New Modeling and Control Solutions for Integrated
Microgrid System with respect to Thermodynamics
Properties of Generation and Demand
by
MASSAGH
Fang-Yu Liu
B.S. Mechanical Engineering
National Taiwan University, 2012
4OLOGY
OCT
I
LIBRA RIES
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2014
@ Massachusetts Institute of Technology 2014. All rights reserved.
A uthor ...................
Signature redacted
Department of Mechanical Engineering
Aug 8, 2014
Certified by.......
Signature redacted
Kamal Youcef-Toumi
Professor of Mechanical Engineering
Thesis Supervisor
Accepted by...................
20%
Signature redacted
David E. Hardt
Professor of Mechanical Engineering
Chairman, Committee on Graduate Students
2
New Modeling and Control Solutions for Integrated Microgrid
System with respect to Thermodynamics Properties of
Generation and Demand
by
Fang-Yu Liu
Submitted to the Department of Mechanical Engineering
on Aug 8, 2014, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
This thesis investigates microgrid control stability with respect to thermodynamics
behaviors of generation and demand. First, a new integrated microgrid model is
introduced. This model consists of a combined cycle power plant, a building with
air-conditioning system and a renewable source. This integrated model allows researchers to study microgrid stability while considering the physical behaviors of the
generation and demand. Specifically, the model takes into account the slow dynamics
existing in both the generation side and demand side.
Second, a model predictive controller (MPC) is implemented for this new integrated
system. The MPC uses a linearized model of this system to generate control commands by predicting the system behavior and optimize the system performance. The
MPC minimizes an objective function that includes the power imbalance between
generation and load, as well as the temperature differences between zone temperatures and the desired temperature setpoints.
Overall, the results in this work proved that excellent grid balance could be achieved
by a model predictive controller for our integrated model. The modeling and control
performance have been verified by simulating several different scenarios.
Thesis Supervisor: Kamal Youcef-Toumi
Title: Professor of Mechanical Engineering
3
4
Contents
1
2
Introduction
13
1.1
Research Motivation .......
1.2
Research Approach and Thesis Structure . . . . . . . . . . . . . . . .
...........................
System Modeling
Introduction ........
2.2
Modeling of the Integrated System
................................
17
Subsystem Modeling . . . . . . . . . . . . . . . . . . . . . . .
19
Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.3.1
Steady State of the System Model . . . . . . . . . . . . . . . .
31
2.3.2
Building Air-conditioning (AC) Subsystem . . . . . . . . . . .
33
2.3.3
Generation Sub-system . . . . . . . . . . . . . . . . . . . . . .
38
Simulation Results - Showing System Behaviors . . . . . . . . . . . .
41
2.4.1
Scenario 1: Effect on Grid Frequency with Increasing Fuel Demand of Generation . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2
2.4.3
43
Scenario 3: Effect on Grid Frequency with Variation in Renewables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4
41
Scenario 2: Effect on Grid Frequency with Changing in the
Building Demand . . . . . . . . . . . . . . . . . . . . . . . . .
45
Scenario 4: Basic Control of Generation and Building under
Renewable Variations . . . . . . . . . . . . . . . . . . . . . . .
2.5
17
. . . . . . . . . . . . . . . . . . .
2.2.1
2.4
15
17
2.1
2.3
13
Sum m ary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
50
54
3
Model Predictive Control for Integrated Microgrid System
3.1
. . . . . . . . . . . . . . . . . .
55
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
D esign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
M PC Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
Simulation Results and Discussion . . . . . . . . . . . . . . . . . . . .
65
3.2.1
3.3
..
Introduction . . . . . . . . . . ..
3.1.1
3.2
55
4 Conclusions and Recommendations
69
4.1
Sum m ary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
4.2
Recommendations . . . . . . . . . . . . . . . . . . . .... . . . . . . .
70
A Tables
71
B MATLAB Code - Steady State Value of the System
75
C MATLAB Code - Model Linearization
81
D MATLAB Code - Model Predictive Control
6
109
List of Figures
..
. . ..... . . .
.
. . . .. .
2-1
Standard 6-buses grid network . . . . . . . . . . . . . . . . . . . . . . .
18
2-2
Brayton cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2-3
Gas turbine block diagram . . . . . . . . . .... . . . . . . . . . . . . .
22
2-4 Steam turbine block diagram . . . . . . . . . . . . . . . . . . . . . . . .
22
2-5
Fuel system block diagram . . . . . . . . . . . . . . . . . . . . . . . . .
23
2-6
Single-shaft combined cycle power plant model . . . . . . . . . . . . . . .
25
2-7
The building + air-conditioning model . . . . . . . . . . . . . . . . . . .
27
2-8
Steady state simulation for generator angles and speeds. 61 62 63
32
2-9
Steady state simulation for generation powers, valve positions and fuel flows.
32
Vp1 W
1
Pml V
2
W1 2 Pm2 ..
.
W1 W2 W3
.
2-10 Steady state simulation for building temperatures. Tzi Twii Tw1 2 Twi3
. . . .
33
2-11 Building AC sub-system verification results of cooling effect. . . . . . . . .
35
2-12 Building AC sub-system verification results of cooling effect. . . . . . . . .
35
Tw14 Tc1 Tf1 Tz 1 Tw 2 1 Tw 22 Tw 23 Tw 24 Tc 2 Tf 2 .
. . . . .. ..
2-13 Building AC sub-system verification results with ambient temperature change. 36
2-14 Building AC sub-system verification results with ambient temperature change. 36
2-15 Building AC sub-system verification results with power input change.
. .
37
2-16 Building AC sub-system verification results with power input change.
. .
37
2-17 Generation sub-system verification results with fuel increase . . . . . . . .
39
2-18 Generation sub-system verification results with fuel increase. The zoom-in
version for Figure 2-17. . . . . . . . . . . . . . . . . . . . . . . . . . ... .
2-19 Generation sub-system verification results with fuel decrease.
7
. . . . . . .
39
40
2-20 Generation sub-system verification results with fuel decrease. The zoom-in
version for Figure 2-19. . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2-21 Scenario 1: Time domain simulation of the mechanical power of generation
for the integrated system with increasing fuel demand. . . . . . . . . . . .
42
2-22 Scenario 1: Time domain simulation of the mechanical power of generation
for the integrated system with increasing fuel demand. The zoom-in version
of Figure 2-21.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2-23 Scenario 1: Time domain simulation of the generator (motor) angles and
speeds for the integrated system with increasing fuel demand. . . . . . . .
43
2-24 Scenario 2: Time domain simulation of the generator (motor) angles and
speeds for the integrated system with changing in the building demand.
.
44
2-25 Scenario 2: Time domain simulation of temperature of the building for the
integrated system with changing in the building demand.
. . . . . . . . .
2-26 Scenario 3-1: Fast and small variation in the power generated by renewables.
44
46
2-27 Scenario 3-1: Time domain simulation of the generator (motor) angles and
speeds for the integrated system with fast and small renewable power variation. 46
2-28 Scenario 3-2: Slow and small variation in the power generated by renewables. 47
2-29 Scenario 3-2: Time domain simulation of the generator (motor) angles and
speeds for the integrated system with slow and small variation in renewables. 47
2-30 Scenario 3-3: Fast and large variation in the power generated by renewables.
48
2-31 Scenario 3-3: Time domain simulation of the generator (motor) angles and
speeds for the integrated system with fast and large variation in renewables.
2-32 Scenario 3-4: Slow and large variation in the power generated by renewables.
48
49
2-33 Scenario 3-4: Time domain simulation of the generator (motor) angles and
speeds for the integrated system with slow and large variation in renewables.
49
2-34 Governor steady-state power-frequency characteristics . . . . . . . . . . .
51
2-35 On-off + proportional controller algorithm . . . . . . . . . . . . . . . . .
51
2-36 Scenario 4: Control commands for the generations and building with basic
control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
52
2-37 Scenario 4: Time domain simulation of the generator (motor) angles and
speeds for the integrated system with basic control. . . . . . . . . . . . .
53
2-38 Scenario 4: Time domain simulation of the mechanical power of the generation for the integrated system with basic control . . . . . . . . . . ... . .
53
2-39 Scenario 4: Time domain simulation of temperature of the building for the
integrated system with basic control. . . . . . . . . . . . . . . . . . . . .
54
3-1
Principle of MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3-2
Time domain simulation of the generators angles for both nonlinear
and linear m odel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-3
61
Time domain simulation of the generator speeds for both nonlinear and
linear m odel.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
3-4 Time domain simulation of the mechanical power of generation for both
nonlinear and linear model.
3-5
. . . . . . . . . . . . . . . . . . . . . . .
61
Time domain simulation of the zone temperatures for both nonlinear
and linear m odel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
3-6
Structure of the MPC
. . . . . . . . . . . . . . . . . . . . . . . . . . .
63
3-7
The control commands calculated from the MPC. . . . . . . . . . . . . .
65
3-8
Time domain simulation of the generator (motor) angles and speeds for the
integrated system under MPC. . . . . . . . . . . . . . . . . . . . . . . .
3-9
66
Time domain simulation of the mechanical power of generation for the integrated system under MPC. . . . . . . . . . . . . . . . . . . . . . . . . .
66
3-10 Time domain simulation of temperature of the building for the integrated
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
3-11 The power difference over time . . . . . . . . . . . . . . . . . . . . . . .
67
system under MPC.
9
10
List of Tables
3.1
x, z and u for our integrated system . . . . . . . . . . . . . . . . . .
3.2
x, y and u for our integrated system modified for MPC . . . . . . . . 64
60
A.1 Notation for the zone model . . . . . . . . . . . . . . . . . . . . . . .
72
A.2 Steady state value of the integrated model . . . . . . . . . . . . . . .
73
11
12
Chapter 1
Introduction
1.1
Research Motivation
The renewable energies are playing an important role today and more so in the future.
Due to its sustainability and little environmental impact, utilizing renewables become
an ideal solution to the depletion of fossil fuel sources. Therefore, government and
industries are looking for ways to efficiently manage the renewables.
As the rapid development of the renewable energy techniques, the penetration of
variable energy resources has increased tremendously in recent years. The high penetration of renewable resources into the grid changes the conventional structure of the
grid. A motivation of dividing grid network into smaller networks that allow local
controller implementation has emerged.
Microgrids concept is then introduced which is very different from the conventional
grid structure. Microgrid presents a cluster of loads and distributed energy resources
(DERs) operating as a single engineering system, which is capable of supplying its
owanlocal load in both grid-connected and autonomous (island) mode
{I1J.
The mi-
crogrid can act autonomously without being connected to the utility, or it can also
connect to the utility under huge disturbances. This small semi-autonomous system
offers increased reliability and efficiency for the power system network.
The microgrids concept has great capabilities but also present new challenges. As a
small-scale power units, the fluctuation of load and the intermittency of renewable
13
energy - particularly wind and solar - will make a huge impact on the stability of microgrid, especially during island mode. The microgrid stability has been investigated
in many research papers [17]. Much effort by researchers has been spent on developing methods to address such great challenges in order to make microgrids become a
reality.
Different modeling and control structures have been proposed in literatures [4] [3] [12].
Generally, microgrids are modeled as a small-scale version of the conventional grid
that can be connected or disconnected to the utility grid. The major control structures can be summarized into three categories, island mode control [22], hierarchical
control [27] and agent based control [6]. In island mode control, the microgrid is
designed to run autonomously with local controller on each micro-sources. In hierarchical control, three control levels are used for maintaining reliable operation, using
a centralized controller at upper level to control all the local controllers. In agentbased control, agents can represents the physical or virtual entities to communicate
and react to each other and the environment.
A gap in the existing literatures is the integration of physical models that include slow
dynamics from both the generation and load sides. The grid dynamics considered in
the previous works are basically the coupled electromechanical dynamics among synchronous generators, which is the most relevant to grid frequency.
However, the
time-scale of this kind of dynamics is really fast when compared to the thermal response (ex. the ramping effect of generation and the thermal capacitance of building
load). Therefore, it is important to take into account the physical characteristics of
generation and load in order to realize the grid dynamics.
An integrated model that considers those important characteristics of generations
and loads is introduced in this thesis. Additionally, a model predictive controller is
implemented based upon this model. Chapter 2 describes the detailed modeling of
each component in this system using MATLAB language. Those components are connected by a standard grid model to form the integrated system. Verification testing
of each component and whole system confirmed that the model behaves as expected.
Several simulation results are shown in this chapter to demonstrate the system be14
havior under different actions. Chapter 3 covers the development of model predictive
control for the integrated system in the MALAB environment. Simulation results
demonstrate the performance of the proposed control approach.
1.2
Research Approach and Thesis Structure
To address these emerging challenges mentioned above, the microgrid requires a more
comprehensive model to investigate the important characteristics of generation and
load, as well as a more generalized control method to achieve better grid stability.
And given the small size of a microgrid and advances in computational and communication speeds, the possibility of a purely centralized real time control that could
get information from both generation and load sides and communicate between each
other is much greater. Therefore, we propose a new integrated modeling solution for
microgrid with power plant generation and building load, as well as a model predictive
control structure based upon this new model.
15
16
Chapter 2
System Modeling
2.1
Introduction
The purpose of this work is to investigate the thermodynamic effects of both the
generation and load on the microgrid. Here we introduce an integrated model that
consists of a combined cycle power plant subsystem as the dispatchable generation,
as well as a building air-conditioning subsystem as the controllable load. The main
objective is to generate a physical-based model that captures both the fast and slow
dynamics of these subsystems.
The details of the different subsystems are first described. Then simulation results
are presented to show the verification of the system response. Finally, the chapter
demonstrates the overall system behaviors under three different scenarios.
2.2
Modeling of the Integrated System
The modeling objective of this section is to develop & generalised iateggAted model
of a microgrid. This. integrated model consists of a standud -6&b
rniegwrid system
used as the grid network [251. In addition, it incorporates a dispatchable generation
subsystem, a variable generation subsystem and a controllable load subsystem. These
are all connected to different buses in the grid network. The combination of all the
subsystem forms an integrated model of the microgrid system. Figure 2-1, inspired
17
from [25], shows the whole structure of the system, which contains two dispatchable
generations (Buses 1 and 2), one variable generation - renewables (Bus 3) that can
represent solar or wind energy generation, one building (Bus 4), and two static loads
(Buses 5 and 6).
Dispatchable
Generation
Dispatchable
Grid
0Generation
Comind CdeFo
I
Combined Cyde
I
Power Plant
Powe Plat
2
I
Controllable Load
Buding + Air-conditioning System
.
5
3
Variable
Generation
RenewaNes
Static Load
0
Static Load
Figure 2-1: Standard 6-buses grid network
Bus 1: Dispatchable Generation (Combined Cycle Power Plant)
Bus 2: Dispatchable Generation (Combined Cycle Power Plant)
Bus 3: Variable Generation (Renewables)
Bus 4: Controllable Load (Building + Air-conditioning System)
Bus 5: Static Load
Bus 6: Static Load
18
As shown in Figure 2-1, the overall system consists of five subsystems: the dispatchable generation, the variable generation, the controllable load, the static load, and
the grid subsystem. Details of each subsystem modeling are stated in what follows.
2.2.1
Subsystem Modeling
Generation Subsystem
1. Dispatchable Generation - Combined Cycle Power Plant
In power generation, the combined cycle power plant (CCPP) plays an important role due to its increased efficiency and low emissions. Therefore, our model
uses a combined cycle power plant model to present the characteristics of the
dispatchable generation. The combined cycle power plant consists of a gas turbine and a steam turbine. The steam is generated by firing hot exhaust gases
from the gas turbine using the heat recovery steam generator.
Gas Turbine
The most commonly used model for a gas turbine in CCPP is Rowen's model
[24]. Rowen's model has provided a starting point for the development of gas
turbine models but it doesn't include the physical behaviors of the gas turbine.
Therefore, several gas turbine models have been introduced in the literature
with different degree of complexity to get deeper insight into internal processes.
There are some reviews that give a brief comparison of those models [26] [9] [21].
The model used in this work to capture the physical behavior of the gas turbine of a CCPP is based on the Detailed model [10] [19]. The thermodynamic
characteristics of the gas turbine are modeled by algebraic equations in the Detailed model. These equations axe corresponding to the adiabatic compression
and expansion, as well as the isobaric heating within the combustor of a gas
turbine.
A schematic of the gas turbine is shown in Figure 2-2. It is based on a Brayton thermodynamic cycle. The actual Brayton cycle consists of four processes:
1-2 adiabatic compression, 2-3 isobaric heat addition, 3-4 adiabatic expansion,
19
2
Chwnber
34TM.t
Turbkn*
CaMMp M 9 W
4
Figure 2-2: Brayton cycle
4-1 isobaric heat rejection. The algebraic equations describing these thermody-
Td=Ti(1+
)
xc = (PR
,
namic processes are presented below (Equation (2.1)):
Xh =
(PR
Te =
Tf(l
-
7
(1 -
)
fe bnLHV
TI=Td+(%Cph
Wf + W
Wf
Wf +W
'Yh-
) -h
1
)7t)
Pg = (W + Wf)Cph(Tf - Te)
20
-
WCpc(Td
-
T)
(2.1)
where
Td
the temperature at compressor outlet
T
the temperature at compressor inlet
Tf
gas turbine firing temperature
Te
exhaust gas temperature
77C
compressor efficiency
77t
turbine efficiency
71eoma
combustion efficiency
PR
compression ratio
7c
cold end ratio of specific heats
W
air flow
Wn
air flow at nominal operation
Wf
fuel flow
Wfn
fuel flow at nominal operating condition
LHV
lower heating value of natural gas
hot end ratio of specific heats
Cph
specific heat of exhaust gas flow
CCc
specific heat of air flow
Pg
the power setpoint
For this model, the input is the fuel flow labeled by W and the output is the
mechanical power setpoint labeled by Pg. This model has no dynamics, we are
adding a first order differienotial eque*1oaWdemonstrate the dynamic response
of the gas turbine..The dynamics is-ralatingthe power setpoint and the actual
mechanical power output of the gas turbine. The overall block diagram of the
gas turbine is shown in Figure 2-3. And for the overall gas turbine model, the
input is the fuel flow labeled by Wf and the output is the mechanical power
21
generated by the gas turbine labeled by Png.
Gas turbine
Gas Turbineb
d
I + STCD
Algebraic Equations
Figure 2-3: Gas turbine block diagram
where
Wf
fuel flow
Pmg
the -power generated by the gas turbine
TCD
gas turbine time constant
Steam Turbine
In this thesis, a simplified steam turbine model, which only takes into account
the dynamics within the heat recovery steam generator, is used. The dynamics
is modeled as a second order response shown in Figure 2-4. The input is the
power setpoint from the algebraic equations Pg, and the output is the mechanical
power generated by the steam turbine Pm,,.
Stus Turbine
P
PM
(1 +sT,.)(1+sTb)
Figure 2-4: Steam turbine block diagram
22
where
PM
the power generated by the steam turbine
Tm
steam turbine time constant
Tb
heat recovery boiler time constant
Fuel System
The fuel system in the CCPP is modeled as a second order differential equation
as shown in Figure 2-5. The input is the fuel demand labeled as F and the
output is the fuel flow labeled as Wf, which is also the input to the gas turbine
model.
Fuel qutsin
i
1d
Wr
Figure 2-5: Fuel system block diagram
where
Fd
fuel demand
V
valve position
TV
valve position r time conota4t
TF
fte1aytem time constat
At steady state, the fuel flow is exactly the same as the fuel demand.
23
The Synchronous Generator Model
There are various sophisticated and detailed models in the literature [20]. For
our purpose, we choose the most commonly used second order simplified model
that captures the electromechanical properties of synchronous generators. The
second order model describes the acceleration (deceleration) of the synchronous
generator due to any imbalance between the mechanical power and electrical
power.
For each generator i, we have
Wi =
7rf
(Pmi(pu)- Pe(pu) - Diwi)
where
rotor angle
rotor speed
Pm(pu), Pe(pu)
per unit mechanical power, electrical power
D
damping coefficient
H
generator rotor inertia
f
nominal frequency
The overall CCPP model consists of a gas turbine, a steam turbine, a fuel
system, and a synchronous generator. The block diagram for the overall CCPP
model is shown in Figure 2-6.
24
Fuel systern
F
IW
I
- AfgebraWe equations
of energy traufwrm
1 + sTF
1 + STV
P
Gas turbine
Synchronous
P
P6
+
P,
Generator
1+ s
1
;CI1
(1 +srT)(1+s
b)
Heat recvery/Steam turbine
Figure 2-6: Single-shaft combined cycle power plant model
There are seven state variables in the CCPP model representing the valve position V, the fuel flow Wf, the mechanical power output from gas turbine P,,
the mechanical power output from steam turbine Pn, the rate change of the
mechanical power output from steam turbine R,&., as well as the angle and the
speed of the synchronous generator 5,, w. The input to the overall CCPP model
Js the fue demadinto the4uel ystem labeledas F&ndoutput isithe net electrical power output from the synchronous generator
ii Ieled&
the ideal
cases, the net power output of our CCPP model is twice of the power output of
a single gas turbine. This is not the case in practice, but this simplified model
is sufficient for the purpose of this thesis.
2. Variable Generation - Renewables
The variable generation is presented as a simple power injection into the grid
with forecast. The forecast could be a square signal, a sinusoidal signal or the
25
real-time data obtained from solar or wind resources.
Load Subsystem
1. Controllable Load - Building with Air-conditioning System
For the building models, there are three main types of methods in the literature
[2]: nodal (multi-zone) method, zonal method and CFD (Computational Fluid
Dynamics) method. The resulting models are all based on solving the equations
describing the thermal behaviors of the building. Among these three, the CFD
method is the most complete approach. It is appropriate for investigating the
detailed local effect inside the building. The zonal method is a first degree of
simplification of the CFD method. It's a two-dimensional method and can be
used to study the spatial and temporal distribution of the building temperature.
However, both the CFD and zonal methods require huge computation time. And
both methods requires knowledge related to flow dynamics.
In this work, since we are only concerned with the temperature behaviors, the
nodal (multi-zone) approach is chosen to describe the building load. The nodal
method is a one-dimensional method, with a main assumption that each building
zone is a homogenous volume described by uniform variables. Therefore, each
zone is treated as a node with a unique temperature. The heat transfer equations
are solved for each node of the system. The nodal method is good for multi-zone
building modeling and it's easier to implement with reasonable computation
time [5], [15], [18], [1]. Figure 2-7 shows the whole structure of the building
including the air conditioning model, which consists of two zones and an airconditioning system.
26
Outside air
i
M
M*ut
*
Supply air
T
Cooling coil
Return air
Zone 1
Zone 2
Tn
T
Ta
Figure 2-7: The building
+ air-conditioning model
The Zone Model
The governing equations derived from mass and energy balance of each zone i
are given by,
d Tz.
hMsaiCp(Tsa - Tzi)
Czi dt
4
+ 1hwij Awi (Twij - Tzi)
j=1
+ he Aes(Tej - Tzi) + hfpAfp(Ti - Tzi)
(2.2)
Equation (2.2) states that rate change of energy in a zone is equal to the difference between the power transferred to the zone by convection and the power
removed from the zone. The time derivative of the zone temperature Tz is given
by this energy equation. This temperature depends on the wall temperature
Tw, the ceiling temperature Tc, the floor temperature Tf, the supply air temperature Ta, and the ambient temperatures Tamb. The ambient temperature is
set by the environment. The other associated temperatures are coming from
the equations stated below.
27
For the internal walls of each zone i, we have
0w13
d Tw
dt
=hj
j(z- wj
+ hwij Awij (Tz-adjacent
Tw )
-
(2.3)
Equation (2.3) shows the rate changes of the energy in internal walls are equal
to the energy transferred through walls due to temperature difference between
the indoor temperature and the adjacent zone temperature.
For the external walls of each zone i, we have
- hwl Awj(Tzi - Twi
)
Cwd dt
+ hwij Avij(Tamb - Twi)
(2.4)
Equation (2.4) shows the rate changes of the energy in external walls are equal
to the energy transferred through walls due to temperature difference between
the indoor temperature and the ambient temperature.
For the ceiling and floor of each zone i, we have
Ce d T'd = hci Aci(Tzi - Te ) + hciAci(Tamb - Tdi)
dt
Cf
+ hfiAfi(Tamb -Tf)
dt =hA(Tzi-T)
(2.5)
where the notation is given in Table A.1. Equation (2.5) shows the rate changes
of the energy in ceilings and floors are equal to the energy transferred through
ceilings and floors due to temperature difference between the indoor tempera,
ture and the ambient temperature.
28
The Air-conditioning System
The air-conditioning system contains a mixing box and a cooling coil.
The energy balance of the air in the mixing box is given by,
Tmix
T o'Tamb + EN
=
mfaiTz
ha=z
2.6)
26
Msa
Equation (2.6) shows the air temperature comes out of the mixing box Tmi,
depends on the ambient temperature Tamb and the return air temperature from
each zone Ti.
The energy consumption of the cooling coil is calculated as
follows,
Pcwling = COP (mix
Tsa)
(2.7)
Equation (2.7) shows the power input to the cooling coil Pcooing has effect on the
supply air temperature Ta. The performance of the cooling coil is quantified
by the coefficient of performance (COP) in the equation.
2. Static Load
The static load is presented as constant power consumption over time.
Grid Subsystem
The grid itself is modeled by a set of algebraic equations, which come from the power
flow analysis shown in Equation (2.8).
VAVjGi cos(i - 93)
Pe= + ViGr
2 2 + (
+ Bij sin(9 2 - 9,)]
Qei =
--
1 VV[Gi
3 sin(Oi - Oj)
3
VViB +
- Bij cos( 2 - 93)]
29
(2.8)
where
9
bus voltage angle
V
bus voltage magnitude
Pe, Qe
active electrical power and reactive electrical power
G, B
real and imaginary part of the system admittance matrix
Detailed formulation of this equation can be found in [8]. The renewables and static
loads are modeled as static power injections, which affect the system admittance
matrix.
30
2.3
Model Validation
This section presents simulation results to ensure that the components and subsystems
in the integrated model behaves as they axe intended to.
2.3.1
Steady State of the System Model
The integrated model has a set of differential equations associated with 30 states
shown in Table A.2. The steady state values can be calculated by setting all the
derivatives in the differential equations to zeros. Here we calculate the steady state
value from the equations in the previous section and run the simulation to verify it.
The MATLAB code for finding the steady state value is in Appendix B.
From calculation, the steady state value of the system states are listed in Table
A.2. Simulation results show the system behavior when setting the calculated steady
state values in Table A.2 as initial conditions. Figure 2-8 shows the time domain
simulation of the generator angles and speeds, Figure 2-9 shows the time domain
simulation of the power output and the fuel flow of the generation, and Figure 2-10
shows the time domain simulation of the building temperatures. From these three
figures, the calculated steady state values are verified. Nothing changes in the 1200
seconds simulation duration. The whole system rests in equilibrium.
31
1 01
1
1
I 1
1
Phase Angle Difference of Synchronous Generators Over Time
1
I
I
1
0
-0)-
-
-
< -20
-30
I
I
120
I
60
II
240
I
180
I
II
II
I
I
I
540 600 660 720
Time (sec)
I
I
I
I
420 480
I
I
I
300 360
Gen1 angle
Gen2 angle
Loadi angle
II
840
II
780
II
900
II
II
II
I
960 1020 1080 1140 1200
Speed of Synchronous Generators Over Time
61
I
1
1
- Geni speed
- Gen2 speed
-Load1
speed
6C .5-
60
U)
55 9.5-
60
120
I
I
I
I
r
0
180 240
300
480
540 600 660
Time (sec)
I
I
I
I
I
360 420
720
780
840
I
I
900 960 1020 1080 1140 1200
Figure 2-8: Steady state simulation for generator angles and speeds. 61 62 63
W1 W2 W3
Mechanical Power Over Time
1.8r
- GenI mechPower
-Gen2 mechPowerJ
-1.6L
CL 1.4-
~1.21j
200
0
600
Time (sec)
800
1000
1200
Valve Position/Fuel Flow Over Time
-Geni
Valve Position
- Gent Fuel Flow
-Gen2 Valve Position
- Gen2 Fuel Flow
-
0.85r-
400
0.8
0.75
0.7
0 0.654
.J-I
I
200
0
400
600
Time (sec)
j
800
1000
1200
Figure 2-9: Steady state simulation for generation powers, valve positions and fuel flows.
Vp1 Wf1 Pm1 V
2
Wf2 Pm 2
32
Temperature Over Time for bus 4 (building)
244-Z.r.I2.P0.11
-
24.
24.2
-
24.1
-
24
23,
-
E
800
00
00400
0
10
1000
Time (sec)
Temperature Over Time for bus 4 (building)
2 7.5 --
I2, P24 1 6
r1
Z41.H I.3P1p0041
27-22_
Ztadl
-
state s at f. T T2T
24.5
-1222
E
Time
ag IO
0401.
seeiigc)-MU
242
Tw14
2.
Buidin
2.2
Figure 2-10:
1200
2.040406010
2
Tc1
Time (sec)
f1
Ai-wnaxo
T
Tw2
z1T2012T222Tw23
T
Tc2
f
(AC)Subsste
n
building temperatures.
foroftebidnlCsbsse
a etdwt min emertr Tz
Steady
~ ~~~~~" behllo
~ ~~ ~simulation
The
~ ~~~ state
Twig Tw 1 2 Tw 1 3
Tw 14 Tcj Tf1 Tz, TW 2 1 TW 22 Tw 23 Tw 24 Tc 2 Tf 2
eprtr
Inta
2.3.2
ftewoebidn
s3
Ther
ambien temperauetruh
Building Air-conditioning (AC) Subsystem
The behavior of the building AC sub-system was tested with ambient temperature
change and power input (the power input to the air-conditioning unit) change.
Case 1: Figures 2-11 and 2-12 show the expected cooling event of a building. The
initial temperature of the whole building is 300. The ambient temperature through-
out the whole simulation period is also 300. When the air-conditioning is off, the
whole building remains at 300. At 240 seconds, the air-conditioning is turned on.
As expected, the zone temperatures, the wall temperatures as well as the ceiling and
floor temperatures ramped down due to the cooling effect. The ramping rates of the
wall, ceiling and floor temperatures are slower than the zone temperature due to their
larger capacitances. We also assumed that the airflows to these two zones are the
same in this case. Therefore, we can see from Figure 2-12 that the final temperature
of zone 1 is lower than zone 2 due to its smaller volume.
33
Case 2: Figures 2-13 and 2-14 show the building behaviors subject to ambient
temperature change.
The room temperature goes down (goes up) as the ambient
temperature decrease (increase) under the same power input.
Case 3: Figures 2-15 and 2-16 show the building behaviors subject to input power
change. The room temperature goes lower as the power input increase. And the room
temperature ramped back to the ambient temperature (300) when the air-conditioner
is off.
The steady state temperatures of the zone temperatures as well as the wall, ceiling and floor temperatures under three cases can be calculated from the MATLAB
code in Appendix B. The calculated steady state temperatures match the simulation
results.
34
Ambient Temperature Over Time
31
30.8-
30.6
-
30.4
30.2-
E) 29.829.4 -29.2 --
0
20
40
1200
Time (sec)
Power input to the air-condition unit
0.5
0.45 --
0.4
0.35 --
0.3 -0.25
0.2 -o- ..,5 --
0.1 --
0.05 -010
2
Boo
400
Time (sec)
1200
Iwo0
Figure 2-11: Building AC sub-system verification results of cooling effect.
Temperature Over Time for bus 4 (building)
C
E
28 --
lr
Znl
mprur
Time (sec)
Temperature Over Time for bus 4 (building)
-2o1000o.00
29-
Zon.1 3004902p1ra41r
. .040
. .I01mperatur0
....
-23n31 00404i 10,mporaIur.
-
-..
Zn31
3
-
--
Z4ne2
-
Zon.2
w40
w004
Oloraor
Ioomperaur
-
27
'
23
-
E
25
-
Zooo2o:
loo
pro.ur
3)
0
200
400
6Tm
Time (sec)
800
I
1000
1200
Figure 2-12: Building AC sub-system verification results of cooling effect.
35
Ambient Temperature Over Time
29.5
E329
2 21.5
28
E CL27.5
F-
27
26.5
0
260
Time (sec)
Power input to the air-condition unit
O 5I
0
CL
a
-0.5-
I
1000
2000
3TMe
4000
S=0
5000
Time (sec)
Figure 2-13: Building AC sub-system verification results with ambient temperature change.
Temperature Over Time for bus 4 (building)
24 -
Z-2
temperature
23,5 --23 -22.522-
CL
21.5
-
-
E S21
20Tec1.0
1
30DO
2
400M
Time (sec)
60 0
5000
Temperature Over Time for bus 4 (building)
27
26
-
25
-.
- tlm erlr
wa
~~~Z0.e1
al: temperalure
mprlr
CL 23
-
E
---
Zo
-
22
Zonel1wal tempsrlure
ZonI wa]l
temperature
I floor Iemperalur.
-- Zons2 wall lemperature
-
----
Ioe
calng temperalurm
Z.n
0
1000
2M00
3000
4D0
5600
Zo2
wall
lempemture
uper
lm
- 0Zon2 w11
Zo.2 I rlng Imspersture
--- Zon2
,
20
-
ilor temperature
Figure 2-14: Building AC sub-system verification results with ambient temperature change.
36
Ambient Temperature Over Time
-
31
30.630.4302302
E
29.8-
E)
29 .6-
29.4
-
29.2
0
l0
2000
00
300
Time (sec)
DO
s00
60
5000
6000
Power input to the air-condition unit
0.9
0.8
0.7
0.6
0.5
0,4
0.3
0.2
0.1
0
100D
2000
4000
Time (sec)
Figure 2-15: Building AC sub-system verification results with power input change.
Temperature Over Time
for bus 4 (building)
-
28
26
02 22-
WONm
tempertur
ZonI
E
C)20
-
18
Zn-1 wEAl
---
0
1000
4000Woo
2000
---
3000
Time (sec)
4000~Z
tempemilure
Z"ne wall
temporaiure
Z"n2
I-pmpeiure
-Zone1wl
tMp9ralur.
I00
f--Znt.
waN-p-rtur.
-2 _n11.nn tepMW
- Z_2e W.oo tempmraur
.
1.
Temperature Over Time for bus 4
(buldng
ejmemes
wall tempwralure
n2
-- -Z
n2 wallr
-
28
wall
- -Z
- -Zn2
-n2
tempoaum
mprur
Celr1emerau
I-oo
22
24
E
22
Ill 0
1000
2M00
Time (sec)
4000
5"
60DO
Figure 2-16: Building AC sub-system verification results with power input change.
37
2.3.3
Generation Sub-system
The behavior of the generation sub-system was tested with the fuel demand input.
Figures 2-17 - 2-20 show the simulation results of the verification test, which confirmed the expected system behavior. When there's an increase in the fuel demand,
both fuel flow and power output of the power plant increase accordingly at different
rates due to different time constants. In contrast, when there's a decrease in the fuel
demand, both fuel-flow and power output of the power plant decrease accordingly at
different rates due to different time constants.
Figure 2-17 shows the simulation results when encountered a fuel demand increase
at 50 seconds. Figure 2-18 is the zoom-in version of Figure 2-17. From the graphs,
we can see that the valve position and fuel flow can quickly catch up with the fuel
demand due to its small time constant (T, = 0.05s and TF = 0.4s). The mechanical
output power goes up as a a combination of first order and second order response.
The time constant associated with the first order response is the gas turbine constant
TCD
=
0.2s, and the time constants associated with the second order response are the
steam turbine constants Tm = 5s and T = 20s.
Figure 2-19 shows the simulation results when encountered a fuel demand decrease
at 50 seconds. Figure 2-20 is the zoom-in version of Figure 2-19. From the graphs,
we can see that the valve position and fuel flow can quickly catch up with the fuel
demand due to its small time constant (T, = 0.05s and TF = 0.4s). The mechanical
output power goes down as a a combination of first order and second order response.
The time constant associated with the first order response is the gas turbine constant
TCD
=
0.2s, and the time constants associated with the second order response are the
steam turbine constants Tm = 5s and T = 20s.
38
Valve Position/Fuel Flow
-
2.5
I
I
I
10
20
30
-- Valve Position
-Fuel
Flow
2
21.5
0
CL
> 0
40
50
Time (sec)
60
I
1
1
70
80
90
70
80
100
Mechanical Power Over Time
4
3.5 -0
Ca
-- Gen mechPower
0
20
10
30
40
60
50
90
100
Time (sec)
Figure 2-17: Generation sub-system verification results with fuel increase
Valve Position/Fuel Flow
--- Valve Position
-Fuel
Flow
I
25-
Ca
48
49
50
Time (sec)
51
52
.
-Gen
52
53
Mechanical Power Over Time
2.8-
5 2.6a S2.4c 2.2a
247
48
I
49
I
.
50
Time (sec)
51
52
mechPower
53
Figure 2-18: Generation sub-system verification results with fuel increase. The zoom-in
version for Figure 2-17.
39
Valve Position/Fuel Flow
1
2
-Valve
-Fuel
0
LL
o.5H
0
0
Position
Flow
0
CU5-0.
0
>
10
20
30
40
50
Time (sec)
60
70
80
Mechanical Power Over Time
I
I
I
I
90
100
-- Gen mechPower
1.5
0
0.5
Ca
0
-0.5
0
I
10
I
20
I
30
I
40
I
50
Time (sec)
I
60
I
70
80
90
100
Figure 2-19: Generation sub-system verification results with fuel decrease.
Valve Position/Fuel Flow
-Valve
Position
- Fuel Flow
0.5-
IL
S00
as -0.5
47
48
49
50
Time (sec)
51
52
53
Mechanical Power Over Time
-- Gen mechPower
Z51.50
CL
C:
0.5
47
48
49
50
Time (sec)
51
52
53
Figure 2-20: Generation sub-system verification results with fuel decrease. The zoom-in
version for Figure 2-19.
40
2.4
Simulation Results - Showing System Behaviors
This integrated system has three manipulated inputs: the fuel demand to both generations (Fdl, F2), and the power input to the building (Pooiing).
Scenario 1 and 2 show the system behaviors subject to input change.
Scenario 1
demonstrate the effect on grid frequnecy with generations' fuel demands change. Scenario 2 demonstrate the effect on grid frequnecy with building's power demand change.
Scenario 3 demonstrate the effect on grid frequency with different level of renewables
power penetration.
2.4.1
Scenario 1: Effect on Grid Frequency with Increasing
Fuel Demand of Generation
To study the impact of increasing fuel demand of generation on grid stability, the
power consumption of the building remains unchanged as the fuel demand increase.
In this case, each generation has increased its fuel demand by 0.2 per unit. The
mechanica1imr of ea
g
*mon
ramped up due to the increasingfuel demand as
shown in Figur2-21. Figure 2-22 is the zoom-in version of Figure 2-21, which shows
the fuel system dynamics as described in Section 2.2.1 and verified in Section 2.3.3.
The increase of power generated by generations due to raising fuel demand causes
a power imbalance within the grid, which results in the oscillations in the generator
angles and speeds (frequency). Figure 2-23 shows the results of the time domain
simulation in generator angles and speeds. Total of 40% raise in the fuel demands
results in approximately 0.5 Hz frequency oscillation.
41
Mechanical Power Over Time
2
--
Genl mechPower
--- Gen2 mechPower
1.8
1.6-
0
C.
.0 1.4
(
1.2II
0
200
400
1
1
600
Time (sec)
800
1200
1000
Valve Position/Fuel Flow Over Time
-Valve
-Fuel
Position
Flow
c 0.8-
>
0
200
400
600
Time (sec)
800
1000
1200
Figure 2-21: Scenario 1: Time domain simulation of the mechanical power of generation
for the integrated system with increasing fuel demand.
Mechanical Power Over Time
1.8
-Gent
-Gen2
1.6
mechPower
mechPower
'L 1.4
1.2
1
c
-7
0
2
3
4
5
Time (sec)
6
7
8
9
10
Valve Position/Fuel Flow Over Time
1.1
-Valve Position,
Fuel Flow
1
0
- 0.9
ILL
0.8
0
>0.7
-a
> 0
1
2
3
4
5
Time (sec)
7
8
9
10
Figure 2-22: Scenario 1: Time domain simulation of the mechanical power of generation
for the integrated system with increasing fuel demand. The zoom-in version of Figure 2-21.
42
Phase Angle Difference of Synchronous Generators Over Time
1
401
1
-- Gen1 angle
- - Gen2 angle
20
-- Lodl
angle
-o0
<-20-4011
0
60
120
180 240
300
360 420
480
540 600 660 720
Time (sec)
780
840 900
960 1020 1080 1140 1200
840 900
960 1020 1080 1140 1200
Speed of Synchronous Generators Over Time
61
60.5
IZ
Q
Ce
en
--- Gen2 speed
-Load1
speed
60
59.5-
0
Figure 2-23:
60
120
180 240
300
360 420
480
540 600 660 720
Time (sec)
780
Scenario 1: Time domain simulation of the generator (motor) angles and
speeds for the integrated system with increasing fuel demand.
2.4.2
Scenario 2: Effect on Grid Frequency with Changing in
the Building Demand
To study the impact of power demand changes, Figure 2-24 and 2-25 show the results
of the time domain simulation with changing in the building demand corresponding to
the temperature of the zones. In this case, the building air-conditioning unit is turned
on and off trying to maintain the zone temperatures around the temperature setpoint
270. When the controller is turned on, the building consumes 50 MW power. When
the controller is turned off, the building consumes 0 MW power. The performance
of the on and off controller can be seen from Figure 2-25. The controller presented
here is just used to change the consumed power in load in order to investigate the
grid response to load changes. Therefore, we are not going to talk about this control
in detail. The change in building consumed power results in the oscillations in the
generators angles and speeds as shown in Figure 2-24.
The figure shows about 50
MW changes in building power demand can result in approximately 2 Hz frequency
43
oscillations. In this case, the power generated by the generation remains unchanged.
150
6
100-
- Gen1 angle
-- Gen2 angle
Load1 angle
50-
1
1
Phase Angle Difference of Synchronous Generators Over Time
3
4
4
5
6
6
7
7
8
9
2
3
240
300
1
1
1
0)
*0
0
0)
00
-50
-00 60
120
180
360
420
480
540 600 660 720
Time (sec)
780
840
900
960 1020 1080 1140 120 0
Speed of Synchronous Generators Over Time
64
1
1
1
1
1
1
1
1
540 600 660
Time (sec)
720
780 840
900
6260
U)
60
120
180
240
300
360
420 480
960 1020 1080 1140 1200
Figure 2-24: Scenario 2: Time domain simulation of the generator (motor) angles and
speeds for the integrated system with changing in the building demand.
Temperature Over Time for bus 4 (building)
-2=2
-I
29.5
2.9
28
0.
27
-400
27.5
E26
25.
0
2W
600
400
Time (sec)
JIM
00
1000
12
Temperature Over Time for bus 4 (building)
wall I-mrture
- -Zonl
-Z-
1W
tigpeaure
29
28.5
~-Z"n2
28
cw
al emp ure
wa: tempe :a ume
E
W 275
27
26s1
0
200
400
Time (sec)
800
Ioo
1200
Figure 2-25: Scenario 2: Time domain simulation of temperature of the building for the
integrated system with changing in the building demand.
44
2.4.3
Scenario 3: Effect on Grid Frequency with Variation in
Renewables
To study the impact of renewables penetration, Scenario 3-1 to 3-4, described below,
show the variation in the power generated by the renewables at different levels and
rates. A square wave was chosen to demonstrate the extreme case of energy variation.
The time constant of the building model is approximately 2 minutes. To represent a
fast variation, the power generated by the renewables changes every 1 minutes. And
to present a small variation, the power generated by the renewables changes every 5
minutes. The net load of this system is approximately Pet = 600 MW. To represent a
large variation, the power generated by the renewables changes between +0.2 Pet and
-0.2 Pat. And to present a small variation, the power generated by the renewables
changes between +0.8 Poet and -0.8 Pet. The power generated by the dispatchable
generation and the power consumed by the building remain unchanged.
Scenario 3-1 and 3-2 investigate the effect on grid frequency with small renewable
power variation. Figure 2-26 and Figure 2-28 show the power variation profiles of the
renewables over time. Figure 2-27 and Figure 2-29 show the resulting generator angles
and speeds due to renewable power variation. We can see from both cases, about 0.8
Hz deviation of frequency occurs due to the 120 MW change of the renewable power.
45
* Scenario 3-1: Fast and small variation in the power generated by
renewables
Power Variation of Renewables Over Time
I
I
I
I
300
360
I
I
I
i
I
100
50
0
0
CL
-WOF
-100
0
60
120
180 240
420 480
540
600 660
time (s)
720
780
840
900 960 1020 1080 1140 1200
Figure 2-26: Scenario 3-1: Fast and small variation in the power generated by renewables.
Phase Angle Difterence of Synchronous Generators Over Time
0)
eI angle
Gen2 angle
-
<40
0
60
180
120
240 300
360
420 480
540 600 660
Time (sec)
720
780
840
900
960 1020 1080 1140 1200
Speed of Synchronous Generators Over Time
G1
--
60.5
NZ
a,
Gen1 speed
-Gen2
-Loadi
speed
s
60
59.5
"0
60
120
180
240 300
360
420 480
540 600 660 720
Time (sec)
780
840
Y T -F I
900
960 1020 1080 1140 1200
Figure 2-27: Scenario 3-1: Time domain simulation of the generator (motor) angles and
speeds for the integrated system with fast and small renewable power variation.
46
e
Scenario 3-2: Slow and small variation in the power generated by
renewables
1 50[
I
1
60
120
Power Variation of Renewables Over Time
1
1
1
1
1
1
I
1
1
1
1
100
50
0
a
a0
-50-
-10oo-
00
180
240
300
360 420
480 540
600
time
660 720
780
960 1020 1080 1140 1200
840 900
(s)
Figure 2-28: Scenario 3-2: Slow and small variation in the power generated by renewables.
Phase Angle Difference of Synchronous Generators Over Time
I
I
60
120
I
I
III
I
I
I
I
I
I
720
780
840
I
I
I
I
200
-
-20
-G0
180 240
300
360 420
480
540 600 660
Time (sec)
900
960 1020 1080 1140 1200
Speed of Synchronous Generators Over Time
6
a
60
Co
U)--Gn
59.5spe
speed
60
120
180 240
300
360 420
480
540 600 660
Time (sec)
720
780 840 900
960 1020 1080 1140 1200
Figure 2-29: Scenario 3-2: Time domain simulation of the generator (motor) angles and
speeds for the integrated system with slow and small variation in renewables.
47
* Scenario 3-3:
Fast and large variation in the power generated by
renewables
Power Variation of Renewables Over Time
500400300200100-
00i
ca
-100-200-300-
-500
F__-- __FN
60 120 180 240
0
E
F-
300
360 420
__
480
I
540 600
time (s)
t--[-1
660 720
780
840
900
~
-
-400-
960 1020 1080 1140 1200
Figure 2-30: Scenario 3-3: Fast and large variation in the power generated by renewables.
Phase Angle Difference of Synchronous Generators Over Time
,x 107
CD
5D
0)
(D
R,
0-5-GenI
-Gen2
-Load1
-10- 15
1
0
120
180
240
300
angle
angle
angle
360
-
420
480
LIIII
540 600 660
Time (sec)
720
780 840
900
960 1020 1080 1140 1200
900
960 1020 1080 1140 12 00
Speed of Synchronous Generators Over Time
200
I
5,
5)
0~
U)
0
- -- Gen speed
-200- -Gead~speed
-4001
0
60
120
180
240
300
360
420 480
540 600 660
Time (sec)
720
780 840
Figure 2-31: Scenario 3-3: Time domain simulation of the generator (motor) angles
and
speeds for the integrated system with fast and large variation in renewables.
48
e
Scenario 3-4: Slow and large variation in the power generated by renewables
Power Variation of Renewables Over Time
I
I
360
420
I
I
I
400300-
-
200
-
100
0
0
100k-200
-300-400
-50'
0
60
120
180
240
300
480
540
600
time (s)
660
720
780
840
900
960 1020 1080 1140 1200
Figure 2-32: Scenario 3-4: Slow and large variation in the power generated by renewables.
X 107
51
1
1
1
Phase Angle Difference of Synchronous Generators Over Time
1
1
-5-
CD
---
< -10-1
0
5
60
120
180
240
300
Gen1 angle
Gen2 angle
Load1 angle
360
420
480
600 660 720
Time (sec)
540
780 840
900
960 1020 1080 1140 1200
Speed of Synchronous Generators Over Time
40
2000-
a. -200
-
-
C/)
__.
-- Gen1 speed
-- Gen2 speed
-Loadi
speed
-400-600
0
60
120
180
240
300
360
420
480
600 660
Time (sec)
540
720
780
840
900
960 1020 1080 1140 1200
Figure 2-33: Scenario 3-4: Time domain simulation of the generator (motor) angles and
speeds for the integrated system with slow and large variation in renewables.
49
Scenario 3-3 and 3-4 investigate the effect on grid frequency with large renewable power variation. Figure 2-30 and Figure 2-32 show the power variation
profiles of the renewables over time. Figure 2-31 and Figure 2-33 show the
resulting generator angles and speeds due to renewable power variation. Under large variation of renewables, the frequency deviates significantly from the
nominal 60 Hz.
Scenario 3-1 to 3-4 demonstrate the influence of renewables penetration on the
grid stability. Small and large renewables variation results in different levels
of frequency deviation. The oscillations in frequency under small renewable
power variations are relatively small and basically symmetric about the nominal frequency (60 Hz), which are generally considered acceptable for reliable
operation.
In contrast, very different transient behavior is observed with large variations.
Under large power variation, the frequency deviates significantly from nominal
frequency (60 Hz) and settle at a higher or lower frequency.
The microgrid
system goes completely unstable due to large power imbalance within the grid.
2.4.4
Scenario 4: Basic Control of Generation and Building
under Renewable Variations
To address the existing control problem with this new integrated model, we choose the
most common control method for both generation and building. Two decentralized
controllers are presented: (1) automatic generation control for generation (2) on-off
controller + proportional controller for building air-conditioning system.
Automatic Generation Control - Turbine Governor
The turbine governor is designed to maintain the desired system frequency as the load
changes by adjusting the mechanical power output of the turbine. Figure 2-34 is the
steady-state characteristic of a turbine governor. The change in speed (frequency) is
sensed by the governor and it will act to adjust the turbine input controlled signal in
50
order to change the mechanical output power.
AI
af
lA
R
Pv~
Figure 2-34: Governor steady-state power-frequency characteristics
On-off Controller + Proportional Controller
The on-off controller is a bang-bang controller while the proportinal controller is
calculated control output proportional to the error signal, which is the difference
between the setpoint and the process variable. In this case, a hybrid controller is
implemented. The temperature setpoint is set to be 270. Therefore, the error signal
for proportional controller is the temperature difference between the current zone
temperature and the temperature setpoint. Figure 2-35 shows the complete algorithm
for this hybrid controller.
If Pm.
-=
T....
0 thim
> Ta..,.g+1
thu
=-k
- Twa
P..Ih, =
(T'-
sog
T0 ..
eke
I T,.,.. < T&... - 1 thu
Pm"UM. = 0
elm
T... -T..)
P.... =~
-
Mnd V
Figure 2-35: On-off + proportional controller algorithm
The reason for this, kind of hybrid controller is becaase the cmrrent building airconditioning systems are generally equipped with variable speed fan that allows the
51
power input be adjusted with respect to temperature measurements.
In this case, the generator control and building control are completely decoupled.
Figure 2-37, 2-38, 2-39 show the time domain simulation results of the basic control.
Figure 2-36 shows the control actions by the AGC and the building controller. The
building controller adjust the input power to the building depends on the temperature
difference. However, the performance is not ideal due to the on and off behaviors.
The temperature fluctuates within a temperature range of [260 29']. On the other
hand, the AGC controller tends to adjust the fuel demand to reduce the frequency
deviation.
Overall, the intermittency of renewable energy, and the power demand needed to
maintain the building within a comfort zone contribute to a large fluctuation of the
net load, which makes the system frequency deviates away from its nominal value.
With the basic control, the system is not completely loss it stability, where the system
still recovers and tends to go back to nominal 60 Hz, but the peak deviation is too
large and unacceptable.
Power Demand of Building Over Time
0.5
0.40.3
00.2EL
0.1-
0
60
120
180 240
300 360
420
480
540 600 660
Time (sec)
720 780
840
900 960 1020 1080 1140 1200
780 840
-Gent
Fuel Demand
Gen2 Fuel Demand
900 960 1020 1080 1140 1200
Fuel Demand to Generations Over Time
1.5
a'0.5
LL0
60
120
180 240
300
360 420 480
540 600 660
Time (sec)
720
Figure 2-36: Scenario 4: Control commands for the generations and building with basic
control.
52
Phase Angle Difference of Synchronous Generators Over Time
100
I
I
I
I
I
I
I
I
720
780
840
900
I
50W
12
-50
-150
0
60
120
180 240 300
"
GelangleW
-Gen2 angle~
-Load
angle
-100
360
420 480
540 600 660
Time (sec)
960 1020 1080 1140 1200
-
Speed of Synchronous Generators Over Time
63
62-
r61
60
Wl 59
58
571
0
Figure 2-37:
540 600 660
Time (sec)
720
Scenario 4: Time domain simulation of the generator (motor) angles and
speeds for the integrated system with basic control.
-Gent
-Gen2
Mechanical Power Over Time
2.5r
mechPower
mechPower
a
1.5
-
a) 0.5
0
200
400
600
Time (sec)
800
1000
Valve Position/Fuel Flow Over Time
1.5
- Gent Valve Position
-Gent
Fuel Flow
- Gen2 Valve Position
- Gen2 Fuel Flow
F--
IL
L---F
0.5
0
0
-J
U
200
400
600
Time (sec)
800
1000
1200
Figure 2-38: Scenario 4: Time domain simulation of the mechanical power of the generation
for the integrated system with basic control.
53
Temperature Over Time for bus 4 (building)
95
28.5C2 28
27 --
26.5202
4.
100
Time (sec)
-
Zone1 wall
---
o
-Z0n-
Temperature
Over Time for bus
4 (building)
11,Il *eln
-~~
ZOnM2 wed:
-
29,5 -
--
2.5
wa
Z-n2
temperature
wall tmperaturm
tmp-ratur.
mperau
floor tempqralure
Summary
Through the modeling and simulation of the integrated system, we could find that
the fluctuation with generation, renewables and load influence the system frequency
greatly. And basic uncoordinated controls of generation and building are not sufficient
to maintain the system frequency within an acceptable range.
In the meantime,
the building temperature performance is not ideal either. Therefore, there's a need
to provide a more generalized control method that can take the characteristic of
generation and building into account for this new introduced system model.
54
Chapter 3
Model Predictive Control for
Integrated Microgrid System
3.1
Introduction
A more generalized control method is now needed to control the integrated system
and taking into account the physical characteristics of the generation and building.
In this Chapter, a model predictive controller is presented and evaluate the system
behavior from a linearized model of the integrated system. The simulation results
show that the controlled system exhibits a satisfactory performance without violating
the system's physical constraints.
3.1.1
Background
Model Predictive Control (MPC) is an advanced control method thakhms been used
in several industries over several decades. In recent years, MPC has been used for
power system models. MPC is a look-ahead method to optimize system outputs based
on the knowledge of system behavior. It relies on a descriptive model of the system
to predict the system's future behavior in order to compute the control commands
by optimizing an objective function over a finite receding horizon. The advantage
of the MPC method is that it can easily be incorporated in the control of MIMO
55
(multi-input, multi-output) systems. Additionally, MPC is capable of setting constraints on the inputs and outputs of the system in the calculation of future control
actions. This is why MPC was chosen for this work. Due to the physical limitations
on generation and demand, the ability to compute control actions without violating
system's constraints and maintain proper system operation is very important.
There are two main breakthroughs in our work:
I. Generally, most of the MPC applications in power system or energy field are based
upon economic dispatch. The objective of an economic MPC is to minimize the cost
function comprising the total electricity cost of the system [23 [7]. In our work,
the performance index involves physical variables in order to address the imbalance
between generation and demand.
II. Some of the literature shows model predictive control on trajectory deviation on
certain variables such as voltage or frequency [13] [14]. However, these studies used
mostly grid models that consider the generation and load as simple power injections.
The mathematical model developed and used in this thesis to predict the microgrid
system behavior under MPC is a new integrated model. As indicated in the previous
chapter, namely section 2.2, this model includes the thermodynamic characteristics
of generation and demand as well as the grid topology.
56
3.2
3.2.1
Design
MPC Setup
The basic principle of MPC is graphically depicted in Figure 3-1, inspired by [16].
past
future
I
4d
I
1
Trajectory
-Reference
--
Predicted future output
Past output
UA n
_J7
Optimal future input trajectory (time k)
Past input
- ---
Optimal future input trajpctory (time k+1)
k
k+I
Control horizon (p)
Prediction horizon (m)
k p
k+Fm
JI
Figure 3-1: Principle of MPC
At every time step t-k, the MPC solves an optimization problem over a finite prediction horizon [k, k+m] with respect to an objective function to ensure that the
predicted outputs can stay as closer to the reference trajectory. The control action
is computed over a control horizon [k, k+p], where p<m. However, only the first
control solution is implemented for each time step. The next control solution comes
from redoing the whole optimization over the next finite time window [k+1, k+m+11.
The optimization problem is solved by minimizing an objective function with the
general form as shown in Equation (3.1).
Ny
J
Nu
Wy(yi - ri) 2
=
i=1
+ Zwui AUi 2
=
57
(3.1)
The first term in Equation (3.1) represents the difference between expected output and
output reference. The second represents the control signal change. The parameters
wyi and w,,i are weights. This representation also assumes that there are N, outputs
and N. inputs.
System Linearization
Our nonlinear multi-input multi-output (MIMO) model developed in Section 2.2 was
built in MATLAB environments. However, the Model Predictive Control Toolbox
provides a control function available in MATLAB that is limited to linear system
models. In order to use the MPC, a linear approximation model of our nonlinear
system is generated around some operating points.
The non-linear differential and algebraic equations of each individual subsystem model
of our system can be put together to form an overall system-integrated model. The
general algebraic-differential equations of the integrated system model are shown
below:
Eb f(x, z, u)
0
g(x, z, u)
where x represents the system state variables, z represents the algebraic variables
that typically appear in the algebraic equations associated with the model and the
power flow equations The variable u designates the control inputs.
The function
f is the
nonlinear function that represents the system differential equa-
tions, and g is the function that represents the system algebraic equations.
The differential equations of our integrated system models are described in Section
2.2 and the algebraic equations of the system are stated as follows
58
For each bus i:
n
o = -Pe + VsG + E
VVj[Giy cos(Oi - 03) + Bij sin( 2 - Oj)I
7=1
o=
-Qe
-VsB
VV[Gi 3 sin(i - 0,) - Bi3 cos( 2 - 8,)]
+
j=1,joi
For each generator i:
0 = Eisetpojft -- Ej
-
V E.
Xdi
sin(i - bi)
V2
V E-
0 =-Qei --+
cos(
Xdi
Xdi
2 -i6
)
0 =-R
where
0
bus voltage angle
V
bus voltage amplitude
j
generator voltage angle
E
generator voltage magnitude
Xd
generator synchronous reactance
The state matrix A, of the linearized system is obtained by eliminating the algebraic
variables.
fA
f g;'
A(.
-9
gz
AzJ
-
0 J
A, =Af - fzgz 9M
59
(3.2)
This is assuming that g- is nonsingular.
From Equation (3.2), we can get a linearized system:
b= A~x+ Bu
Table 3.1 shows the state, intermediate algebraic and the input variables of our system. The linearization of the nonlinear model was calculated at every time step and
fed into the MPC. The system is only linearized around one operating point for coding simplification. The system studied in this thesis is subject to small disturbances.
The operating point was chosen to be the steady state conditions listed in Table A.2.
And the MATLAB code for deriving the linearized model is' in Appendix C.
Table 3.1: x, z and u for our integrated system
x
Vp 1 Wf 1 Pmg1 Pms1 Rms1 61 w 1 Vp 2 Wf 2 PMg 2 PMs 2 Rms 2 62 w 2 63 w 3
Tz1 Twn Tw 12 Tw 13 Tw1 4 Tc1 Tf1 Tz 2 Tw 2 1 Tw 2 2 Tw 23 Tw 2 4 Tc 2 Tf 2
V1 V2 V3 V4 V5 V6 E1 Pe1 Qe1 E2 Pe2 Qe2 E3 Pe3 Qe3
z
01 02 03 04 05 06
u
Fd1 , Fd2 , Pcoouing
Simulations were conducted to verify the validity of the linear model. The nonlinear
and linear model were both simulated and the results are shown in Figure 3-2 - 3-5.
From these four figures, we can see that the linearized model gets the same resutls as
the nonlinear model except for a little offset in the zone temperature. But the offset
is approximately 0.05', which is acceptable for the MPC design.
60
Ph2s
9
Angle Di 6n66
of Synchron6us G0n66
Sp..d
aWM
6 vWr Tim
-0G62 2629 (6006620)
-L06269W62.6
60,8
.Synro Gou
Gnraor 0-
Tim
--
-
L6ad66.p2(
n-a
-
6084
606
10
ql n i w~G
1604
$9.659.4-
-%
90
0
260
66 (0
0
620
Iwo0
2
40
SW
T66
Tim. (-2)
100
120
Figure 3-2: Time domain simulation of theFigure 3-3: Time domain simulation of the
generators angles for both nonlinear andgenerator speeds for both nonlinear and
linear model.
linear model.
i
L55
Mechacal Powe
0e
Tlmpeatue
Tim
-h
G.4
-
Ome Timne 1wr bu
4(building
1:-
1.35-
12
1.15-
20
40
260
Bo0
1000
1200
T6m6
60
)
0
Figure 3-4: Time domain simulation of theFigure 3-5: Time domain simulation of the
mechanical power of generation for bothzone temperatures for both nonlinear and
nonlinear and linear model.
linear model.
Input-Output Choices
The system's control inputs are limited to three controllable attributes of the system:
the fuel demand of both generators (ui
= Fdl, U 2 = Fd2),
and the power input of
the building air-conditioning unit (U 3 = Pcooling). The outputs are chosen to be the
power difference between generation and demand (yi = Pmi + Pm2
the temperature of both rooms in the building
(y2
-
Pcooling), and
= Ti and Y2 = Tz 2 ). Because this
choice of outputs can ensure the control system to maintain the power balance and
building temperature within an acceptable range.
61
Therefore, the objective function in Equation (3.1) becomes,
J
= Wy1(Y1 - r1 ) 2
+ w, 2 (y 2 - r2 ) 2 + wy 3 (y3 - r3 ) 2 + WU1AU1
2
+ Wu 2 AU 2 2 + W 3 AU 3 2
where r1 is the power difference reference signal, r2 and r3 are the zone temperature
reference signal. Aui represents the control signal difference with control input in
steady state.
MPC Structure
Figure 3-6 shows the structure of the MPC setup for the integrated system. At every
time t=k, MATLAB runs the nonlinear integrated system model first over the prediction horizon to get the continuous linearized system model. And then create an MPC
based on the linearized model. The Model Predictive Control Toolbox discretizes a
continuous-time plant with sampling time ts automatically. The MPC controller will
calculate the future control actions upon a weighted objective function. The objective function has two parts: one to minimize the power imbalance between generation
and demand, the other one minimizes the temperature difference of the room temperatures with the desired temperature. The weights on the power balance in the
performance index are set to some higher values than those associated with the zone
temperatures. This will make the power imbalance small and consequently eliminates
the oscillations in the grid frequency. The lower weights on the temperatures may
result in some error in temperature tracking. However, the zone temperatures can be
allowed to vary from their references and still have a comfortable environment.
62
MPC
Predic wd
Outpu
Future
Control Inputs
Fu Ure
En
SySfteM
Reference
Trajectory
System
outputs
Objective Constraints
Function
Figure 3-6: Structure of the MPC
The main purpose of this MPC control is to show if we control the imbalance first
with respect to the thermodynamic effects on both the generation and demand. Even
thought the frequency of the system is not predicted and controlled by the MPC
controller, the frequency oscillation of the grid can still be significantly reduced.
Tunning
The MATLAB Model Predictive Control Toolbox has a limitation on the model.
Namely, the direct feed-forward from the control inputs to any output is not permitted. In our system, the first output that represents the power imbalance is a function
of the input power. Therefore, a first order behavior, with a small time constant
(T), is added as shown in Equation (3.3). Tp is chosen as approximately 100 times
smaller than the smallest time constant in system model. The actual power input to
the building (Pcooing) becomes a state and the power demand becomes an input (Pd).
.1
Pcooing
=
(Pd -
63
Pcooling)
(3.3)
Table 3.2: x, y and u for our integrated system modified for MPC
x
y
Vp 1 Wf1 Pmg1 Pms1 Rms1 61 w1 Vp 2 Wf 2 PMg 2 PMs2 Rms 2 62 W2 63 w 3
Tz 1 Twnl Tw1 2 Tw 13 Tw1 4 Tc 1 Tf1 Tz 2 Tw 2 1 Tw 22 Tw 23 Tw 24 Tc 2 Tf 2 PCOOun
(P1 + Pm2 -Pd) Tz Tz 2
u
Fd1 Fd2 Pd Tamb
where
Pd
power demand for the building
T,
time constant = 0.0001s
In order to take into account the influence of the ambient temperature, we also add one
more input as a measured disturbance Tamb. Then the system becomes a model with
31 states, 4 inputs and 3 outputs (2 manipulated inputs and 1 measured disturbance)
shown in Table 3.2 The simulation sampling time is set to be 30 seconds. This was
the smallest sampling possible due to computer limitations. Although this control
sampling rate is a little larger than the time constant of the generation, but it is
proved to be sufficient for the simulation results.
Each simulation iteration has a unique linear plant for the MPC. This linear plant
is calculated from the nonlinear model in real-time.
And the reference value for
the system is also updated. Both the inputs and outputs are constrained, and with
different weights in the objective function. The fuel demand inputs (Fdl, F2) are
constrained to [0 2] (per unit) and the power demand input (Pd) is constrained to
[0 1] (MW). The temperatures of the building are set to be positive and not greater
than Tamb.
The weight structure for inputs is set to [0 0 01, since we are not concerned about
the control cost in this case. The weight structure for outputs is set to [3 1 1], which
emphasizes the weight for the first output term (the power imbalance of the system).
For the prediction horizon and control horizon, several different values are tested with
simulation. Tests indicated that the prediction horizon cannot be too long and the
64
control horizon cannot exceed a portion of the prediction horizon in order to avoid
some numeric issues during calculation. Therefore, the final value of the prediction
horizon is 60 sampling time and the control horizon is 5 sampling time.
Details of the controller in MATLAB code can be found in Appendix D. The next
section shows the simulation results for the MPC.
3.3
Simulation Results and Discussion
The MPC is implemented and simulated on the 6-bus system and under the same
renewables power variation as described in Section 2.4.4. The simulation results show
the system behavior under MPC. The controller tends to maintain the building room
temperature around the temperature-setpoint (Tdesired = 270) while ensuring the
minimum power difference between generation and demand at the meantime.
Control Commands Calculated from the MPC.
- -- Fuel Demand for Generation 1
200
0.5
H
400
_j
__j
j
L'
600
time (s)
L_j
-L
L
L-
L
800
--L__
_
F
1000
'
H
%
__
_
L 1L--
1200
Control Commands Calculated from the MPC.
0.20
-Power
Input for Building AC Systeml
-
0.26
-
0.24
0.22 -0.2 --.10200
400
600
(s)
time
800
1000
Figure 3-7: The control commands calculated from the MPC.
65
1200
Phase Angle Difference of Synchronous Generators Over Time
40
Q
-
1
1
20 -
.h..
.....
1
....
IV
-20
-40
60
120
180
240
300
360
I
I
420
480
540 600 660
Time (sec)
720 780
840
900
960 1020 1080 1140 1200
Speed of Synchronous Generators Over Time
61
60.5-
I
I
I
I
I
I
I1
.. k .JL I
Gent speed
-- Gen2 speed
60
CL)
59 .5
-
)
59
60
120
180
240
300
360
420
480
540 600 660
Time (sec)
720 780
840
900
960 1020 1080 1140 1200
Figure 3-8: Time domain simulation of the generator (motor) angles and speeds for the
integrated system under MPC.
Mechanical Power Over Time
2 -Gn
ec
w
-
1.5
0.5
0
200
400
600
Time (sec)
800
1000
1200
Fuel Flow Over Time
1.5-
-Gent
-Gen1
L[
L
1
--
Gen2
fuelDemand
fuelFlow
fuelDemand
Gen2 fuelFlow
0L
0 .57-rLL
0
200
400
600
Time (sec)
800
1000
1200
Figure 3-9: Time domain simulation of the mechanical power of generation for the integrated system under MPC.
66
Temperature Over Time for bus 4 (building)
30.5
30
29.5
28.5
C.
28
E
a2 27.5
27
26.5
400
Soo
Time (sec)
100O
Temperature Over Time for bus 4 (building)
all
=3
1
---
---Zon
wdB
Zo
-
-
29
'mperature
iall
Z,0110 6.p0811
2 wag18
315p*12301
-
a)
Q.
E
.mp.rr
wall b.mperatur.
tomp.rure
Z o allIigtmperatur
Z"
temp ratur.
Zone
29,5
Z"o2 .Oog 30sp-t-o
o-Z021
or30,0p53030
28.5 -
-
28
0
2W
4W
No
Time (sec)
I0m
1200
Figure 3-10: Time domain simulation of temperature of the building for the integrated
system under MPC.
Power Difference Over Time
I,
--
-
--
Power Difference Output
7 - Reference Power Difference
--
.-
~l
.~
3
1.5
,
-
I
,
II
2
I
1
I
0
0.5
I
-
COL
I
I
I
I
II
I
0
I
200
400
600
Time (sec)
800
1000
1200
Figure 3-11: The power difference over time
The control solutions optimized by the MPC are shown in Figure 3-7. And Figure 3-9
shows the mechanical power generated by the generators due to the control inputs.
67
From 3-10, we can see that the MPC finds a perfect power input to the building
that could maintain the temperature around 28
0
without fluctuation. Although the
room temperatures are not exactly at the desired temperature 270, the performance
is acceptable for a building climate control since generally people can withstand the
temperature changes within a certain range.
From Figure 3-11, we can see that the power difference (yi) is tracking the reference
value r, as expected. The reference signal is computed by the grid loss with renewables
forecast information. The reference power difference is fed into the MPC to minimize
the power imbalance in the microgrid. Figure 3-8 shows the time simulation results
of generator angles and speeds (frequency), the grid frequency is reduced to
0.5 Hz
oscillation, which is considered acceptable (< 0.2% of nominal 60 Hz = 1.2 Hz). In
the MPC setup, we didn't provide the MPC controller with any information about the
grid frequency. However, by predicting the power imbalance from the system model,
we can reduce the oscillation in the grid frequency successfully. The results proved
that we could achieve better grid stability from minimizing the power imbalance based
on the thermal characteristics of the generation and building.
68
Chapter 4
Conclusions and Recommendations
4.1
Summary
This thesis has introduced a new physical model for microgrid systems. The model
integrates detailed dynamics of each sub-system in the microgrid. The simulation
results of the integrated system demonstrate the effects of the coupled thermodynamic
and electromechanical behaviors. This choice of system boundary not only enhances
the overall knowledge of microgrid behaviors but also provides a more realistic model
for understanding control performance.
The second part of this thesis presents the design and implementation of a Model
Predictive Controller (MPC) along with its performance evaluation.
A linearized
model is generated from the original nonlinear system. This linear model is used in
the desigi and testing of the MPC. The model integrates the thermodynamic model
of generation and demand within the overall system model, and allows the MPC to
use the information from both sides to optimize the control actions while minimizing
power imbalance. The simulation results show that the model predictive controller
successfully reduces the grid frequency deviation and at the same time maintains the
building temperature within acceptable limits in the face of variations in renewables.
69
4.2
Recommendations
There are two main ways to build upon this thesis work. First, the modeling concept
can be used to study different sub-systems that are connected to a microgrid. Specifically, physical models of the combined cycle power plant and building air-conditioning
sub-systems were introduced to help in understanding the behavior and effects of generation and demand. One can incorporate other physical dynamic system models into
this microgrid system to study grid dynamic and control performance issues. Furthermore, this thesis focused only on the frequency stability issues concerning this
microgrid model. The voltage limitations and other constraints have yet to be investigated.
One approach among others to demonstrate the control formulation and performance
evaluation of such an integrated microgrid system. Other control method could be
attempted specifically ones that can handle constrained optimization requirements
for nonlinear physical systems with large dynamic and disturbance effects.
70
Appendix A
Tables
71
Table A.1: Notation for the zone model
Tzi
C_,
Tij
A* y
h,, 3
Tei
Aci
hci
Tf
Apj
hf
the
the
the
the
the
the
the
the
the
the
the
Tamb
the ambient temperature
Ta
C,
the temperature of supply air
specific heat of air
mass flow rate of supply air to each zone i
maai
temperature of each zone i
capacitance of each zone i
temperature of four walls of each zone i, j = 1-4
area of four walls of each zone i, j = 1-4
heat transfer coefficient of four walls, j = 1-4
temperature of the ceiling of each zone i
area of the ceiling of each zone i
heat transfer coefficient of the ceiling of each zone i
temperature of the floor of each zone i
area of the floor of each zone i
heat transfer coefficient of the floor
hoa
mass flow rate of outside air
mrai
mass flow rate of return air from each zone i
72
Table A.2: Steady state value of the integrated model
V1
Wf1
Pmgi
Pms1
Rms1
0.6363
0.6363
0.5268
0.5268
0
61
0
Wi
V,2
62
60
0.8493
0.8493
0.7500
0.7500
0
5.7124
w2
60
63
w3
Tz 1
Twil
-26.9004
60
23.7478
26.8739
26.8739
26.8739
24.0226
26.8739
26.8739
24.2974
27.1487
27 1487
27.1487
24.0226
27.1487
27.1487
Wf
2
Pmg2
Pms 2
Rm8 2
Tw1 2
Tw13
Tw14
Tc,
Tfi
Tz 2
Twai
Tw 22
Tw 23
Tw2 4
Tc2
Tf 2
73
74
Appendix B
MATLAB Code
-
Steady State Value
of the System
75
1
clear all
2
close all
3
clC
4
5
Pcooling = 0.5;
6
Pfan = 0.5;
7
Tamb = 30;
8
9 syms xl x2 x3 x4 x5 x6 x7 x8 x9 x10 xll x12 x13 x14
10
11
Ni = 2;
12
Nj = 4;
13
Ns
14
hw = zeros (Ni,Nj);
15
Aw = zeros(Ni,Nj);
16
Cw = zeros (Ni,Nj);
17
hc = zeros(Ni,l);
18
hf = zeros(Ni,l);
19
Ac = zeros (Ni,l);
20
Af = zeros (Ni,l);
21
Cz = zeros(Ni,l);
22
Cc'= zeros (Ni, 1);
23
Cf = zeros(Ni,l);
24
Cp = 1.005;
7;
=
25
26
Msa = 0.5;
27
msa = zeros (Ni,1);
28
msa(l)
= Msa*Pfan;
29
msa(2)
= Msa*(l-Pfan);
30
Moa
31
Mra = Msa-Moa;
32
mra = zeros (Ni,1);
33
mra(l) = Mra*0.5;
34
mra(2) = Mra*0.5;
35
COP
=
=
0.1;
2.5;
76
Tsa = -COP /Msa/Cp*Pcooling
+ (Moa*Tamb+mra (1) *xl+mra (2) *x8) /Msa;
for j=1:Nj
hw (1,
j)
=
0. 02;
Aw (1,
j)
=
1;
Cw (1,
j)
=
5;
0.02;
end
for j=1:Nj
hw (2,
j)
=
Cw (2,
j)
= 5;
end
Aw (2, 1)
= 2;
Aw(2,2)
= 1;
Aw(2,3)
= 2;
Aw(2,4)
= 1;
Cz(1)
= 2.5;
Cc (1)
= 5;
Cf (1)
= 5;
hc(1)
= 0.01;
Ac (1)
= 1;
hf(1)
= 0.01;
Af (1)
= 1;
Cz(2) = 2.5;
Cc (2)
= 5;
Cf (2)
= 5;
hc(2) = 0.01;
Ac (2)
= 2;
hf(2)
= 0.01;
Af (2)
= 2;
i=1;
fl = 0 == 1/Cz(i)*(msa(i)*Cp*Tsa-(msa(i)*Cp+hw(i,1)*Aw(i,1)+...
hw(i, 2) *Aw(i, 2) +hw(i, 3) *Aw(i, 3) +hw (i, 4) *Aw(i, 4) +hc (i) *Ac (i) +..
77
72
hf (i) *Af(i))*xl+hw(i,1)*Aw(i,1)*x2+hw(i,2)*Aw(i,2)*x3+...
73
hw (i,
74
hf (i)*Af (i)*x7);
75
f2 = 0 == 1/Cw(i,1)*(hw(i, 1)*Aw(i,1)*xl-2*hw(i,1)*Aw(i,1)*x2+...
76
hw (i,
77
f3 = 0 == 1/Cw(i,2)*(hw(i, 2)*Aw(i,2)*xl-2*hw(i,2)*Aw(i,2)*x3+...
78
hw(i, 2) *Aw(i, 2) *Tamb);
79
f4 = 0 == 1/Cw(i,3)*(hw(i, 3)*Aw(i,3)*xl-2*hw(i,3)*Aw(i,3)*x4+...
80
hw(i, 3) *Aw(i, 3) *Tamb);
81
f5 = 0 == 1/Cw(i,4)*(hw(i, 4)*Aw(i,4)*xl-2*hw(i,4)*Aw(i,4)*x5+...
82
hw (i,
83
f6 = 0
==
1/Cc(i)*(hc(i)*Ac(i)*x1-2*hc(i)*Ac(i)*x6+hc(i)*Ac(i)*T amb);
84
ff7 = 0
==
1/Cf(i)*(hf(i)*Af (i)*x1-2*hf (i)*Af (i)*x7+hf (i)*Af (i)*Tamb);
85
i=2;
86
f8 = 0
3)*Aw(i,3)*x4+hw(i,4)*Aw(i,4)*x5+hc(i)*Ac(i)*x6+...
1) *Aw(i,1)*Tamb);
4) *Aw(i, 4) *x8)
1/Cz(i)*(msa(i)*Cp*Tsa-(msa(i)*Cp+hw(i,1)*Aw(i,1)+...
87 hw(i,2)*Aw(i,2)+hw(i,3) *Aw(i,3)+hw(i,4)*Aw(i,4)+hc(i)*Ac(i)+.
88
hf(i)*Af(i))*x8+hw(i,1)*Aw(i,1)*x9+hw(i,2)*Aw(i,2)*xlO+...
89
hw(i,3)*Aw(i,3)*xll+hw(i,4)*Aw(i,4)*x12+hc(i)*Ac(i)*x13+...
90
hff(i)*Aff(i)*x14);
91
f9
92
hw(i,1)*Aw(i,1)*Tamb);
93
fl0 = 0 == 1/Cw(i,2)*(hw(i,2)*Aw(i,2)*x8-2*hw(i,2)*Aw(i,2)*xlO+.
94
hw(i,2)*Aw(i,2)*Tamb);
95
fl1
96
hw(i,3)*Aw(i,3)*Tamb);
97
f12 = 0 == 1/Cw(i,4)*(hw(i,4)*Aw(i,4)*x8-2*hw(i,4)*Aw(i,4)*x12+.
98
hw(i, 4) *Aw(i, 4) *x1)
99
f13 =
0 == 1/Cw(i,1)*(hw(i,1)*Aw(i,1)*x8-2*hw(i,1)*Aw(i,1)*x9+...
=
= 0 == 1/Cw(i,3)*(hw(i,3)*Aw(i,3)*x8-2*hw(i,3)*Aw(i,3)*x11+.
0
==
...
1/Cc(i)*(hc(i)*Ac(i)*x8-2*hc(i)*Ac(i)*x13+hc(i)*Ac(i)*Tamb);
1o
f14 =
0
==
...
1/Cf(i)*(hf(i)*Af(i)*x8-2*hf(i)*Af(i)*x14+hf(i)*Af(i)*Tamb);
101
102
103
[xl,
x2,
x3,
x4,
x5,
x6,
x7,
solve(fl,,f2,ff3,ff4,f5,ff6,ff7,
104
x8,
x9,
xlO,
x1l,
x12,
x13,
x14]
...
f8,f9, flO,ffll,ff12, fl3,ffl4,xl,x2,x3,...
x4,x5,x6,x7,x8,x9,xlO,xli,x12,x13,x14);
78
105
X-steady =
[xl, x2,
x3,
x4, x5,
x6,
x7,
x8,
x13, x14];
106
save ( 'steadyValueBuilding.mat', 'X_steady');
107
108
syms Fd
109
500;
110
Wn
ill
Wfn = 10;
112
W
113
Wf
=
Fd*Wfn;
115
Ti
=
298;
116
PR
=
15.4;
117
rc
=
1.4;
118
rh
=
1.33;
119
Ec
=
0.86;
120
Et
=
0.89;
121
Ecomb = 0.99;
122
Cpc = 1.005;
123
Cph = 1.157;
124
LHV = 47141;
=
1*Wn;
=
114
125
126
Xc =
127
Td = Ti*(1+(Xc-1) /Ec);
128
Tf = Td + (Ecomb*LHV)/Cph*Wf/(Wf+W);
129
Xh =
130
Te = Tf*(1-(1-(1/Xh))*Et);
131
Pg =
(PR*W/Wn)^((rc-1)/rc);
(PR* (Wf+W)/(Wfn+Wn) )^((rh-1)/rh);
(W+Wf) *Cph* (Tf-Te) -W*Cpc* (Td-Ti);
132
133
disp ('Solving');
134
[F&c.genl]
- solve(Pg
==
1.0536, Fd);
135
[Fdgen2]
= solve(Pg
==
1.5000, Fd);
136
Fdgen =
137
save
('
[Fdgenl Fdgen2];
steadyValueGeneration.mat', 'Fdgen')
79
x9,
x10,
x1l,
x12,
80
Appendix C
MATLAB Code - Model Linearization
81
1 function Amatrix =
myEvents,
calculateAmatrix(myPowerDGrid, myPowerGrid,
...
spFlag, x0)
2
f = myPowerDGrid.frequency;
3
Y = myPowerGrid.YBus;
4
5 G = real(Y);
B = imag(Y);
6
E = myEvents(spFlag).eGensMag;
7
V = myEvents(spFlag).eLoadsMag;
8
Xd =
(0.20 0.20 0.25);
9 theta = myEvents(spFlag).eLoadsAng;
10
theta = theta*pi/180;
%degree ->
%degree
rad
11
12
%review where it can get the number of total nodes
13
n=6;
14
%review if dyns is the number of generators + other dynamic devices
15
dyns=length(myPowerDGrid.dynDev.dynDevNum);
16
generators
17
buildings
18
%review if this is the order of each dynamic device
19
orders=3;
20
for i=l:dyns
21
H(i)
22
D(i)
23
M(i)
24
end
25
A=theta;
=
1;
=
=
2;
myPowerDGrid.dynDev.DMData(i).h;
myPowerDGrid.dynDev.DMData(i) .d;
=
2*H(i);
26
27
Tv = 0.05;
28
Tf = 0.4;
29
Tcd = 0.2;
3o
Tm = 5;
31
Tb
32
f = 60;
33
H =
[5 4 5];
34
D =
(0.001 0.001 0.001];
=
20;
82
35
M
36
Tp
2*D;
=
0.0001;
=
37
38
Ni =
2;
39
Nj
=
4;
40
hw
=
zeros(Ni,Nj);
41
Aw
=
zeros (Ni, Nj);
42
Cw
=
zeros (Ni,Nj);
43
Cp
=
1.005;
44
for j=1:Nj
0.02;
45
hw(1,j)
46
Aw(1,j) = 1;
47
Cw(1,j)
48
end
49
for j=1:Nj
=
= 5;
0.02;
50
hw(2,j)
=
51
Cw(2,j)
= 5;
52
end
53
Aw(2,1) = 2;
54
Aw(2,2) = 1;
5s
Aw(2,3) = 2;
56
Aw(2,4) = 1;
57
Czl = 2.5; Cz2 = 2.5;
58
Cfl = 5;
59
hcl = 0.01; hfl = 0.01; hc2 = 0.01;
60
Acl = 1; Afl = 1; Ac2
61
Msa
62
Pfan
=
0.5;
63
msal
=
Msa*Pfan;
64
msa2
=
Msa*(1-Pfan);
65
Moa =
66
Mra = Msa-Moa;
67
mral = Mra*0.5;
68
mra2 = Mra*0.5;
69
COP = 2.5;
70
Aone = 1/Czl*(msal*Cp/Msa*mral-(msal*Cp+hw(1,1)*Aw(1,1)+...
Ccl = 5; Cf2
=
=
5;
Cc2
2; Af2
=
=
5;
hf2
=
0.01;
2;
0.5;
=
0.1;
83
71
hw (1, 2) *Aw (1, 2) +hw (1, 3) *Aw (1, 3) +hw (1, 4) *Aw (1, 4) +.
72
hcl*Acl+hfl*Afl));
73
Bone = 1/Czl*(msal*Cp/Msa*mra2);
74
Atwo = 1/Cz2* (msa2*Cp/Msa*mra2- (msa2*Cp+hw (2, 1) *Aw (2, 1) +...
75
hw (2, 2) *Aw (2, 2) +hw (2, 3) *Aw (2, 3) +hw (2, 4) *Aw (2, 4) +.
76
hc2*Ac2+hf2*Af2));
77
Btwo = 1/Cz2*(msa2*Cp/Msa*mral);
78
79
80
Fx
Fx =
[-1/Tv
0
0
0
0
0 0 ...
0
0
0
0 ...
0
0
0
0 0
0...
0
0 ...
0
0 ...
0
0 ...
0
0
0 ...
0
0 ...
0
0;
0
81
0
0
0
0
-1/Tf
1/Tf
0
0
0
0 0
0 0
0
0
0
0 ...
0
0
0
0...
0
0...
0
0
0...
0
0...
0
0;
0
82
0
0
0
0
-1/Tcd
1/Tcd
0
0
0
0
0
0 0...
0 0
0
0 ...
0
0...
0
0...
0
0
0
84
0...
0 ...
0
0
0
0
83
0
0
0
0
0;
0
0
:1.
0
0
0
0
0 0
0
0 0 ...
0
0
...
0
0
0 ...
0
0
0
0
0 ...
0
0
0
0
84
0
1/(Tm*Tb)
0
0;
0
0
-1/ (Tm*Tb)
0
0
-(Tm+Tb)/(Tm*Tb)
0
0
0 0
0
0 0 ...
0 ...
0
0 ...
0
0
0
0...
0
0
0 ...
0
0
0
85
0
0
0
0;
0
1
0
0
0
0
0
0
0
0 0
0
0 0
0 ...
0
0 ...
0
0 ...
0
0
0 ...
0
0 ...
0
0 ...
0
0
86
0
0
0
0
0;
pi*ff/H (1)
-D(1)*pi*f/H(1)
pi*f/H(1)
0
0
0 0
0
85
0
0
0 0 ...
00 ...
0
0...
0
0
0
0...
0...
0
0...
0
0
0
87
0
0
0
0...
0;
0
0
0
0
0 0
0
0
0
0
-1/Tv
0
0 0
0...
0
0
0
0...
0
0
0
0
0...
0
0
88
0
0
0
0
0;
0
0
0
0
1/Tf
0
0
-1/Tf
0 0
0
0 0...
0 ...
0
0
0...
0
0
0
0
0
0
0...
0
0..
0
0;
0
0
0
0...
0
0
89
0
0
0
1/Tcd
0 0
0
0
-1/Tcd
0 0...
0
0
0
0
0
0...
0
0...
0
0..
0
0
86
0
90
0
0
0
0;
0
0
0
0
0
0
0
0 0...
0 0
1
0 ...
0
0
0...
0
0 ...
0
0
0
0
0
0...
0
0 ...
0
0...
0
91
0;
0
0
0
0
0
-1/ (Tm*Tb)
1/ (Tm*Tb)
0 0...
0
0
0...
0
0 ...
0
0
0
0
0
0
0 ...
0
0...
0
0
0;
0
0
0
0
0
0
0
0 0...
0 ...
0
0...
0
0 ...
0
0
0
0
0 ...
0
0...
pi*f/H(2)
0;
0
0
0..
0
0
0
0
0 1
0
0
93
0
0 0
-(Tm+Tb)/(Tm*Tb)
0
92
0
0
0
0
0
0
pi*f/H(2)
0 -D(2)*pi*f/H(2)
0
0 0...
0 ...
0
87
0
0 ...
0
0 ...
0
0
0 ...
0
0 ...
0
0;
0
94
0
0
0
.0
0
0
0
0
0
0 1
0 0
0
0
0
0 ...
0
0
0
0 ...
0
0
0
0..
0
0
0
0
0
95
0;
0
0
0
0
0
0
0
0
0
0
0
0 0
0
-D(3)*pi*f/H(3)
0
0
0
0 ...
0
0
0...
0
0
0
0 ...
0
0 ...
0
0
96
-pi*f/H(3);
0
0
0
0
0
0
0
0
0
0
0
0 0
0
Aone
0 0 ...
hw(1,1)*Aw(1,1)/Czl
hw(1,2) *Aw(1,2) /Czl
hw(1, 4) *Aw(1, 4) /Czl
hfl*Afl/Czl
...
hw(1,3)*Aw(1,3)/Czl
...
hcl*Acl/Czl
0 ...
Bone
0 ...
0
0 ...
0
0
-msal*COP/Msa/Czl;
88
9-r
0
0
0
0
0
0
0
0
0
0
0
0
0 0...
0 0
hw(1,1)*Aw(1,1)/Cw(1,1)
-2*hw(1,1)*Aw(1,1)/Cw(1,1)
...
0
0
0
0 ...
0 ...
0
0
0 ...
0
0 ...
0
0
98
0
0
0
0;
0
0
0
0
0
0
0
0
0
0 0 ...
0 0
hw(1,2)*Aw(1,2)/Cw(1,2)
0
-2*hw(1,2) *Aw(1,2) /Cw(1,2)
0 ...
0
0
0
0
0
0
0
0
0 ...
0
0
0
99
0;
0
0
0
0
0
0
0
0
0
0
0 0 ...
0 0
hw(1,3)*Aw(1,3)/Cw(1,3)
0 ...
0
-2*hw(1,3)*Aw(1,3)/Cw(1,3)
0...
0
0
0
0 ...
0 ...
0 ...
0
0
100
0
0
0
0
0;
0
0
0
0
0
0
0
0
0 0
0 0
hw (1, 4) *Aw (1, 4) /Cw (1, 4)
0
0
-2*hw(1,4)*Aw(1,4)/Cw(1,4)
89
0
0 ...
hw (1, 4) *Aw (1, 4) /Cw (1, 4)
0
0
0..
0
0.
0
101
0
0;
0
0
0
0
0
0
0
0
0
0
0.
0 0.
0 0
0
hcl*Acl/Ccl
0...
0.
0
0
-2*hcl*Acl/Ccl
0
0
0..
0
0.
0;
0
102
0
0
0
0
0
0
0
0
0
0
0 0
0 0
0
0
0
0
0
hfl*Afl/Cfl
...
0
0
0...
0
-2*hfl*Afl/Cfl
0
0
0...
0
0...
0
0;
0
103
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0 0...
Btwo
0...
0
0
0...
0
Atwo
0
hw(2, 1) *Aw(2, 1) /Cz2
hw(2,2) *Aw(2,2) /Cz2
hw(2, 3) *Aw (2,3) /Cz2
hw(2, 4) *Aw(2, 4) /Cz2
hc2*Ac2/Cz2
104
0
0
0
0
hf2*Af2/Cz2
0
-msa2*COP/Msa/Cz2;
0
0
0
0
90
0
0
0
00
00 0
0
0 ...
0
0.
0
0..
0
...
hw (2, 1) *Aw (2, 1) /Cw (2, 1)
-2*hw(2,1)*Aw(2,1)/Cw(2,1)
0
0.
0
0.
0
105
0
0
0
0;
0
0
0
0
0
0
0
0
0 0
0
0 0.
0
0 ...
0
0.
0
0
0
...
hw(2,2)*Aw(2,2)/Cw(2,2)
0
-2*hw(2,2) *Aw(2,2) /Cw(2,2) 0
...
0
0.
0
106
0
0
0
0;
0
0
0
0
0
0
0
0
0 0
0
0 0.
0
0 ...
0
0
0
0
0
hw(2,3)*Aw(2,3)/Cw(2, 3) 0
0
-2*hw(2,3) *Aw(2, 3) /Cw(2, 3)
0
0
107
0
0
0
0
0;
0
0
0
0
0
0
0
0
0
0 0
hw (2, 4) *Aw (2, 4) /Cw (2, 4)
0 0
0 ...
0
0 ...
0
0
0 ...
hw(2, 4)*Aw(2, 4)/Cw(2,4)
0
91
0 ...
0 ...
-2*hw(2,4)*Aw(2,4)/Cw(2,4)
0;
0
0
0
108
0
0
0
0
0
0
0
0
0
0
0 0
0
0 0
0 ...
0
0
0
0
0.
0
0
hc2*Ac2/Cc2
0.
0
0
-2*hc2*Ac2/Cc2
109
0
0
0
0
0
0;
0
0
0
0
0
0 0
0 0
0
0
0 ...
0
0
0.
0
0..
0
0
0
0
0
0..
0
0.
0
0;
0
0
0
0
0
0
0
0 0.
0 0
0
0...
0
0
0.
0
0..
0
0
0..
0
0.
-1/Tp];
111
%
Fy
113
Fy
=
114
[0 0 0 0 0 0 0
0
0.
0
0
112
0.
hf2*Af2/Cf2
-2*hf2*Af2/Cf2
110
0
0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
92
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
-pi
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
S0
0 0
0 0 0 0 0;
0 0
0 0
0 0
-pi*f/H(2)
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 -pi*f/H(3) 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
S0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0;
0 0
0 0
0 0
0 0 0 0 0);
/H(1) 0 0 0 0 0 0 0;
0 0 0 0;
144
145
%% The Fx, Fy,
146
rad
=
148
%
Fx-p
149
Fx-p =[-1/Tv
Gx,
Gy in PSAT's unit
2*pi*f;
147
0
0
0
0 ...
0
0
0
93
0
0
00..
0 0
0
0..
0
0 ..
0
0 ..
0
0
0 ..
0
0..
0
0..
0
150
1/Tf
0;
-1/Tf
0
0 0
00
0
0
0
0
0
0 0
00..
0
0.
0
0..
0
0
0..
0
0..
0
0..
0
0
151
1/Tcd
0
0..
0;
-1/Tcd
0 0
0
0
0
0
0
0
0
0 0
00..
0
0 ...
0
0..
0
0
0..
0
0..
0
0..
0
0
0
0
0
0;
0
0 0
0..
0
1
0
0
0
0
0 0
0
00..
0 ..
.
152
.
0
.
0
0
0..
0
0
0..
0
0
94
0..
0..
0 ...
0
0
153
0
1/ (Tm*Tb)
0
0
0;
0
-1/ (Tm*Tb)
0
0
0
0
(Tm+Tb) / (Tm*Tb)
-
0 0.
0 0
0
0
0
...
0.
0
0 ...
0
0 ...
0
0
0 ...
0
0 ...
0
0;
0
154
0
0
0
0
0
0
rad
0
0
0
0
0 0
0 0
0
0 ...
0
0
0
0...
0
0
0 ...
0
0
0
0 ...
0
0
155
pi*f/H (1)
0
0
0
0;
0
-D(1)/M(1)
0
1/M(1)
0
0
0
0 0 ...
0 0
0
0 ...
0
0 ...
0
0 ...
0
0
0
0
0...
0
0 ...
0
0
156
0
0
0
0
0;
0
0
0
0
0
0
-1/Tv
0 0
0
0 0 ...
0 ...
0
95
0
0
0
0.
0
0
0.
0
0
0;
0
157
0
0
0
0
0
0
0
0
0
1/Tf
0
-1/Tf
0
0 0
0
0 0.
0 ...
0
0
.
0..
0
0
0
0
0
0
0
0
0
0
0
158
0
0
0
0;
0
0
0
0
0
0 0
0
0 0
0
0 ...
0
0
0
0.
0
0
0
0
0
0.
0
0
0;
0
0
0
0
0
0
0
159
-1/Tc d.
1/Tcd
0
0
0
0
0
0 0
1
0
0 0
0 ...
0
0
0
0.
0
0
0
0.
0
0
0;
0
96
160
0
0
0
0
0
0
1/ (Tm*Tb)
0
0
-1/ (Tm*Tb)
0 0...
0 0
(TM+Tb) / (Tm*Tb)
-
0
0
0
0 ...
0
0 ...
0
0 ...
0
0
0 ...
0
0 ...
0
0;
0
161
0
0
0
0
0
0
0
0 0
0 rad
0
0
0
0
0
0
0
...
0
0
0 ...
0
0 ..
0
0
0 ...
0
0 ...
0
0;
0
162
0
0
0
0
0
0
0
1/M (2)
1/M(2)
0
0
0 0
0 -D(2)/M(2)
0
0 ...
0
0
0
0 ...
0
0
0 ...
0
0...
0
0
0
0
0
0;
0
0
0
0
0...
0
0
183
...
0
0
0
0 rad
0 0
0
...
0...
0
0
0
97
0 ...
0...
0
0
0...
0
0..
0
164
0;
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0...
0
-D (3) /M (3)
.
0
0
0 ...
0
0
0...
0
0
0..
0
0
0 ...
0
0
165
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0 0 ...
Aone
hw(1, 1) *Aw(1, 1) /Czl
hw(1, 2) *Aw(1, 2) /Czl
hw(1,3)*Aw(1,3)/Czl
hw(1,4)*Aw(1,4)/Czl
hfl*Afl/Czl
hcl*Acl/Czl
0 ...
Bone
0
0...
0
0
0
166
0
0
0
0
-msal*COP/Msa/Czl;
0
0
0
0
0
0
0
0 0
0
0 0 ...
hw(1, 1) *Aw(1, 1) /Cw(1, 1)
0
-2*hw(1, 1) *Aw(1, 1) /Cw(1,1)
0.
0 ...
0
0
0 ...
0
0
0
0
0
167
0
0
0
0
0.
0;
0
0
0
0
0
98
...
0
0
0
0 0
0 0.
hw (1, 2) *Aw (1, 2) /Cw (1, 2)
0
...
0.
-2*hw(1,2) *Aw(1,2) /Cw(1,2)
0
0..
0
0
0
0..
0
0.
0
168
0
0;
0
0
0.
0
0
0
0
0
0
0
0
0
0 0.
0 0
hw(1, 3) *Aw(1, 3) /Cw(1, 3) 0 ...
0
2*hw(1,3)*Aw(1,3)/Cw(1,3)
0
0
0...
0
0...
0
0
0
0
169
0
0;
0
0
0...
0
0
0
0
0
0
0
0
0 0
0
0 0...
hw(1, 4)*Aw(1, 4) /Cw(1, 4) 0
0
0...
-2*hw(1,4)*Aw(1,4)/Cw(1,4)
0
hw(1,4)*Aw(1,4)/Cw(1,4)
0
0 ...
0
0...
0;
0
0
0
0
0
0
0
170
0 ...
0
0
0
0
0
0
0
0 0...
0 0
hcl*Acl/Ccl
0...
0
0...
0
-2*hcl*Acl/Ccl
0
0
0
99
0 ...
0 ...
0
0...
0;
0
171
0
0
0
0
0
0
0
0
0
0
0
0 0...
0 0
0
0
hfl*Afl/Cfl
...
0
0
0
0
0
-2*hfl*Afl/Cfl
0...
0
0
0
0
0
172
0
0
0
0
0
0;
0
0
0
0
0
0
0 0
0
0 0...
0 ...
Btwo
0...
0
0
0...
0
hw(2, 1) *Aw(2, 1) /Cz2
hw(2, 2) *Aw(2, 2) /Cz2
hw(2, 3) *Aw(2, 3) /Cz2
hw(2, 4) *Aw(2, 4) /Cz2
hc2*Ac2/Cz2
173
Atwo
hf2*Af2/Cz2
0
0
0
-msa2*COP/Msa/Cz2;
0
0
0
0
0
0
0
0
0 0
0
0 0
0
0...
0
0
0
0...
0
hw(2,1)*Aw(2,1)/Cw(2,1)
-2*hw(2,1)*Aw(2,1)/Cw(2,1)
0...
0
0
0
174
0
0
0;
0
0
0
0
0...
0
0
0
0
0
0 0
0
0 0...
0 ...
0
100
0
0...
0
0...
0
hw(2,2) *Aw(2,2)/Cw(2,2) 0
-2*hw(2,2) *Aw(2,2) /Cw(2,2)
0
0
0
0
175
0;
0
0
0
0
0
0
0
0
0
0
0
0
0 0...
0 0
0
0 ...
0
0
0
0...
0
hw(2,3)*Aw(2,3)/Cw(2,3)
0
0
0
-2*hw(2,3)*Aw(2,3)/Cw(2,3)
0
176
0;
0
0
0
0...
0
0
0
0
0
0
0
0
0
0 0
.hw(2, 4) *Aw (2, 4) /Cw (2, 4)
0
0
0 ...
0
0
0
0
0
0
0
0
0 0
0
0 0
0...
0
0
0
hc2*Ac2/Cc2
0
0;
101
0 ...
0 ...
0
0
...
0 . ..
0
-2*hc2*Ac2/Cc2
0
0;
0
0
...
hw (2, 4) *Aw (2, 4) /Cw (2, 4) 0
-2*hw(2, 4) *Aw(2, 4) /Cw(2, 4)
0
0
0 ...
0
177
...
0 ...
0
0
0 0
0
178
0
0
0
0
0
0
0
0
0
0
0
0 0
0 0
0 ...
0
0
00..
0
0
hf2*Af2/Cf2
00 ...
.
0
0...
0
0...
0
-2*hf2*Af2/Cf2
0
179
0
0
0
0
0
0..
0;
0
0
0
0
0
0
0 0
0
0 0...
0 ...
0
0...
0
0
0...
0
0...
0
0...
0
0
0...
-1/Tp];
180
181
%
182
Fy-p
Fy-p
[0 0 0 0
=
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
183
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
184
0
185
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
186
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0;
187
0 0 0 0 0 0
0 0 0 0 0
188
0 0 0 0 0 0 0 0 0 0 0 0 0 -1/M(1)
189
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0;
190
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0;
191
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
192
0 0
0 0 0 0
0 0 0 0 0
0 0 0 0 0
193
0 0
0 0 0 0
0 0 0 0 0
0 0 0 0 0 0 0 0 0 0;
194
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0 0 0 0 0;
195
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1/M(2)
0 0 0- 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0
102
0 0 0 0 0;
0 0 0 0 0 0 0;
0 0 0 0 0;
0 0 0 0;
196
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
197
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1/M(3)
198
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
199
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
200
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
201
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
202
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
203
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
204
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
205
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
206
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
207
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
208
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
209
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
210
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
211
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
212
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0];
0;
0;
213
214
215
P,
(partial
216
%Km
217
for i=l:n
Partial
theta)
for j=1:n
218
if
219
j#i
220
Km(i,j)=V(i)*V(j)*(G(i,j)*sinf(A(i)-A(j))-.
221
B(i,
end
222
end
223
224
end
225
for i=l:n
226
Km(i,i)=0;
227
for j=1:n
228
if
jsi
Km(i,i)=Km(i,i)-Km (i,
229
j)
end
230
231
j)*cos(a(i)-A(j)));
end
103
232
end
233
234
%Lm
235
for i=l:n
(partial Q,
236
for j=l:n
237
if
partial
theta)
jsi
238
Lm(i,j)=V(i)*V(j)*(-G(i,j)*cos(A(i)-a(j) )-...
239
B (i,
)
)
-A (j)
end
240
end
241
242
end
243
for i=1:n
244
Lm(i,i)=O;
245
for j=l:n
246
if
ji
247
Lm(i,i)=Lm (i,i)-Lm(i,j);
end
248
end
249
25o
j) *sin (A (i)
end
251
252
%Mm
253
for i=1:n
(partial P,
partial V)
for j=l:n
254
255
if
j#i
j)*cos(A(i)-a(j))+...
256
Mm(i, j)=V(i)*(G(i,
257
B(i, j)*sin(a(i)-a(j) ));
end
258
end
259
260
end
261
for i=l:n
262
Mm (i,
263
for j=l:n
264
if
i)=2*V (i) *G (i, i);
j#i
265
Mm(i,i)=Mm(i,i)+V(j)*(G(i,j)*cos(A(i)-a(j))+.
266
B(i,j)*sin(a(i)-a(j)
267
));
end
104
end
end
%Nm (partial Q,
for
partial V)
i=l:n
for j=l:n
if
jsi
Nm(i, j)=V(i)*(G(i, j)*sin(a(i)-a(j))-
B(i, j)*cos(a(i)-a(j)));
end
end
end
for i=l: n
Nm(i, i) =-2*V(i) *B (i,i);
for j=l:n
if
jsi
Nm(i, i) =Nm(i, i) +Nmr(j, i)
end
end
end
% Gy
for i=l:n
for j=l :n
Gyl (i, j) =Km (i,
j);
Gyl (n+i, j)=Lm(i,
j)
Gyl (i,n+j)=Mm(i, j)
Gyl (n+i,n+j)=Nm(i, j)
end
end
if orders == 2
for i=l:dyns
Gyl(i,2*n+((i-i)*4+3))=-l;
Gyl(i+n,2*n+((i-i)*4+4))=-1;
end
elseif orders ==
3
for i=l:dyns
105
Gyl(i,2*n+((i-i)*3+2))=-1;
Gyl(i+n,2*n+((i-i)*3+3))=-1;
end
end
og=
[];
for i=1:(n/2)
Ag =
[Ag
xO((i-1)*2+1)];
end
if orders == 2
Gy2 = zeros(2*n,2*n+4*dyns);
for i=1:dyns
Gy2((((i-i)*4)+3),i)=
-V (i) *E (i) *cos (theta
(i) -Ag (i) ) /Xd (i)
i) =
-V (i) *E (i) *sin (theta
(i)-Ag (i) ) /Xd (i)
Gy2 ((((i-1)
*4) +4)
,
Gy 2 ((((i-i)*4)+3), (n+i)
=-E () *sin
Gy2((((i-i)*4)+4), (n+i)
=-(2*V(i) /Xd(i))+.
(E (i) *cos (theta
(i) -Ag(i)
Gy2 ((((i-1) *4) +3),
(theta
(i)-Ag (i) ) /Xd (i)
)/Xd(i));
(2*n) + (((i-1) *4) +1))) =-V(i) *sin (theta (i)-..
Ag(i) )/Xd(i);
Gy2 ((((i-1) *4)+4),
(2*n) + (((i-1)
*4) +1)) )=V(i) *cos (theta
(i) -.
Ag(i))/Xd(i);
end
for i=1:4*dyns
Gy2 (i, 2*n+i) =-1;
end
elseif orders ==
3
Gy2 = zeros(2*n,2*n+3*dyns);
for i=1:dyns
Gy 2 ((((i-i)*3)+2),i)=
-V (i)
*E (i) *cos (theta
Gy2((((i-i)*3)+3),i)= -V (i) *E (i) *sin
(theta
(i)-Ag (i))
/Xd (i);
(i)-Ag (i) )/Xd (i);
Gy2((((i-1)*3)+2), (n+i)) =-E(i)*sin(theta(i)-Ag(i))/Xd(i);
Gy2((((i-1)*3)+3), (n+i)) =-(2*V(i) /Xd(i))+.
(E (i) *cos (theta(i)
-Aq(i) )/Xd(i));
106
e)*3)+2),-((2*n)+(((i-l)*3)+)))=...
Gy2((((i-
-V (i) *sin (theta (i) -Ag (i) ) /Xd (i);
Gy2((((i-l)*3)+3), ((2*n)+(((i-1)*3)+1)))= ...
V(i)*cos(theta(i)-Ag(i))/Xd(i);
end
for i=l: 3*dyns
Gy2 (i, 2*n+i) =-1;
end
end
Gy=[Gyl;Gy2];
Gyp=Gy;
% Gx
Gx =
zeros(21,31);
Gx(14,6)
= V(1)*E(1)*cos(theta(1)-Ag(1)) /Xd(1)
Gx(15,6)
= V(1)*E(1)*sin(theta(1)-Ag(l)) /Xd(1)
Gx(17,13)
= V(2)*E(2)*cos(theta(2)-Ag(2) /Xd(2)
Gx(18,13)
= V(2)*E(2)*sin(theta(2)-Ag(2) /Xd(2)
Gx(20,15)
= V(3)*E(3)*cos(theta(3)-Ag(3) /Xd(3)
Gx(21,15)
= V(3)*E(3)*sin(theta(3)-ag(3) /Xd(3)
Gx-p=Gx;
Amatrix = Fx -
Fy*(Gy_p\eye(size(Gy-p)))*Gx_p
Amatrix = Fx-p -
Fy-p*(Gy_p\eye(size(Gy_p)))*Gx_p;
end
107
108
Appendix D
MATLAB Code
-
Model Predictive
Control
109
1 addpath('rsgtssimv2\')
2
addpath('DataMPC\')
3 addpath('rsgtssimv2\matpower4.l\')
4
addpath('rsgtssimv2\powerGrid.mindnode\')
5 addpath('rsgtssimv2\utilities\')
6
7
8
clear all
9
close all
10
clC
11
12
load('xint.mat');
13
load('xint-b.mat');
14
load( 'steadyValueGeneration.mat');
15
16
%% Initialization
17
%initial states for linearized model
18
xO =
19
xO(1) = Fd2Pg(xint(1));
20
xO(2)
21
xO(8) = Fd2Pg(xint(8));
22
xO(9)
23
% initial ocntrol signal for MPC control
24
uintc
25
% initial control signal for runSimulator
26
uints
(MPC control)
[xint;xint-b;0.5];
Fd2Pg(xint(1));
=
Fd2Pg(xint(8));
=
[x0(l)
=
xO(8)
0.5];
[xint(1) xint(8)
=
0.5];
27
[;
28
UMPC =
29
% total simulation time
30
ttotal = 1200;
31
% time period to apply every control move
32
tcontroller = 30;
33
% number of MPC simulation steps
34
Nsimulation = t total/t_controller;
35
% time to get each linearized plants
110
30;
36
tlin
37
% time for iteration
38
Nlin = t total/t_controller;
=
39
40
% start time and final time to get
41
Ts =
0;
42
Tf =
0;
43
% start time and final time to
44
Tsu = 0;
45
Tfu = 0;
linearized model
implement
new control signal
46
47
%% Iteration code:
48
for i=l:Nlin
MPC
49
50
Ts = Tfu;
51
Tf = Tfu+t_lin;
52
iter = i;
53
54
[labelVec,
timeVec,
stateVec,
b-stateVec,
btimeVec,
rl,
r3]=runSimulatorMPC(1,Ts,Tf,xint,xintb,uints,iter);
55
56
%%
get the
57
Ni
= 2;
58
Nj
=
59
hw =
60
Aw = zeros(Ni,Nj);
61
Cw =
zeros(Ni,Nj);
62
Cp =
1.005;
63
for j=l:Nj
linearized system
4;
zeros(Ni,Nj);
64
hw(1,j)
=
0.02;
65
Aw(1,j)
=
1;
66
Cw(1,j)
=
5;
67
end
68
for j=l:Nj
69
hw(2,j)
=
0.02;
70
Cw(2,j)
=
5;
111
r2,
71
end
72
Aw(2,1)
=
2;
73
Aw(2,2)
=
1;
74
Aw(2,3)
=
2;
75
Aw(2,4)
=
1;
76
Czl =
77
Cfl = 5;
78
hcl = 0.01;
79
Acl =
80
Msa = 0.5;
81
Pfan = 0.5;
82
msal
=
83
msa2
= Msa*(1-Pfan);
84
Moa
85
Mra = Msa-Moa;
86
mral = Mra*0.5;
87
mra2 = Mra*0.5;
88
COP = 2.5;
89
Tv
90
H
=
[5 4 5];
91
D
=
[0.001 0.001
92
M
=
93
freq = 60;
94
Tp
=
2.5;
1;
Cz2 = 2.5;
Ccl = 5; Cf2 = 5;
Cc2 = 5;
hfl = 0.01; hc2 = 0.01; hf2
Afl = 1;
Ac2
2;
=
Af2
Msa*Pfan;
0.1;
0.05;
=
0.001];
2*D;
=
0.0001;
95
96
load('Amatrix.mat')
97
A = As;
98
B =[1/Tv 0
0
0;
99
0
0
0
0;
100
0
0
0
0;
101
0
0
0
0;
102
0
0
0
0;
103
0
0
0
0;
104
0
0
0
0;
105
0
1/Tv
0
0;
106
0
0
0
0;
112
=
2;
=
0.01;
107
0
0
0
0;
108
0
0
0
0;
109
0
0
0
0;
110
0
0
0
0;
111
0
0
0
0;
112
0
0
0
0;
113
0
0
0
0;
114
0
0
0
1/Czl* (msal*Cp*Moa/Msa);
115
0
0
0
1/Cw(1,1)*hw(1,1)*Aw(1,1);
116
0
0
0
1/Cw(1,2)*hw(1,2)*Aw(1,2);
117
0
0
0
1/Cw(1,3)*hw(1,3)*Aw(1,3);
118
0
0
0
0;
119
0
0
0
1/Cc1*hcl*Acl;
120
0
0
0
1/Cfl*hfl*Afl;
121
0
0
0
1/Cz2* (msa2*Cp*Moa/Msa);
122
0
0
0
1/Cw(2,1)*hw(2,1)*Aw(2,1);
123
0
0
0
1/Cw(2,2)*hw(2,2)*Aw(2,2);
124
0
0
0
1/Cw(2,3)*hw(2,3)*Aw(2,3);
125
0
0
0
0;
126
0
0
0
1/Cc2*hc2*Ac2;
127
0
0
0
1/Cf2*hf2*Af2;
128
0
0
1/Tp
0];
129
C
0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
=[
0 0 -1;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
130
0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0..
131
0 0];
132
D
[0
=
0 0
0;
133
0 0 0 0;
134
0 0 0 0];
135
136
sys = ss(A,B,C,D);
137
138
%%
139
ts =
create MPC object
30;
113
140
p
=
60;
141
m
=
5;
142
% MPC weights structure
143
wghts = struct('ManipulatedVariables', [0.0
'OutputVariables',
0.0 0.0],
[3 1 1]);
144
% MPC manipulated variable 1 (Fd genl)
145
MV(l)
146
% MPC manipulated variable 2 (Fd gen2)
147
MV(2)
148
% MPC manipulated variable 3 (Pcooling)
149
MV(3)
iso
% MPC output variable 1 (Pmg1+Pmsl+Pmg2+Pms2-Pcooling)
151
OV(l)
152
% MPC output variable 2 (Tzl)
153
OV(2)
154
% MPC output variable 3 (Tz2)
155
OV(3)
156
% set the forth input
157
sys.InputGroup.MD = 4;
158
% 4 inputs
159
MPCobj = mpc(sys, ts,
= struct('Min',0,'Max',2);
= struct('Min',0,'Max',2);
= struct('Min',0,'Max',1);
= struct('Min',0,'Max',10);
= struct('Min',0,'Max',30);
= struct('Min',0,'Max',30);
(3 MV,
(Tamb) as measured disturbance
1MD), 3 outputs
p, m, wghts,
MV,
OV);
160
161
%% MPC controller
162
T
163
% reference signal
164
r =
=
10;
[rl r2 r3];
165
166
set imulation options w/ initial values
167
SimOpt = mpcsimopt(MPCobj);
168
% define MPC controller state
169
xmpc = mpcstate (MPCobj,x0, [], [],uintc);
170
171
% set simulation options w/ initial values
172
set(SimOpt, 'PlantInitialState',x0,'ControllerInitialState',xmpc);
173
% specifies the measured disturbance signal v (Tamb), that
...
has as many columns as the number of measured disturbances.
114
30;
=
174
v
175
% do the MPC simulation
[y-mpc,
176
t-mpc,
u-mpc,
xp-mpc, xmpcqmpc,
SimOptions]
=
sim(MPCobj,T,r,v,Sim~pt);
% y-mpc:
177
outputs,
variables,
t-mpc:
sequence,
u:
manipulated
...
xpmpc:
% state of the model,
178
time
xmpc-mpc: state of the controller
179
step of the control strategy by MPC controller
180
% save the first
181
u-new-s = u mpc(,:);
182
unews(l) = Pg2Fd(unew_s(l));
183
unews(2)
184
u_MPC =
= Pg2Fd(unews(2));
[uMPC;u-new-s];
185
186
%% run simulation and apply the first
187
Tsu =
188
Tfu = Ts+tcontroller;
Ts;
[labelVec, timeVec,
189
step of u
stateVec, b-stateVec, btimeVec]
...
runSimulatorMPC(2,Tsu,Tfu,xint,xintb,u_new-s,iter);
190
%%
191
grab the
initial
last
state for the
state for the next
192
xint =
193
xintb = bstateVec(size(b
194
uints = unew-s;
195
% set
196
uintc
197
xO =
198
xO(1)
= Fd2Pg(xint(1));
199
xO(2)
= Fd2Pg(xint (1));
200
xO(8)
= Fd2Pg(xint (8));
201
xO(9)
= Fd2Pg(xint (8));
next simulation
simulation)
stateVec(size(stateVec,l),:)';
stateVec,l), :)';
initial state for next MPC control
= u_mpc(1,:);
[xint;xintb;uintc(3)];
202
end
203
save('uMPCtimeseries.mat', 'uMPC');
115
(as the
...
116
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