimal solution.
While a model of the grid has been used in the design of the algorithm, the
online controller does not need to know the grid parameter and to solve the nonlinear equations that are generally a critical issue in offline ORPF solvers. On
the contrary, the computational effort required for the execution of the proposed
algorithm is minimal. These features are extremely interesting for the scenario
of low voltage or medium voltage power distribution networks, where real time
measurement of the loads is usually not available, the grid parameters are only
partially unknown, and many buses are unmonitored.
Another feature that becomes apparent from this feedback interpretation, is
the guarantee that the algorithm can provide regarding the eventual satisfaction
of the voltage constraints. Because the output of the function K2 (u) defined
in (24) is integrated, it is guaranteed that, if the algorithm converges, then
|uh | ≥ Umin for every node h ∈ C. This is true independently of the choice of the
parameters γ and also of the coefficients Ghk .
p, p, q integrator
+
q
1
z−1
+
Grid
u
K1
delay
1
z
saturated integrator
2 sin θ
λ
≥0
γ
z−1
K2
Fig. 5. A block diagram representation of the algorithm proposed in Section 4. The
two feedback functions K1 and K2 are defined element-wise in (23) and (24).
274
S. Bolognani, G. Cavraro, and S. Zampieri
Finally, a control-theoretic approach to the problem of optimal power flow
enables a number of analyses on the performance of the closed loop system that
are generally overlooked when tackling the problem with the tools of nonlinear
optimization. Examples are L2 -like metrics for the resulting losses in a timevarying scenario (see for example the preliminary results in [25]), robustness to
measurement noise and parametric uncertainty, stability margin against communication delays. These analyses, still not investigated, are also of interest for
the design of the cyber architecture that has to support this and other real time
algorithms, because they can provide specifications for the communication channels, the communication protocols, and the computational resources that need
to be deployed in a smart distribution grid.
A
Proof of Theorem 1
As shown in Section 5, the algorithm (19) is a dual-ascent algorithm for the
solution of the constrained quadratic problem (17). As the expression (18) for
the voltage constraint (17b) is an affine function of the decision variables, strong
duality holds. We thus have zero duality gap and therefore if the dual ascent
algorithm converges, it converges to the optimal solution of the problem.
In order to characterize converge of the algorithm via a condition on the
parameter γ, we need to define the two quantities
x(t) = q(t) − q ∗
and y(t) = λ(t) − λ∗ ,
where q ∗ and λ∗ are the optimal value of the primal and dual variables, respectively. Notice that, because of the constraint 1lT q = c in (17), we have that
1lT x(t) = 0,
∀t.
The following two necessary conditions descend from the Uzawa saddle point
theorem [26]:
q∗
∂L(q ∗ , λ∗ ) = MN
(25)
− 2 sin θM λ∗ = α1l
q
∂q
for some α, and
∂L(q ∗ , λ∗ )
2
= Umin
1l − v(q ∗ ) ≤ 0
∂λh
(26)
with
∂L(q ∗ , λ∗ )
2
= Umin
1l − v(q ∗ ) < 0
∂λh
The update for x(t) is, given (19a),
⇔
λ∗h = 0.
x(t + 1) = q(t + 1) − q ∗
q ∗ + x(t)
= x(t) − G M N
+ 2 sin θλ(t) + ke1
q
∗
q
= (I − GM )x(t) − G M N
+ 2 sin θλ(t) + ke1 ,
q
(27)
A Distributed Feedback Control Approach to the ORPF Problem
275
which, by using (25) and the fact that G1l = 0 becomes
x(t + 1) = (I − GM )x(t) − 2 sin θGM λ∗ + 2 sin θλ(t) + ke1
= e1 1lT x(t) − 2 sin θ(I − e1 1lT )λ∗ + 2 sin θλ(t) + ke1
= 2 sin θy(t) + k e1 ,
where k = 2 sin θ1lT λ∗ + k and where we used Lemma 1 and the fact that
1lT x(t) = 0.
We then consider the update equation for y(t). By using the fact that, according to (18),
v(q(t)) = v(q ∗ ) + 2 sin θM x(t),
and according to (19b), we have that
y(t + 1) = λ(t + 1) − λ∗
2
= λ(t) + γ Umin
1l − v(q(t + 1)) + − λ∗
2
= y(t) + λopt + γ(Umin
1l − v(q ∗ )) − 2γ sin θM x(t + 1) + − λ∗ .
By using the fact that λ∗ ≥ 0 together with (26) and (27) we have that
2
λ∗ = [λ∗ ]+ = [λ∗ + γ(Umin
1l − v(q ∗ ))]+
Therefore, by plugging in the expression for x(t + 1) and by using the fact that
M e1 = 0, we have
2
y(t + 1) = (I − 4γ sin2 θM )y(t) + λ∗ + γ(Umin
1l − v(q ∗ )) +
2
− [λ∗ + γ(Umin
1l − v(q ∗ ))]+ .
Then, by using the fact that
a+ − b+ ≤ a − b,
we have that
y(t + 1) ≤ (I − 4γ sin2 θM )y(t) ,
and therefore y(t) converges to zero if
γ≤
1
,
4 sin2 θρ(M )
where ρ(M ) is the spectral radius of M .
Finally, because of (11), we have that
Mhh ≥ Mhk ≥ 0 ∀h, k,
276
S. Bolognani, G. Cavraro, and S. Zampieri
and thus
2
ρ(M ) = max
Mhk vk
≤ max
v=1
h
v=1
k
M
h
hh
2
vk
k
2
√ 2 √
Mhh m ≤
≤
max Mhh m = max Mhh m.
h
h
h
h
By using the fact that, via (9), Mhh = |Z1h |, we then finally have the sufficient
condition for convergence
γ≤
1
.
4 sin θ maxh |Z1h |m
2
References
1. Katiraei, F., Iravani, M.R.: Power management strategies for a microgrid with
multiple distributed generation units. IEEE Trans. Power Syst. 21(4), 1821–1831
(2006)
2. Prodanovic, M., De Brabandere, K., Van den Keybus, J., Green, T., Driesen, J.:
Harmonic and reactive power compensation as ancillary services in inverter-based
distributed generation. IET Gener. Transm. Distrib. 1(3), 432–438 (2007)
3. Zhao, B., Guo, C.X., Cao, Y.J.: A multiagent-based particle swarm optimization
approach for optimal reactive power dispatch. IEEE Trans. Power Syst. 20(2),
1070–1078 (2005)
4. Lavaei, J., Rantzer, A., Low, S.H.: Power flow optimization using positive quadratic
programming. In: Proc. 18th IFAC World Congr. (2011)
5. Lam, A.Y.S., Zhang, B., Dominiguez-Garcia, A., Tse, D.: Optimal distributed voltage regulation in power distribution networks. arXiv [math.OC] 1204.5226 (2012)
6. Tenti, P., Costabeber, A., Mattavelli, P., Trombetti, D.: Distribution loss minimization by token ring control of power electronic interfaces in residential microgrids.
IEEE Trans. Ind. Electron 59(10), 3817–3826 (2012)
7. Bolognani, S., Zampieri, S.: Distributed control for optimal reactive power compensation in smart microgrids. In: Proc. 50th IEEE Conf. on Decision and Control
and European Control Conf. (CDC-ECC 2011), Orlando, FL (2011)
8. Green, T.C., Prodanović, M.: Control of inverter-based micro-grids. Electr. Pow.
Syst. Res. 77(9), 1204–1213 (2007)
9. Lopes, J.A., Moreira, C.L., Madureira, A.G.: Defining control strategies for microgrids islanded operation. IEEE Trans. Power Syst. 21(2), 916–924 (2006)
10. Bolognani, S., Zampieri, S.: A distributed control strategy for reactive power compensation in smart microgrids. IEEE Transactions on Automatic Control 56(11)
(November 2013)
11. Phadke, A.G.: Synchronized phasor measurements in power systems. IEEE Comput. Appl. Power 6(2), 10–15 (1993)
12. Kundur, P.: Power system stability and control. McGraw-Hill (1994)
A Distributed Feedback Control Approach to the ORPF Problem
277
13. Costabeber, A., Erseghe, T., Tenti, P., Tomasin, S., Mattavelli, P.: Optimization
of micro-grid operation by dynamic grid mapping and token ring control. In: Proc.
14th European Conf. on Power Electronics and Applications (EPE), Birmingham,
UK (2011)
14. Dorfler, F., Bullo, F.: Kron reduction of graphs with applications to electrical
networks. IEEE Transactions on Circuits and Systems I 60, 150–163 (2013)
15. Ciobotaru, M., Teodorescu, R., Rodriguez, P., Timbus, A., Blaabjerg, F.: Online
grid impedance estimation for single-phase grid-connected systems using PQ variations. In: Proc. 38th IEEE Power Electronics Specialists Conf., PESC (2007)
16. Ciobotaru, M., Teodorescu, R., Blaabjerg, F.: On-line grid impedance estimation
based on harmonic injection for grid-connected PV inverter. In: Proceedings of the
IEEE International Symposium on Industrial Electronics (ISIE), pp. 2437–2442
(2007)
17. Bertsekas, D.P.: Nonlinear programming, 2nd edn. Athena Scientific, Belmont
(1999)
18. Lavaei, J., Low, S.H.: Zero duality gap in optimal power flow problem. IEEE Trans.
Power Syst. (2011)
19. Baran, M.E., Wu, F.F.: Optimal sizing of capacitors placed on a radial distribution
system. IEEE Trans. Power Del. 4, 735–743 (1989)
20. Turitsyn, K., Šulc, P., Backhaus, S., Chertkov, M.: Options for control of reactive
power by distributed photovoltaic generators. Proc. IEEE 99(6), 1063–1073 (2011)
21. Gómez-Expósito, A., Conejo, A.J., Cañizares, C.: Electric energy systems. Analysis
and operation. CRC Press (2009)
22. Kersting, W.H.: Radial distribution test feeders. In: IEEE Power Engineering Society Winter Meeting, vol. 2, pp. 908–912 (2001)
23. Bolognani, S., Cavraro, G., Carli, R., Zampieri, S.: A distributed feedback control
strategy for optimal reactive power flow with voltage constraints. arXiv:1303.7173
[math.OC] (2013)
24. Wang, J., Elia, N.: A control perspective for centralized and distributed convex
optimization. In: Proceedings of the 50th IEEE Conference on Decision and Control
(CDC), Orlando, FL, USA, pp. 3800–3805 (2011)
25. Bolognani, S., Zampieri, S.: A distributed optimal control approach to dynamic
reactive power compensation. In: Proc. 51st IEEE Conf. on Decision and Control,
CDC (2012)
26. Uzawa, H.: The Kuhn-Tucker theorem in concave programming. Studies in linear
and nonlinear programming, pp. 32–37. Stanford University Press (1958)