by Bharath R. Vellaboyana - University of Toronto T-Space · 2017-03-20 · Bharath R. Vellaboyana...

Decentralized scheduling and control in power systems

by

Bharath R. Vellaboyana

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Graduate Department of Electrical and Computer EngineeringUniversity of Toronto

c© Copyright 2016 by Bharath R. Vellaboyana

Abstract

Decentralized scheduling and control in power systems

Bharath R. Vellaboyana

Master of Applied Science

Graduate Department of Electrical and Computer Engineering

University of Toronto

2016

Decentralized control reduces the need for expensive and delay inducing communica-

tion infrastructure. In this thesis we will present new algorithms for decentralized control

of DC-segmented power systems and energy storages. DC-segmentation is a process in

which a transmission system is divided into isolated AC subsystems connected to each

other only by DC lines. It improves transient stability and increases the transmission

capacity of a transmission system. We construct a poset-causal optimal decentralized

controller for this system, which only requires neighbor-to-neighbor communication. In

the second part of this thesis, we focus on multiple energy storages located in a distri-

bution system with renewable generation. We use dynamic programming and inventory

control theory to obtain optimal scheduling policies for the energy transactions of the

storages. These policies are optimal and only require local information.

ii

I dedicate this work to my caring parents, Nirmala and Jayaram.

Acknowledgements

I want to thank Prof. Joshua A. Taylor for the guidance that he provided throughout

my research. His broad range of interests in the field of control systems and optimization

helped me in finding my own interest in decentralized control. His keen interest in my

work helped me in keeping my research on the right track. His inputs were invaluable in

moving my work forward.

I want to thank Prof. Reza Iravani for the discussions about Voltage sourced con-

verters which proved very useful in crafting our DC line model.

Thanks to Dariush Fooladivanda for all the help with solving the problem of optimiz-

ing storage schedules. Discussions with him cleared many a doubt that I had. And also

thanks to him for helping me write my first paper.

iii

Contents

1 Introduction 1

1.1 DC-segmented systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Network of energy storages . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

I DC-segmented systems 5

2 Decentralized control of DC-segmented power systems 6

2.1 Motivation and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Chapter organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Poset causal systems . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.2 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.3 Complete model and costs . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1 Proof of poset causality . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.2 Controller design procedure . . . . . . . . . . . . . . . . . . . . . 23

2.5 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

iv

II Network of energy storages 29

3 Optimal scheduling of networked energy storages 30

3.1 Motivation and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Chapter organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4 Convexity of the function Gt . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5 Computing the optimal policy . . . . . . . . . . . . . . . . . . . . . . . . 36

3.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

III Conclusion 39

4 Conclusion and future work 40

Appendix A Linearization of the DC line model 41

Appendix B Calculation of the decentralized controller 44

B.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

B.2 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

B.3 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Appendix C Proof of Theorem 2 49

Bibliography 59

v

List of Tables

2.1 Parameters of the generators . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Parameters of the DC lines . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Parameters of the VSCs . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Values of constants at the operating point . . . . . . . . . . . . . . . . . 25

2.5 Comparison of controller performances on System 1 . . . . . . . . . . . . 26

A.1 Values of constants at the operating point . . . . . . . . . . . . . . . . . 42

vi

List of Figures

2.1 Example of a graph G ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 The DC line model. States to the left of the vertical dotted line are

associated with the ith subsystem. States to the right are associated with

the mth subsystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 G ′ graph of the example . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Disturbance at the first bus . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Generator frequency deviation at the first bus . . . . . . . . . . . . . . . 27

2.6 Power flow from the subsystem 2 from 1 . . . . . . . . . . . . . . . . . . 27

2.7 VDCI of the DC line joining 1 and 2 . . . . . . . . . . . . . . . . . . . . . 28

3.1 ith bus model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 The resultant policy (y1, y2) in a two bus system after solving Pt . . . . 38

vii

Chapter 1

Introduction

This chapter introduces DC-segmented transmission systems and energy storage net-

works, and explains the need for and benefits of each application.

1.1 DC-segmented systems

In the early days of electricity grid, AC and DC technologies contended to become the

standard for generation, transmission and distribution. Though DC had already spread

widely through out the world, AC won out in the end and supplanted it in most places

because of

• The need for higher voltage levels to reduce losses in transmission, which was easily

achieved with AC because of the transformer technology.

• The simpler construction of AC motors and generators.

• Ease of construction of AC circuit breakers.

But now-a-days High Voltage Direct Current (HVDC) lines are preferred over AC lines

for certain applications like asynchronous interconnections, long underwater transmission

cables and bulk power transmission [1], [2]. They provide many benefits like improved

1

Chapter 1. Introduction 2

wire utilization, reduced losses and controllable power flow. We can control the power

flow through a DC line via power electronic converters at either terminal buses, unlike

the flow through an AC line which is completely determined by the voltages at the

terminals. This controllability can be used to enhance the transient stability of a power

system [3], [4], [5].

Controllability of DC lines can be used to solve the challenges faced by the, now

ubiquitous, interregional grids, where neighbouring utilities in a grid are interconnected

to each other to enhance the grid security by enabling utilities to provide emergency

assistance and to improve the economy of system operation by reducing the need for

high reserve capacity. But these large inter-regional grids are hindered by their inherent

limitations from providing cost-effective solutions to the problems like securing the grid

from cascading network outages, keeping up with the demands for increased transmission

capacity and enabling complex market operations [6].

In interregional grid, with all the utilities running on uniform system-wide frequency,

contingency at any point will force a system wide response. And when this response

crosses voltage and angular stability limits it can cause disastrous consequences. The

likelihood of such contingencies is increasing because of the aging equipment, climate

change and the increasing complexity of grid operation. Complexity raises the risk of

political sabotage and operator error. And global warming causes failures by way of

extreme weather excursions.

And the need for the transmission service to handle complex market operations will

continue to grow as the number of third-party generators and the utilization of distant

energy sources increases. But parallel flows, ubiquitous in an AC network, cause a

hindrance in the conformance between contractual paths and the power-flow paths. Power

trading over AC systems will also be hampered by phenomena of inadvertent flows and

loop flows.

The above problems can be solved by decomposing existing large inter-regional net-


works into a set of asynchronously operated sectors connected to each other exclusively

by DC lines. Segmenting the transmission system will increase its reliability as the ex-

tent of the response of a region to a contingency in another region can be limited by the

DC line connecting both. DC lines can avoid the parallel, inadvertent and loop flows

allowing for efficient enforcement of contracts. And DC-segmentation also increases the

transmission capacity of a transmission system [6].

Other applications of DC-segmentation include using HVDC line supplementary con-

trols to damp inter-area oscillations [7]. They are low frequency power-system oscillations

in which a coherent group of generators in an area swing against those of other areas and

can cause a large variations in the tie-line power flow.

In this thesis we will consider the problem of controlling a DC-segmented power

system.

1.2 Network of energy storages

The second application we focus on is using a network of energy storages to reduce

variability in renewable energy generation.

Renewable energy sources are non-polluting, sustainable alternatives to fossil fuels for

electricity generation. Coupled with the rapidly growing energy needs of the world [8],

increasing the penetration of renewable energy sources is a logical proposition. In fact,

according to [8] renewable energy is the fastest growing source of energy, growing at 2.6%

per year.

But 82% of that growth in OECD (Organization for Economic Cooperation and De-

velopment) countries was from non-hydropower sources like wind and solar energy which

are intermittent. This causes problems of reliability and stability in power systems [9].

Because of their variability we may need to increase the generation from conventional

sources such as fast ramping generation when they fall short of meeting the demand


and curtail the generation from conventional sources when there is an excess of supply.

Demand response [10] and energy storage [11], [12] offer alternative mechanisms of bal-

ancing renewable supply with demand. In particular, using energy storage is a better

alternative to conventional energy sources in terms of damage to environment and it also

brings down the system cost of curtailment [13], [14].

Renewable energy sources are often small in scale and distributed over multiple lo-

cations in power systems. Mitigating their variability locally by co-locating the stor-

age [15], [16] brings in the additional benefits of reducing the demands on transmission

infrastructure. Independent storage agents with multiple storages can also play an active

role in such scenarios by absorbing the variability.

In this work we will consider the problem of scheduling the energy transactions of

this network of co-located storages.

1.3 Organization

The rest of the thesis will present optimal decentralized solutions to these problems in

two parts. The first part will present an algorithm to design an optimal decentralized

controller for a DC-segmented power system using poset-causal control theory. And the

second one will present optimal scheduling policies for storages. Each part will explain the

motivation for our approach and its scope, and after providing the necessary theoretical

background will present the solution.

Part I

DC-segmented systems

5

Chapter 2

Decentralized control of

DC-segmented power systems

2.1 Motivation and scope

Stable operation of such DC-segmented systems using centralized control setups involves

burdensome communication requirements because of the presence of large number of

components spread over large geographical area. Wide area control setups [17] that

are generally used in these scenarios can be expensive and introduce delays [18]. Thus

decentralized control, with its reduced communication requirements, is beneficial in this

regard [19].

Designing decentralized controllers is a computationally hard problem [20], [21]. In

fact, Witsenhausen proved in his seminal paper [22] that the optimal decentralized con-

troller for a simple LQG (linear dynamics, quadratic cost function and gaussian noise)

problem is non-linear. It was also proved in [23] that linear controllers for this problem

could be arbitrarily suboptimal. Several other classes of decentralized control problems

were proven to be computationally intractable [24–26].

But it was also shown by many authors that certain classes of problems are tractable

6

Chapter 2. Decentralized control of DC-segmented power systems 7

[27–29]. These classes are distinguished from the intractable classes by the structure of

communication between their component subsystems. Reference [30] proves that decen-

tralized H2-optimal control is a tractable problem if the system has one such structure

called poset-causal communication structure.

In [31], sparsity-promoting decentralized control was used to obtain suboptimal de-

centralized controllers for power systems with HVDC lines. In [32], a voltage-droop

method decentralized controller is obtained for multiterminal HVDC systems. In [33], a

DC-segmented system with a simple model is shown to have the poset-causal structure.

In this study, we extend the work of [33] by showing that DC-segmented power systems

are poset causal even with a more complex model. We validate the controller through

simulation, and find that it performs almost as well as the centralized controller.

There are two basic types of converter technologies that are used in HVDC transmis-

sion systems. One is the conventional line commutated converter (LCC) and the other

is the voltage sourced converter (VSC). While VSCs are a relatively recent technology,

they are being increasingly adopted for implementation in certain scenarios because they

allow independent control of real and reactive power and can provide voltage support to

AC systems [34]. We focus on VSC-based HVDC systems in this work.

Our main results are the following:

• We show that DC-segmented systems are poset-causal.

• We give a procedure for using this structure to obtain optimal decentralized con-

trollers. These controllers only require neighbor-to-neighbor communication be-

tween AC subsystems.

• We validate the decentralized controller on a realistic system model. The decen-

tralized controller performs almost as well as the optimal centralized one.


2.2 Chapter organization

This chapter is organized as follows. We define poset-causal systems and present our

system model in Section 2.3. We prove that the system is poset-causal in Section 2.4.

In Section 2.5 we examine the performance of the H2-optimal decentralized controller on

an example system.

2.3 Preliminaries

2.3.1 Poset causal systems

A partially ordered set or poset Ψ consists of a set P and a binary relation [35]. It has

following properties for all a, b, c ∈ P:

• Reflexivity: a a

• Antisymmetry: a b and b a⇒ a = b

• Transitivity: a b and b c⇒ a c

In this work we will make use of the following results relating graphs and posets

Result 1. We can always choose directions of edges of an undirected graph with no self

loops such that the resulting directed graph is acyclic [36].

Result 2. The resultant Directed Acyclic Graph (DAG) specifies a unique poset [37].

For a particular a ∈ P , we refer to elements b ∈ P that satisfy a b as downstream

of a. We refer to the set of all such elements as ↓a. The relations between elements of a

poset can be encapsulated by a function f : P× P→ < such that f(a, b) = 0 when a 6

b for all a, b ∈ P. A set of such functions for a poset Ψ is called the incidence algebra of

Ψ and is denoted by I(Ψ).


For a finite poset Ψ (|P|, number of elements in P, being finite), each element of I(Ψ)

can be represented by a matrix. Specifically, for any f ∈ I(Ψ) we can construct a matrix

M whose rows and columns are indexed by the elements of P and M(i, j) = f(g(i), g(j)),

where function g : ℵ → P is defined such that g(i) and g(j) are elements of P and they

index row i and column j of M respectively. We can equivalently say M ∈ I(Ψ).

Example 1. Consider a poset Ψ with a set P = 1, 2, 3, 4 and the relation ≤ (less than

or equal to). Then one matrix representation of an element in I(Ψ) is

Γ =

1 0 0 0

1 1 0 0

1 1 1 0

1 1 1 1

.

Rows and columns of Γ are indexed by elements of P in the order from 1 to 4. Here,

↓1 = 2, 3, 4, ↓2 = 3, 4 and ↓3 = 4.

The poset causality of a system is determined by it’s physical flow of information.

Consider the following linear time invariant (LTI) system

x(t) = Ax(t) + Fw(t) +Bu(t) (2.1)

z(t) = Cx(t) +Du(t) (2.2)

where x(t), u(t) and w(t) are states, inputs and disturbances respectively. We assume

CTD = 0, CTC 0, DTD ≺ 0, F is block diagonal and that our system has full state

feedback.

Let the system consist of p subsystems. We partition x(t) into [x1(t), x2(t), · · · , xp(t)]T

(where x(t) ∈ <n and xi(t) ∈ <ni such that∑

i ni = n) such that xi(t) are states of

subsystem i. Similarly we divide inputs, u(t) into [u1(t), u2(t), · · · , up(t)]T with u(t) ∈ <m

and ui(t) ∈ <mi such that∑

imi = m. We then write the matrix A as [Aij]i,j∈1,··· ,p


(where Aij is the block indexed by the ith and jth partition of x(t)). Matrices B,F,C

and D can be similarly written as [Bij], [Fi], [Ci] and [Di] respectively.

Before defining poset-causality we first extend the concept of incidence algebra to

include block matrices. Consider a poset Ψ = (P,), |P| = p and a function f in

its incidence algebra. The matrix B belongs to the block incidence algebra IB(Ψ) if

Bij = 0ni×mjwhenever f(g(i), g(j)) = 0 (where 0ni×mj

is a ni ×mj matrix of zeroes). A

similar definition holds for IA(Ψ).

Example 2. Let p = 4 with x1(t), x2(t), x4(t), u1(t),

u3(t), u4(t) ∈ <1 and x3(t), u2(t) ∈ <2 and let the poset be Ψ from 1. Then, an example

of an element from IB(Ψ) is:

B =

1 0 0 0 0

1 1 0 0 0

1 1 0 1 0

1 0 1 0 0

1 1 1 1 1

Writing (2.1)-(2.2) as a map from w and u to z and x we get

z = P11w + P12u (2.3)

x = P21w + P22u (2.4)

where

P11 = C(sI − A)−1F (2.5)

P12 = C(sI − A)−1B +D (2.6)

P21 = (sI − A)−1F (2.7)

P22 = (sI − A)−1B (2.8)


If P22 ∈ IP22(Ψ), then relations between elements of poset Ψ restrict how subsystems of

(2.1)-(2.2) affect each other. To see this clearly, consider two different subsystems i and

j. If g(i) g(j) then i can affect j through its inputs but because g(j) 6 g(i), (P22)ij

will be 0ni×mjand j cannot affect i.

We call (2.1)-(2.2) Ψ-poset causal (or simply poset-causal) if P22 ∈ IP22(Ψ).

We establish the poset causality of a system by using the following result from [30].

Result 3. If A ∈ IA(Ψ) and B ∈ IB(Ψ), then P22 ∈ IP22(Ψ).

Let Tzw be the transfer function from w to z induced by a controller u = Kx. In this

work our objective is to minimize the H2 norm of Tzw. We reduce our communication

infrastructure by restrictingK to the set IK(Ψ). Then the optimization problem becomes:

minimizeK

||Tzw||

subject to K ∈ IK(Ψ)

Kstabilizing

(2.9)

In general, structural constraints like (2.9) make this problem computationally intractable

[21]. When the system is poset-causal with poset Ψ, this problem is tractable and [30] has

provided the method to obtain the optimum solution. We will summarize this method

to calculate the controller in Appendix B. Note that without the first constraint on K,

this is the standard H2 optimal control problem [38].

2.3.2 System model

In this section, we give models for the network structure, DC lines, and AC lines.

Network structure

We consider power systems which can be decomposed into subsystems, within which the

buses are connected to each other by AC lines. These subsystems are connected to each


other only by HVDC lines.

We represent our system by a graph G = (V , E), where V is the set of buses and E is

the set of AC and DC lines. We divide E into subsets EAC and EDC such that the former

is the set of AC lines and the latter is the set of DC lines. If buses i and j are connected

by an AC line then ij, ji ∈ EAC , and if they are connected by a DC line then either ij

or ji is in EDC depending on VSC configurations at i and j. Thus E = EAC ∪ EDC and

EAC ∩ EDC = φ. Observe that AC lines are undirected and DC lines are directed.

We divide G into subgraphs Gk = (Vk, Ek), k = 1, ..., p, where p is the number of

AC subsystems, Vk is the set of buses in subsystem k, and Ek is the set of AC lines in

subsystem k.

Let Psys be the set of AC subsystems, 1, ..., p. Define the function f : V → Psys such

that f(i) = k if bus i ∈ Vk. We can construct a new graph G ′ = (Psys, E ′) such that if k,l

∈ Psys and kl ∈ E ′, then there exists i, j ∈ V such that f(i) = k, f(j) = l and ij ∈ EDC .

Since G ′ has no self loops, we can choose the directions of its edges so that it is a DAG.

An example is shown in Fig. 2.1.

Figure 2.1: Example of a graph G ′

From Result 2, the DAG G ′ specifies a poset. Define Ψsys = (Psys,), where for

k, l ∈ Psys, k l if kl ∈ E ′. We extend the definition of a downstream set in a poset

to the graph G ′ by defining k to be the upstream subsystem to l and l the downstream

subsystem to k.

Remark 1. In this work we are concerned with DC lines that are part of the DC cuts

of the graph G = (V , E), that is, those DC lines removing which the graph will become


Figure 2.2: The DC line model. States to the left of the vertical dotted line are associatedwith the ith subsystem. States to the right are associated with the mth subsystem.

disconnected. However, DC lines which are not part of the DC cuts can still be included

in our model as part of an AC subsystem.

DC line model

In this section we present the model for the DC line and the converters connected to it.

Our VSC model is based on the back-to-back HVDC transmission system model of [39],

and our model of DC line dynamics is based on [40].

The DC line in Fig. 2.2 connects two buses i and m through current controlled VSCs.

CDCR and CDCI are the capacitances of the VSCs. The parameters CCDC , LCDCR,

LCDCI , RCDCR and RCDCI specify the characteristics of the DC line. The transformers

connecting the VSCs to the buses are represented by L and R′.

The notation of the line depends upon the configurations of VSCs at i and m. The

line in the figure will be named im as the VSC cim controls the power flow through the

line while cmi controls the DC-voltage VDCI . Both VSCs can provide reactive-power to

their respective buses, but we do not use this capability here.

We assume that the switches and diodes of both converters change their states in-

stantaneously, eliminating switching losses. The conduction losses of the converters are

added to the transformer impedance, R′.

We represent the voltage at the terminal joining the transformer to the bus i by vsi .

We assume that the magnitude of the voltage vsi , vsi , is constant. We also assume that


the frequency deviations at i are small enough so that phase locked loops (PLL), which

are part of the VSCs, can keep track of the AC system frequency and thus render vsdi

approximately constant and vsqi ≈ 0 , where vsdi and vsqi are dq frame quantities. Similar

assumptions hold for vsm.

The power flows from cim to i and from cmi to m are determined by the d-frame

component of currents iim and imi as Viidim and Vmi

dmi respectively, where variables Vi =

1.5vsi and Vm = 1.5vsm.

The dynamics of idim and idmi are similar and are given by the transfer functions

id =idrefτs+ 1

(2.10)

idref =PrefV

(2.11)

where id is the current idim for cim (idmi for cmi), τ is the time constant determined by L

and R′, V is equal to Vi (Vm) and Pref is the reference power.

As can be seen from (2.10)-(2.11), both converters are capable of tracking a given

reference for real-power. But only one controller directly controls the power flow. As

per our notation of the DC-line, cim controls the power by setting Pref equal to ψim.

cmi controls the voltage VDCI by setting its Pref equal to VCDCIICDCI + P ′. P ′ and its

relation to VDCI is defined in the following transfer function

P ′ =n0(s+ p)

s(V 2

DCI − V 2DCref ) (2.12)

where n0 and p are parameters of the PI compensator tracking VDCref , the reference for

VDCI . Writing down (2.10)-(2.12) in state space form we get


idim = −idim

τ+ψimτVi

(2.13)

idmi = x (2.14)

x = −xτ

+ICDCIICDCR − I2CDCI

τVmCCDC+

+V 2CDC −RDCIVCDCICDCI − VCDCVDCI

τVmLDCI

−2n0VmCDCI

idmi + n0pV2DCI − n0pV

2DCref (2.15)

where we used the dynamics of VDCI given by

V 2DCI =

−2VmCDCI

idmi (2.16)

Equation (2.16) is based on simplified model for the dynamics of VDCI presented in

Section 8.6.3 of [39].

Circuit analysis of the DC-line gives the following dynamics [40]:

VDCR =IDCRCDCR

− ICDCRCDCR

(2.17)

ICDCR =VDCRLDCR

− VCDCLDCR

− RDCRICDCRLDCR

(2.18)

VCDC =ICDCRCCDC

− ICDCICCDC

(2.19)

ICDCI =VCDCLDCI

− RDCIICDCILDCI

− VDCILDCI

(2.20)

The relation between DC-line currents IDCR, IDCI and the AC currents iim, imi is given

by

IDCR =3

4(md

imidim +mq

imiqim) (2.21)

IDCI =3

4(md

miidmi +mq

miqiqmi) (2.22)


where mdim and mq

im are dq frame components of mim, the modulating signal at the

converter cim, and are given by

mdim =

2

VDC0

(udim − Lω0iqim + vsdi ) (2.23)

mqim =

2

VDC0

(uqim + Lω0idim + vsqi ) (2.24)

where VDC0 is the DC-voltage across CDCI at the operating point, ω0 is the frequency at

the operating point and udim, uqim are outputs of compensators which enable tracking of

DC reference command idref , the reference for idim and idmi. Equations similar to (2.23)-

(2.24) hold for mdmi and mq

mi.

Equations (2.15), (2.16), (2.21) and (2.22) are nonlinear. To reduce notation, we

linearize around an operating point where no power is transferred through the DC line.

Note that our approach would apply the same at another operating point, simply with

more notation and the more general case (presented in Appendix A) will not affect the

results in Section 2.4.

The DC line joins two AC subsystems. We partition the state and control variables

associated with the line at the dotted vertical line shown in Fig. 2.2, i.e., those left of

the line are associated with the left AC subsystem, and those to the right with the right

AC subsystem.

The above modeling gives us the following model for the states of the DC line im at

bus i

idim =−1

τidim +

1

τViψim (2.25)


The dynamics at bus m of the DC line im are

VDCR

ICDCR

VCDC

ICDCI

VDCI

idmi

x

= Am

VDCR

ICDCR

VCDC

ICDCI

VDCI

idmi

x

+

3mdim0

4CDCR0

0 0

0 0

0 0

0 0

0 0

0 −n0PVmτ

idim

V 2DCref

(2.26)


where

x = idmi

Am =

0 T7 0 0 0 0 0

T8 T4−1

LDCR0 0 0 0

0 1CDC

0 −1CDC

0 0 0

0 0 1LDCI

T6−1LDCI

0 0

0 0 0 0 0 T5 0

0 0 0 0 0 0 1

0 0 T1 0 T2 T3−1τ

T1 =

VDC0

LDCIVmτ

T2 =VDC0(2n0pLDCI − 1)

VmLDCIτ

T3 =2CDCIn0LDCI −RDCI

LDCIτ

T4 =−RDCR

LDCR

T5 =CDCIVmVDC0

T6 =−RDCI

LDCI

T7 =−1

CDCR

T8 =1

LDCR

where mdim0 is the d-frame modulation index at the operating point.

Remark 2. Observe that in (2.25)-(2.26), the state of the bus at i, idim, affects the sub-

system at m, but no states in subsystem m affect idim. We therefore follow the convention

that im ∈ EDC , i.e., the direction of the DC line is the same as the direction of the

physical coupling induced by the VSC configurations. We choose these configurations,

and hence the directions of the DC lines, so that G ′ is a DAG. As discussed in Section


2.3.2, G ′ specifies the poset Ψsys.

Subsystem model

In this section we give several models for the subsystem consisting of the AC network and

its connections with VSCs and corresponding DC lines. We first define a generic model,

which we use for our theoretical results in Section III. We then provide two specific

examples that fit the generic model’s format.

Generic model : The model for the subsystem k is given by:

xk = Alck xk + F lck wk +Blc

k uk +∑l:lk∈E ′

Aupkl xdclk +

∑l:lk∈E ′

Bupkl u

dclk (2.27)

Here, xk contains the states of the generators in AC subsystem k and the DC line and

VSC states on the k-side of the DC line state partition (cf. Section 2.3.2). uk and wk are

the corresponding control and disturbance inputs. xdclk are the states in subsystem l that

influence subsystem k through a DC line. Similarly, udclk are the controls in subsystem l

that also influence subsystem k.

In (2.27) Aupkl , Bupkl can have non-zero entries if l is an upstream subsystem to k. Note

that DC line control variables that are partitioned into subsystem k are included in uk.

The matrices Aup and Bup depend on the DC model. The rest of the system matrices

are defined by both the model of the AC network and the DC model.

As we will see in Section 2.4, the AC network model does not affect the system’s

poset causality. Therefore, it can be made to be as complex as desired. We now give two

examples of models for the AC subsystems.

Example model 1 : The simplest model is obtained from the linearized swing equation


[41]. The dynamics of bus i are given by

Jiωi = di + pi −Hiωi −∑

ij∈EAC

bij(θi − θj) +∑

im∈EDC

Viiim +∑

mi∈EDC

Viiim (2.28)

θi = ωi (2.29)

The dynamics of iim and the DC line states are given in Section 2.3.2. Ji, Hi and bij are

the moment of inertia, damping coefficient and admittance; ωi and θi are the frequency

and voltage angle (xi in (2.27)); di the Gaussian power disturbance (wi); and pi the real

power injection at the bus i. pi is typically the output of power system stabilizers. We

remind the reader that based on the assumptions of Section 2.3.2 Vi is a constant and

therefore (2.28) is linear.

Example model 2 : A more detailed model of the AC network is the linearized multi-

machine model in Chapter 8 of [42]. This model includes the dynamics of turbine-

generator and excitation systems during transients.

The model for a subsystem k, with n buses and m (m ≤ n) generators, can be written

as follows:

x = K3x+K4Vg + Eu (2.30)

0 = K2x+K1Vg +D5Vl + F1W (2.31)

0 = D6Vg +D7Vl + F2W (2.32)


where

x = [xT1 , · · · , xTm]T (2.33)

xi = [δi ωi E′qi E

′di Efdi VRi Rfi xidc]

T (2.34)

Vg = [θ1 v1 · · · θm vm]T (2.35)

Vl = [θm+1 vm+1 · · · θn vn]T (2.36)

u = [uT1 · · · uTm ]T (2.37)

ui = [pi Vrefi uidc]T (2.38)

W = [d1 · · · dn]T (2.39)

In (2.34) δ and ω are rotor angle and frequency, E ′d and E ′q are dq-frame transient stator

voltages, Efd is the field voltage, VR and Rf are excitation system parameters and xidc

represents the states related to the DC line at bus i. v and θ are bus voltage and

phase angle respectively. p, Vref , uidc are real power injections, reference voltages for the

excitation systems and control inputs related to the DC line at bus i. Lastly, in (2.39)

di is the Gaussian power disturbance at bus i.

The model presented in (2.30)-(2.32) is similar to equations (8.49-8.51) of [42]. The

definitions of the system matrices need to be changed to include the DC model. The

extension of K3, K4 and E is straightforward using the models from Section 2.3.2. K2

must be extended to include the effect of real power transfer through the DC line (Viiim

for bus i) on the power balance equations. F1, F2 are 2m× n and 2(n−m)× n matrices

which consider the effect of disturbance at buses and are made up of zeros except at the

indices described in equations below:

F1(2i− 1, i) = 1 for i = 1, · · · ,m (2.40)

F2(2j − 1, j +m) = 1 for j = 1, · · · , n−m (2.41)


Equations (2.30)-(2.32) comprise a differential algebraic system. We can eliminate the

algebraic variables Vg and Vl using standard techniques, resulting in a purely differential

model of AC subsystem k.

2.3.3 Complete model and costs

Equations (2.25)-(2.27) comprise a full state space model of the DC-segmented power

system, where (2.27) can be any linearized AC subystem model. These equations can be

written in the same form as (2.1).

The matrices C and D in (2.2) determine the state and control costs. In our setup, we

choose C such that CTC is diagonal. The entries corresponding to mechanical states such

as ω would be larger than those corresponding to electrical states, such as VDC . Similarly,

We choose D such that DTD is diagonal and entries corresponding to electrical controls

like Ψ and V 2DCref are smaller than mechanical controls like p in (2.28), which incur more

wear and tear and fuel costs. These choices of C and D are realistic and increase the

role of the DC lines in the resulting controller.

2.4 Main Result

2.4.1 Proof of poset causality

We now show that the model (2.25)-(2.27) is poset causal and hence amenable to the

optimal decentralized control framework of [30].

Let us divide our system matrices A and B into p2 block matrices, as in Section 2.3.1,

corresponding to p AC subsystems. Recall from Section 2.3.2 that we have chosen the

converter configurations so that G ′ is a DAG, which specifies the poset Ψsys = (Psys,).

Theorem 1. The DC-segmented system model is poset causal.

Proof. Assume k 6 l for some l, k ∈ Psys. The block matrices Alk and Blk are made up


of zeroes because, irrespective of the presence or absence of a DC line between l and k,

no state and control variable associated with k affects l.

Therefore, if k 6 l, Alk and Blk are composed of zeros, and therefore A ∈ IA(Ψsys)

and B ∈ IB(Ψsys). Therefore, by Result 3, the model (2.25)-(2.27) is poset-causal.

2.4.2 Controller design procedure

The following procedure produces the decentralized controller.

1. Identify all disjoint AC systems and corresponding DC lines.

2. Assign directions to these DC lines so that the graph of DC lines and AC subsys-

tems, G ′, is a DAG.

3. Obtain the A, B, F, C and D matrices of the system following the modelling ap-

proach of Section 2.3.2.

4. Using the formulation provided in [30] obtain the decentralized controller for the

system.

The DAG of Step 2 can be obtained by totally ordering the set Psys, i.e., by as-

signing numbers 1, 2, · · · , p to the AC subsystems and pointing each edge in G ′ from

lower to higher-numbered subsystems. The number of distinct acyclic orientations are

(−1)|Psys|X (−1), where X is the chromatic polynomial of graph G ′ = (Psys, E ′) [43]. Note

that the matrices obtained in Step 3 are poset-causal by design.

2.5 Numerical example

We demonstrate our approach on a system made up of five AC subsystems. The G ′ graph

of the system along with the chosen directions of the DC lines is shown in Fig. 2.3. The

subsystems are based on the IEEE 30 bus system [44]. All DC lines are connected to the


first bus of each subsystem. The parameters of all the generators are the same and are

listed in the Table 2.1. These are taken from the NETS-NYPS 68 bus system [45]. The

DC line parameters in Table 2.2 are taken from [46]. The parameters of the VSCs in

Table 2.3 are from Chapter 8 of [39]. The values of constants in equations (2.25)-(2.26)

at the operating point are given in Table 2.4.

Figure 2.3: G ′ graph of the example

Parameter Value

X ′d 0.3373 ΩX ′q 0.2904 ΩXd 1.4278 ΩXq 1.3649 ΩTf 1 sKf 0.03Ka 40Ke 1Ta 0.02 sTe 0.785 sT ′d 6.56 sT ′q 1.5 sD 9.8M 60.4 s2

Ax 3.18e− 5Bx 1.9596

Base MVA 100

Table 2.1: Parameters of the generators

We will design the decentralized controller (call it Kdc) after modelling the AC sub-

systems with the detailed model presented in the second example of Section 2.3.2 and


Parameter Value

Length 50 KmRDCR 0.6 ΩRDCI 0.6 ΩCCDC 0.0115 FLDCR 0.0043 HLDCI 0.0043 H

Table 2.2: Parameters of the DC lines

Parameter Value

CDCR 0.0048 FCDCI 0.0048 Fτ 1 msn0 1.996p 172

Base AC Voltage 391 VBase DC Voltage 582 V

Table 2.3: Parameters of the VSCs

Parameter Value

VDC0 1 pumdim0 1

Table 2.4: Values of constants at the operating point

the DC lines with, (2.25)-(2.26). We will call this complete model as System 1. The

operating point for this linearized model is obtained from the power flow solution of the

IEEE system.

The cost matrices, Q = CTC and R = DTD, are determined as follows. Since it is less

costly to control power electronics than generation levels, the entries of R corresponding

to generation are set to 100 and for DC line control to 1. And also we represent the higher

cost of deviations in mechanical states by setting the entries of Q matrix corresponding

to generation to 1 and those of DC line to 0.01.


The structure of A and B matrices for System 1 looks as follows:

∗ 0 0 0 0

∗ ∗ 0 0 0

0 ∗ ∗ 0 0

0 ∗ 0 ∗ 0

0 0 ∗ 0 ∗

The decentralized H2-optimal controller, Kdc, calculated using the technique provided

in [30] inherits the same structure. From this structure ofKdc we can infer that controlling

a subsystem requires information (states) only from the upstream subsystems.

Table 2.5 compares the H2 norm of the system with different setups. We see that

the decentralized controller achieves essentially the same performance as the centralized

controller with DC lines, and that both improve over the case without DC lines.

Controller ||Tzw||Open loop 416.0051

Centralized without DC lines 263.1996Centralized with DC lines 260.6004

Decentralized with DC lines 260.6004

Table 2.5: Comparison of controller performances on System 1

The response of System 1 to a disturbance (Fig. 2.4) at the first bus of the first

subsystem is shown in Figs. 2.5,2.6 and 2.7.

Finally, if we were to reverse the direction of all edges in Figure 2.3 then the H2 norm

of the System 1 with a decentralized and centralized controller increased to 260.6009 and

260.6009 respectively. Therefore, the acyclic orientation affects the performance of the

decentralized controller.


Figure 2.4: Disturbance at the first bus

Figure 2.5: Generator frequency deviation at the first bus

Figure 2.6: Power flow from the subsystem 2 from 1


Figure 2.7: VDCI of the DC line joining 1 and 2

Part II

Network of energy storages

29

Chapter 3

Optimal scheduling of networked

energy storages

3.1 Motivation and scope

In this chapter, we obtain optimal scheduling policies for networked energy storages

while considering energy payments and inter-temporal price arbitrage and using inventory

control theory [47], [48]. The current work is also based on [49]. [49] obtained optimal

policies for the case of single storage with temporally uncorrelated energy imbalances and

efficient energy transactions. We extend that study to the case of networked storages.

More precisely, we consider an ideal network of serially connected storage units in which

there is no leakage in each unit, where charging and discharging processes are assumed

to be efficient and the energy imbalances are temporally uncorrelated.

The problem of obtaining optimal scheduling policies for storages has been studied

before, for example in [50], [51]. But we contribute to the literature in the following

ways: (1) We solve the problem for networked storages unlike [50] which dealt with a

similar problem for a single storage. (2) And we assume that imbalances can have arbi-

trary distributions unlike in other past works such as [51] which restricts the imbalance

30

Chapter 3. Optimal scheduling of networked energy storages 31

distribution to a normal distribution and [50] which restricts it to a uniform distribution.

3.2 Chapter organization

The chapter is organized as follows: In Section 3.3, we present the system model, and

formulate the networked energy storage problem. In Section 3.4, we explore the convexity

of the problem. In Section 3.5, we summarize the algorithm to calculate optimal policy.

We provide numerical results and engineering insights in Section 3.6.

3.3 Problem Formulation

We consider a power system consisting of N buses serially connected in chain configura-

tion, and assume that there is a central controller which controls the system transactions.

We assume that an energy imbalance (excess or deficit) at a bus incurs some penalty to

the bus. Therefore, the central controller aims to minimize the energy imbalances in

the network by allowing the buses to buy/sell (transact) energy from their neighbouring

buses. Note that the root bus does not have a successor. Hence, the root bus buys/sells

energy from the market.

We assume that there is an energy storage unit on each bus. We assume that the time

is slotted in time-slots(TS) of size δ, and that a constant energy imbalance rit is applied

over the duration of TS t at node i. The energy imbalance rit is in general random

with some known statistics. We assume that the energy imbalance rit has a continuous

probability density function pdf(rit).We assume that there is an energy deficit at node i

during TS t if rit is negative, and that there is an energy excess if it is positive.

Let sit and zit denote the state of charge (SoC) of the storage at node i at TS t and

the amount of energy the storage chooses to absorb in response to the signal rit. Let uit

denote the amount of energy that the energy storage at node i chooses to transact, and

let pt be the price of energy at time t. We assume that all nodes except the first node


transact with their immediate upstream nodes, and that the first node transacts with

market. We assume that there is an ideal energy storage with energy capacity Si at node

i, and characterize the SoC of the storage at node i with the following equation:

sit+1 = sit + uit − ui+1t + zit (3.1)

We assume that at each bus i, the energy storage supplies/absorbs energy to balance

the energy deficit/excess at bus i, and to balance the residual energy imbalance created

by its downstream buses. To illustrate this, consider bus N at the end of the network.

By conservation of energy, the energy flow from node N −1 to node N is (−rNt +zNt ), rNt

is the random energy imbalance at node N . Therefore, at bus N − 1, the energy storage

is responsible for supplying/absorbing the energy imbalance rN−1t = rN−1t − (zNt − rNt ).

We can show that the total amount of unscheduled energy seen at bus i at time t (call it

rit) is equal to

rit =N∑k=i

rkt −N∑

k=i+1

zkt (3.2)

The total amount of unscheduled energy seen at bus i can be decomposed into the random

injection/extraction at bus i, i.e., rit, plus the residual amount of random energy that the

downstream buses are unable to absorb/supply.

The model for a single bus in the network with the above assumptions is shown in

Fig. 3.1.

We assume that node i will supply/absorb as much of deficit/surplus as it can. This

rule can be represented by zit = sat(rit, sit + vit) where vit = uit − ui+1

t . Defining rit as the

amount of excess energy seen at bus i (i.e. rit ≥ 0) and yit as the amount of energy that


Figure 3.1: ith bus model

the bus currently holds (i.e., yit = sit + vit), the sat function is defined as follows:

sat(rit, yit) =

Si − yit if rit ≥ Si − yit

rit if rit ≤ Si − yit(3.3)

If rit is the amount of deficit energy seen at bus i (i.e., rit < 0) and yit is as above, the

energy that is provided to satisfy the demand by bus i is as follows:

sat(rit, yit) =

rit if rit ≥ −yit

−yit if rit ≤ −yit(3.4)

Our energy management rule at node i can be represented by the following function:

zit = maxminrit, Si − (sit + vit),−(sit + vit) (3.5)

Our objective is to minimize the expected value of the residual energy imbalance at

each node i by controlling energy transacted at each node of the network. To do so, we

formulate a finite horizon stochastic control problem as follows:

minuit

E

[N∑i=1

∑t

git(rit, s

it + vit) + ptv

it

](3.6)

s.t. sit+1 = sit + vit + sat(rit, sit + vit)


where pt is the price of energy at time t and the penalty function git represents portion of

the random energy imbalance not captured by the storage at node i during time-slot t.

The saturation function ensures that the state of charge (sit) remains in the range [0, Si].

In this study, we choose the following penalty function:

git(rit, z

it) =

∣∣rit − sat(rit, sit + vit)∣∣ (3.7)

As the number of time-slots increases, the above minimization increases in complexity.

This complexity can be reduced by optimizing in stages using Dynamic Programming

(DP), which promises to provide the same optimum values as solving the original problem

[47]. Using the DP techniques, the dynamic programming recursion can be written as

follows:

Jt(st) = minvit

E

[∑i

git(rit, s

it + vit

)+ pitv

it + Jt+1 (st + vt + sat(rt, st + vt))

](3.8)

s.t. 0 ≤ (vit + sit) ≤ Si ∀i

where st and vt are vectors whose elements are sit’s and vit’s, respectively. After applying

the inventory control substitution yit = sit + vit, we will have

Jt(st) = minyt

E

[∑i

git(rit, y

it

)+ pity

it + Jt+1 (yt + sat(rt, yt))

]−∑i

pitsit (3.9)

s.t. 0 ≤ yt ≤ S

where S ,yt and st are vectors whose elements are Si’s, yit’s and sit’s, respectively. Let us

define the function Gt as follows:

Gt(yt) =E

[∑i

git(rit, y

it

)+ pity

it + Jt+1 (yt + sat(rt, yt))

](3.10)


We can write the value function and corresponding optimal control policy as follows:

Jt(st) = −∑i

pitsit +Gt(y

?t ) (3.11)

v?t (st) = −st + y?t (3.12)

where y?t is the optimal solution to the following optimization problem:

Pt : minyt

Gt(yt)

s.t. 0 ≤ yt ≤ S

Remark 3. The central controller can solve Pt, and send the optimal solution yit to node

i. Each node i can then calculate vit on its own, independent of other nodes which can

then be added together by the controller to compute ut, i.e., the optimal control policy

is naturally decentralized.

Pt is a nonlinear program which can be solved globally optimally if it is convex. In

the next section we will provide sufficient conditions under which the function Gt(yt) is

convex.

3.4 Convexity of the function Gt

In order to compute Gt(yt) iteratively we assume that the cost of the final stage JN(yN) is

zero. Then, at each step t, we can be assured that Gt+1(y?t+1) has already been computed

in the previous step. Now, under these assumptions the following result provides sufficient

conditions under which the function Gt(yt) is convex. A sketch of the proof is provided

in the Appendix C.

Theorem 2. The function Gt(yt) is convex if the price pt is zero and the amount of

random energy seen at each bus i in TS t is of the same sign.


In general, the function Gt(yt) is not convex. In particular, when there is an energy

deficit at some buses and an energy excess at some others in a TS t, the functionGt(yt) can

be non-convex. To illustrate this, let us consider a 2-bus system with S1 = 5 and S2 = 5.

Let us assume that the amount of energy imbalance at bus 1 and bus 2 equals r1 = 2, r2 =

−3 for some t and pt = pt+1 = 0. We can easily verify that [∑

i git (rit, y

it) + Gt+1

(y?t+1

)]

is not convex in this case. Hence, the function Gt(yt) is not convex in general.

3.5 Computing the optimal policy

Our goal is to compute the optimal control policy in (3.11)-(3.12). To do so, assume the

cost of the final stage JN(yN) is zero. Then beginning from the stage N − 1 compute

the function Gt(yt) iteratively using the probability distribution function of the energy

imbalance rit for all i and the function Gt+1(y?t+1) of the previous stage. If the resulting

Gt(yt) is convex then one can solve the problem Pt to obtain the optimal policy.

3.6 Numerical Results

We consider a two-bus system (i.e., N = 2), and assume that each bus is connected to

a wind farm and an energy storage unit. We take the storage capacities to be equal to

S1 = 5, S2 = 6 units, and assume that the distribution line capacity is infinite. We

focus on a period of length 24 hours, and assume that the time is slotted in time-slot of

size δ = 1 hour. Let us define rit = pi(t) − ì(t), where ì(t) denotes the deterministic

load profile and pi(t) is a random variable representing the power generated by wind

farm at bus i and at time period t. For simplicity, we assume that each bus i has a

constant load which can be the average load estimated from the historical data, i.e.,

ì(t) = `. The random energy seen at bus i at time-slot t (i.e., pi(t)) is assumed to have a

truncated normal distribution with the means (µit) from the Fig. 3.2 and having non-zero

probability from µit − 3 ∗ σit to µit + 3 ∗ σit. The rest of the parameters are pit = 0 ∀t, i,


standard deviation(σit)of rit = 0.3 units ∀i, t and the mean of rit ∀i, t is plotted in the Fig.

3.2.

To compute the expectation of the function Gt, we have used the integral2 and norm-

pdf functions of Matlab with the above assumed means and variances. We have also used

the function fmincon of Matlab in order to obtain optimal solutions to the minimization

problem Pt of section 3.3.

The results from the Fig. 3.2 show that larger yit is needed in the times of high

demand while it reduces when we have enough renewable energy. Recall that yt (the

variable from the problem Pt of section 3.3) is the sum of state of charge and energy

bought from the market. Thus in the periods of high demand a larger y implies that for

a given state of charge a larger amount should be bought to satisfy the demand which

results in reduced penalty on energy imbalance. When there is surplus renewable energy

(rit ≥ 0), the policy requires smaller amount to be bought from the market. This in turn

will help the absorption of the excess energy reducing the penalty. The results show that

when there is energy in excess of capacity at bus 2, the storage at bus 1 tries to absorb

the excess energy, as can be seen from the dip in the value of y1 which allows extra space

in storage at bus 1. This behaviour also shows the effect of battery capacity on optimal

policies.

And when prices pt are zero, the minimization at each TS t is decoupled from the

next TS, t+ 1. Then, as long as a policy allows enough remaining capacity for absorbing

the current surplus or enough energy to supply the current demand, it will be optimal.

Thus, the optimal policy is not necessarily unique in this case.


2 4 6 8 10 12 14 16 18 20 22 24

−5

0

5

10

Hour of the day

En

erg

y

r1tr2ty1

t

y2

t

Figure 3.2: The resultant policy (y1, y2) in a two bus system after solving Pt

Part III

Conclusion

39

Chapter 4

Conclusion and future work

In the first part of this thesis we have shown that a DC-segmented system can be modelled

to be poset-causal if we assume that the DC lines are lossless. We have also presented

an algorithm to design a decentralized controller based on its poset-causal structure. For

an example system with the lossless assumption, we have compared the performance of

the decentralized and centralized controller and found them to be comparable. We found

that even after relaxing the lossless assumption the decentralized controller provides an

acceptable performance.

In the second part we proposed a stochastic formulation for the optimal scheduling

problem in a serial network and using dynamic programming recursion arrived at an

optimal policy. We explored the convexity of the problem of computing the optimal

policy and found that it is not convex in general. We also found some sufficient conditions

under which it is convex.

In our future work related to DC-segmented systems we wish to provide a method

to obtain the optimal acyclic orientation for their DAGs. And in the work related to

energy storages we wish to consider the impact of charging and discharging power limits,

transmission line capacity limits, power flow limits and other network topologies while

calculating optimal schedules.

40

Appendix A

Linearization of the DC line model

The complete linearized model for the DC lines is given by following equations

idim = −idim

τ+ψimτVi

(A.1)

iqim = −iqim

τ+Qimref

τVi(A.2)

udim = (kimi −kimpτ

)(ψimVi− idim) (A.3)

uqim = (kimi −kimpτ

)(Qimref

Vi− iqim) (A.4)

VDCR = idim(3md

im0

4CDCR− 6iqim0Lω0

4VDC0CDCR) + iqim(

3mqim0

4CDCR− 6idim0Lω0

4VDC0CDCR) +

+udim(6idim0

4CDCRVDC0

) + uqim(6iqim0

4CDCRVDC0

) + vsdi (6idim0

4VDC0CDCR) +

+vsqi (6iqim0

4VDC0CDCR)− ICDCR

CDCR(A.5)

ICDCR =VDCR − VCDC −RDCRICDCR

LDCR(A.6)

VCDC =ICDCR − ICDCI

CCDC(A.7)

ICDCI =VCDC −RDCIICDCI − VDCI

LDCI(A.8)

VDCI =CDCIVmi

dmi

VDC0

(A.9)

idmi = x (A.10)

41

Appendix A: Linearization of the DC line model 42

x = −x1

τ+idmiτ

(2CDCIn0 −RDCI

LDCI+CDCIICDCI0

VDC0

) +

VDCIτ

(2n0pVDC0

Vm− 2VDC0

VmLDCI+

VCDC0

VmLDCI)− V 2

DCref (n0p

Vmτ) +

ICDCI(CDCIi

dmi0

VDC0τ) + VCDC(

VDC0

LDCIVmτ) (A.11)

iqmi = −iqmi

τ+Qmiref

τVm(A.12)

udmi = (τkmii − kmip )x (A.13)

uqmi = (kmii −kmipτ

)(Qmiref

Vm− iqmi) (A.14)

where kimi , kimp and kmii , kmip are proportional and integral constants for the compensators

tracking reference currents at cim and cmi respectively; Qimref and Qmiref are reference

commands for the reactive power supplied to the buses i and m. As was said in Section

2.3.2, the operating point for linearization is chosen so that no power is flowing through

the DC lines. Also note that no reactive power is being supplied to the buses i and m

(Qimref = Qmiref = 0). The values of constants in the above equations (variables with a

zero in subscript, except n0) under these assumptions are given in Table A.1. Because

of our assumptions the variables iqim, udim, vsdi , vsqi , uqim, iqmi, udmi and uqmi are either

unchanging or do not affect the system dynamics. And for the purpose of simplification

their dynamics will not be included in the DC-model (2.25)-(2.26).

Parameter Value

idim0 0 puiqim0 0 puVi 1 pumqim0 0

mdim0 1

VDC0 1 puICDCI0 0 puidmi0 0 puiqmi0 0 puVm 1 pu

Table A.1: Values of constants at the operating point

Appendix A: Linearization of the DC line model 43

And we will obtain the following simplified linear equations after ignoring the losses

of the DC line:

idim = −idim

τ+ψimτVi

(A.15)

iqim = −iqim

τ−iqimrefτ

(A.16)

VDC = −idmi(Vm

VDC0CDC) (A.17)

idmi = x (A.18)

x = −x1

τ+ idim

ViVmτ 2

− ψim1

Vmτ 2− idmi

2n0

CDCτ+

+VDC2VDC0n0p

Vmτ− V 2

DCref

n0p

Vmτ(A.19)

Appendix B

Calculation of the decentralized

controller

In this appendix we will summarize the technique presented in [30] to design the controller

for a poset-causal system.

B.1 Notation

We introduce some new notation related to matrices and subsystems. For the system

(2.1)-(2.2) and a matrix M:

• M(i) represents a new matrix constructed with the ith subsystem’s columns of

matrix M .

• M(↓i) represents a new matrix [M(k)]k∈↓i.

• M(↓i, ↓i) is a matrix made up of exactly the rows and columns of subsystems in

the set ↓i.

44

Appendix B: Calculation of the decentralized controller 45

• Ei is a tall block matrix with an identity at ith block:

Ei =

0

0

I

0

0

. (B.1)

• For given matrices M(↓i, ↓i) define diag(M(↓i, ↓i)) as:

diag(M(↓i, ↓i)) =

M(↓1, ↓1)

M(↓2, ↓2)

.

.

M(↓p, ↓p)

. (B.2)

B.2 Components

The controller will be dynamic and its design requires each subsystem to keep track of its

downstream subsystems. We will denote this knowledge about downstream subsystems

as additional states which will be called controller states.

For a subsystem i, we remind the reader that, ni is the number of states. Let us define

↓↓i as a set containing all elements of ↓i except i itself. Then, let n↓↓i =∑

j∈↓↓i nj. We

can define q(i) ∈ <n↓↓i to be the controller states of i which keep track of its downstream

subsystems. And also for a j ∈ ↓↓i, qj(i) ∈ <nj , a component of q(i), will track the


subsystem j. Let q be the combination of such controller states from all subsystems:

q =

q(1)

q(2)

...

q(p)

(B.3)

Now define a new vector v which will be a combination of x and q. It is defined as

follows:

v =

x1

q(1)

x2

q(2)

...

xp

q(p)

(B.4)

Construct new matrices πq and πx made up of 0s and 1s to select q and x from v

respectively:

πxv =

x1

x2...

xp

(B.5)

πqv =

q(1)

q(2)

...

q(p)

(B.6)


Define a matrix Σ whose function is to add up the state of a subsystem, xi, and the

predictions about xi at its upstream subsystems:

Σv =

x1 +∑

1∈↓↓k q1(k)

x2 +∑

2∈↓↓k q2(k)

...

xp +∑

p∈↓↓k qp(k)

(B.7)

B.3 Controller

Solving for the decentralized controller involves calculation of centralized controllers for

each subsystem. For example, for i ∈ Psys the new system matrix to be used to calculate

the centralized controller is A(↓i, ↓i). We calculate the centralized controllers using the

familiar Algebraic Riccati Equation (ARE):

A(↓i, ↓i)TXi +XiA(↓i, ↓i) (B.8)

−XiB(↓i, ↓i)R(↓i, ↓i)−1B(↓i, ↓i)TXi+ (B.9)

Q(↓i, ↓i) = 0 (B.10)

K(↓i, ↓i) = R(↓i, ↓i)−1B(↓i, ↓i)TXi (B.11)

Where K(↓i, ↓i) is the required centralized controller related to i. Now define matrices

A, K, Aφ and Bφ as follows:

A = diag(A(↓i, ↓i)−B(↓i, ↓i)K(↓i, ↓i)) (B.12)

K = diag(K(↓i, ↓i)) (B.13)

Aφ = πqAπTq (B.14)

Bφ = πqAπTx (B.15)


The H2-norm of Tzw can then be calculated as:

||Tzw|| =∑i∈Psys

√trace((EiF (i, i))TXi(EiF (i, i))) (B.16)

And the dynamic decentralized controller can be written as:

K∗ =

Aφ −BφΣπTq Bφ

−ΣK(πTq − πTx ΣπTq ) −ΣKπTx

(B.17)

Appendix C

Proof of Theorem 2

We need the following lemmas for the proof.

Lemma 1. Given y ∈ [0, S] and α ≥ 0, r ≥ 0, we have:

r + α− Sat(r + α, y) ≥ r − Sat(r, y)

Proof. Define

LHS = r + α− Sat(r + α, y)

RHS = r − Sat(r, y)

Suppose α + r ≤ S − y, that implies r ≤ S − y and LHS = 0, then

RHS = 0

= LHS

Now, suppose α + r ≥ S − y, that implies LHS = r + α− S + y.

49

Appendix C: Proof of Theorem 2 50

• Now with this supposition consider a case of r ≥ S − y, then

RHS = r − S + y

≤ LHS

• Now, the other case of r ≤ S − y

RHS = 0

≤ LHS

Hence proved

Lemma 2. Given 0 ≤ a, b, x, y ≤ S, r ≥ 0, and 0 ≤ θ ≤ 1, we have:

Sat(r − θa− (1− θ)b, θy + (1− θ)x)

≥θSat(r − a, y) + (1− θ)Sat(r − b, x)

Proof. Define

LHS = Sat(r − θa− (1− θ)b, θy + (1− θ)x)

RHS = θSat(r − a, y) + (1− θ)Sat(r − b, x)

1. Suppose a ≥ b and y ≥ x.

If r − θa− (1− θ)b ≥ S − (θy + (1− θ)x), then

r − b ≥ S − y

LHS = S − (θy + (1− θ)x)


• Now consider the case of r − a ≥ S − y, r − b ≥ S − x, then

RHS = θ(S − y) + (1− θ)(S − x)

= LHS

• Consider the case of r − a ≤ S − y, r − b ≤ S − x,then

RHS = θ(r − a) + (1− θ)(r − b)

≤ LHS

• Consider the next case of r − a ≥ S − y, r − b ≤ S − x, then

RHS = θ(S − y) + (1− θ)(r − b)

≤ LHS

• Consider the final case of r − a ≤ S − y, r − b ≥ S − x, then

RHS = θ(r − a) + (1− θ)(S − x)

≤ LHS

And if r − θa− (1− θ)b ≤ S − (θy + (1− θ)x), then

r − a ≤ S − x

LHS = r − θa− (1− θ)b



RHS = θ(S − y) + (1− θ)(S − x)

≤ LHS


RHS = θ(r − a) + (1− θ)(r − b)

= LHS


RHS = θ(S − y) + (1− θ)(r − b)

≤ LHS


RHS = θ(r − a) + (1− θ)(S − x)

≤ LHS

Thus for the supposition a ≥ b and y ≥ x, lemma is proved.

2. Now suppose a ≤ b and y ≥ x.


r − a ≥ S − y

LHS = S − (θy + (1− θ)x)


• Now consider the case of r − b ≥ S − x, then

RHS = θ(S − y) + (1− θ)(S − x)

= LHS

• Consider the final case of r − b ≤ S − x, then

RHS = θ(S − y) + (1− θ)(r − b)

≤ LHS

And now if r − θa− (1− θ)b ≤ S − (θy + (1− θ)x), then

r − b ≤ S − x

LHS = r − θa− (1− θ)b

• Now consider the case of r − a ≥ S − y, then

RHS = θ(S − y) + (1− θ)(r − b)

≤ LHS

• Consider the final case of r − a ≤ S − y, then

RHS = θ(r − b) + (1− θ)(r − b)

= LHS

Thus for the supposition a ≤ b and y ≥ x, lemma is proved.

3. Suppose a ≤ b and y ≤ x.



r − a ≥ S − x

LHS = S − (θy + (1− θ)x)


RHS = θ(S − y) + (1− θ)(S − x)

= LHS


RHS = θ(r − a) + (1− θ)(r − b)

≤ LHS


RHS = θ(S − y) + (1− θ)(r − b)

≤ LHS


RHS = θ(r − a) + (1− θ)(S − x)

≤ LHS



r − b ≤ S − y

LHS = r − θa− (1− θ)b


RHS = θ(S − y) + (1− θ)(S − x)

≤ LHS


RHS = θ(r − a) + (1− θ)(r − b)

= LHS


RHS = θ(S − y) + (1− θ)(r − b)

≤ LHS


RHS = θ(r − a) + (1− θ)(S − x)

≤ LHS

Thus for the supposition a ≤ b and y ≤ x, lemma is proved.

4. Now suppose a ≥ b and y ≤ x.



r − b ≥ S − x

LHS = S − (θy + (1− θ)x)

• Now consider the case of r − a ≥ S − y, then

RHS = θ(S − y) + (1− θ)(S − x)

= LHS

• Consider the final case of r − a ≤ S − y, then

RHS = θ(r − a) + (1− θ)(S − x)

≤ LHS


r − a ≤ S − y

LHS = r − θa− (1− θ)b

• Now consider the case of r − b ≥ S − x, then

RHS = θ(r − a) + (1− θ)(S − x)

≤ LHS


• Consider the final case of r − b ≤ S − x, then

RHS = θ(r − b) + (1− θ)(r − b)

= LHS

Thus for the supposition a ≥ b and y ≤ x, lemma is proved.

Hence proved.

Lemma 3. Given y ∈ [0, S], α ≥ 0, and r ≤ 0, we have:

−r + α + Sat(r − α, y) ≥ r − Sat(r, y)

Proof. The proof is similar to that of Lemma 1.

Lemma 4. Given −S ≤ a, b ≤ 0, 0 ≤ x, y ≤ S, r ≤ 0, and 0 ≤ θ ≤ 1, we have:

Sat(r − θa− (1−θ)b, θy + (1− θ)x)

≤θSat(r − a, y) + (1− θ)Sat(r − b, x)

Proof. The proof is similar to that of Lemma 2

Proof of Theorem 2: Let yt and xt be column vectors such that

0 ≤ yit, xit ≤ S ∀i ∈ 1, 2, · · · , n

Since the price pt = 0 for all t, we have:

Gt(yt) = E

[∑i

git

(rit

(yK

i

t

), yit

)]+K

where K is a constant. To show that the function Gt is convex, we only need to show


that each function git is convex. For convenience the subscript t representing time-slot is

not shown in the following.

Consider the bus at the leaf node (i.e., i = 1). At this node, we have: r = r. From

Lemma 2 and Lemma 4, we can show that

g1(r1, θy1 + (1− θ)x1) ≤ θg1(r1, y1) + (1− θ)g1(r1, x1)

Therefore, g1 is convex. Now, suppose gk(rk, yk) is convex. Using Lemmas 1,3 and 4, we

find that

gk+1(rk+1, θyk+1 + (1− θ)xk+1) ≤ θgk+1(rk+1, yk+1)+

+ (1− θ)gk+1(rk+1, xk+1)

Thus, gk+1(rk+1, yk+1) is also convex. By induction, we can show that functions gi(ri, yi)∀i ∈

1, · · · , n are convex. Since Gt is an expectation over the sum of convex functions, Gt is

convex. This completes the proof.

Bibliography

[1] M. P. Bahrman and B. K. Johnson, “The ABCs of HVDC transmission technologies,”

IEEE Power and Energy Magazine, vol. 5, no. 2, pp. 32–44, March 2007.

[2] N. Flourentzou, V. G. Agelidis, and G. D. Demetriades, “VSC-Based HVDC Power

Transmission Systems: An Overview,” IEEE Transactions on Power Electronics,

vol. 24, no. 3, pp. 592–602, March 2009.

[3] P. Kundur, N. J. Balu, and M. G. Lauby, Power system stability and control.

McGraw-hill New York, 1994.

[4] J. Arrillaga, High voltage direct current transmission. IET, 1998, no. 29.

[5] T. Smed and G. Andersson, “Utilizing HVDC to damp power oscillations,” IEEE

Transactions on Power Delivery, vol. 8, no. 2, pp. 620–627, Apr 1993.

[6] H. Clark, A. a. Edris, M. El-Gasseir, K. Epp, A. Isaacs, and D. Woodford, “Softening

the Blow of Disturbances,” IEEE Power and Energy Magazine, vol. 6, no. 1, pp. 30–

41, January 2008.

[7] S. P. Azad, R. Iravani, and J. E. Tate, “Damping Inter-Area Oscillations Based

on a Model Predictive Control (MPC) HVDC Supplementary Controller,” IEEE

Transactions on Power Systems, vol. 28, no. 3, pp. 3174–3183, Aug 2013.

[8] “International energy outlook 2016,” http://www.eia.gov/forecasts/ieo/.

59

http://www.eia.gov/forecasts/ieo/

Bibliography 60

[9] “Western wind and solar integration study,” http://eioc.pnnl.gov/benchmark/

ieeess/NETS68/base.htm, May 2010.

[10] M. Milligan and B. Kirby, “Utilizing load response for wind and solar integration

and power system reliability,” in Wind Power Conference, Dallas, Texas, 2010.

[11] P. Denholm, E. Ela, B. Kirby, and M. Milligan, “The role of energy storage with

renewable electricity generation,” 2010.

[12] E. Bitar, R. Rajagopal, P. Khargonekar, and K. Poolla, “The role of co-located

storage for wind power producers in conventional electricity markets,” in Proceedings

of the 2011 American Control Conference. IEEE, 2011, pp. 3886–3891.

[13] H. I. Su and A. E. Gamal, “Modeling and Analysis of the Role of Energy Storage for

Renewable Integration: Power Balancing,” IEEE Transactions on Power Systems,

vol. 28, no. 4, pp. 4109–4117, Nov 2013.

[14] A. Castillo and D. F. Gayme, “Grid-scale energy storage applications in renewable

energy integration: A survey,” Energy Conversion and Management, vol. 87, pp.

885 – 894, 2014. [Online]. Available: http://www.sciencedirect.com/science/article/

pii/S0196890414007018

[15] J. M. Carrasco, L. G. Franquelo, J. T. Bialasiewicz, E. Galvan, R. C. PortilloGu-

isado, M. A. M. Prats, J. I. Leon, and N. Moreno-Alfonso, “Power-Electronic Sys-

tems for the Grid Integration of Renewable Energy Sources: A Survey,” IEEE Trans-

actions on Industrial Electronics, vol. 53, no. 4, pp. 1002–1016, June 2006.

[16] H. Ibrahim, A. Ilinca, and J. Perron, “Energy storage systems characteristics and

comparisons,” Renewable and Sustainable Energy Reviews, vol. 12, no. 5, pp. 1221 –

1250, 2008. [Online]. Available: http://www.sciencedirect.com/science/article/pii/

S1364032107000238

http://eioc.pnnl.gov/benchmark/ieeess/NETS68/base.htm

http://eioc.pnnl.gov/benchmark/ieeess/NETS68/base.htm

http://www.sciencedirect.com/science/article/pii/S0196890414007018




Bibliography 61

[17] I. Kamwa, R. Grondin, and Y. Hebert, “Wide-area measurement based stabilizing

control of large power systems-a decentralized/hierarchical approach,” IEEE Trans-

actions on Power Systems, vol. 16, no. 1, pp. 136–153, Feb 2001.

[18] B. Chaudhuri, R. Majumder, and B. C. Pal, “Wide-area measurement-based stabi-

lizing control of power system considering signal transmission delay,” IEEE Trans-

actions on Power Systems, vol. 19, no. 4, pp. 1971–1979, Nov 2004.

[19] F. Dorfler, M. R. Jovanovic, M. Chertkov, and F. Bullo, “Sparsity-Promoting Opti-

mal Wide-Area Control of Power Networks,” IEEE Transactions on Power Systems,

vol. 29, no. 5, pp. 2281–2291, Sept 2014.

[20] N. Sandell, P. Varaiya, M. Athans, and M. Safonov, “Survey of decentralized control

methods for large scale systems,” IEEE Transactions on Automatic Control, vol. 23,

no. 2, pp. 108–128, Apr 1978.

[21] V. D. Blondel and J. N. Tsitsiklis, “A survey of computational complexity results

in systems and control,” Automatica, vol. 36, no. 9, pp. 1249 – 1274, 2000.

[22] H. S. Witsenhausen, “A Counterexample in Stochastic Optimum Control,” SIAM

Journal on Control, vol. 6, no. 1, pp. 131–147, 1968. [Online]. Available:

http://dx.doi.org/10.1137/0306011

[23] S. Mitter and A. Sahai, Information and control: Witsenhausen revisited.

London: Springer London, 1999, pp. 281–293. [Online]. Available: http:

//dx.doi.org/10.1007/BFb0109735

[24] C. H. Papadimitriou and J. Tsitsiklis, “Intractable Problems in Control Theory,”

SIAM Journal on Control and Optimization, vol. 24, no. 4, pp. 639–654, 1986.

[Online]. Available: http://dx.doi.org/10.1137/0324038

http://dx.doi.org/10.1137/0306011

http://dx.doi.org/10.1007/BFb0109735

http://dx.doi.org/10.1007/BFb0109735

http://dx.doi.org/10.1137/0324038

Bibliography 62

[25] V. Blondel and J. N. Tsitsiklis, “NP-hardness of some linear control design prob-

lems,” in Decision and Control, 1995., Proceedings of the 34th IEEE Conference on,

vol. 3, Dec 1995, pp. 2910–2915 vol.3.

[26] C. H. Papadimitriou and J. Tsitsiklis, “On the complexity of designing distributed

protocols,” Information and Control, vol. 53, no. 3, pp. 211 – 218, 1982. [Online].

Available: http://www.sciencedirect.com/science/article/pii/S0019995882910348

[27] M. Rotkowitz and S. Lall, “A characterization of convex problems in decentralized

control,” IEEE Transactions on Automatic Control, vol. 50, no. 12, pp. 1984–1996,

Dec 2005.

[28] P. Shah and P. A. Parrilo, “A Partial order approach to decentralized control,” in

Decision and Control, 2008. CDC 2008. 47th IEEE Conference on, Dec 2008, pp.

4351–4356.

[29] Y.-C. Ho and K.-C. Chu, “Team decision theory and information structures in op-

timal control problems–Part I,” IEEE Transactions on Automatic Control, vol. 17,

no. 1, pp. 15–22, Feb 1972.

[30] P. Shah and P. A. Parrilo, “H2-optimal decentralized control over posets: A state

space solution for state-feedback,” in 49th IEEE Conference on Decision and Control

(CDC), Dec 2010, pp. 6722–6727.

[31] S. P. Azad, J. A. Taylor, and R. Iravani, “Decentralized Supplementary Control of

Multiple LCC-HVDC Links,” IEEE Transactions on Power Systems, vol. 31, no. 1,

pp. 572–580, Jan 2016.

[32] M. Andreasson, D. V. Dimarogonas, H. Sandberg, and K. H. Johansson, “Dis-

tributed Controllers for Multi-Terminal HVDC Transmission Systems,” IEEE Trans-

actions on Control of Network Systems, vol. PP, no. 99, pp. 1–1, 2016.


Bibliography 63

[33] J. A. Taylor and L. Scardovi, “Decentralized control of DC-segmented power sys-

tems,” in Communication, Control, and Computing, 52nd Annual Allerton Confer-

ence on, Sept 2014, pp. 1046–1050.

[34] J. Pan, R. Nuqui, K. Srivastava, T. Jonsson, P. Holmberg, and Y. J. Hafner, “AC

Grid with Embedded VSC-HVDC for Secure and Efficient Power Delivery,” in En-

ergy 2030 Conference, Nov 2008, pp. 1–6.

[35] M. Aigner, Combinatorial theory. Springer Science & Business Media, 2012.

[36] C. Godsil and G. F. Royle, Algebraic graph theory. Springer Science & Business

Media, 2013.

[37] J. G. Oxley, Matroid theory. Oxford University Press, USA, 2006.

[38] K. Zhou, J. C. Doyle, and K. Glover, Robust and optimal control. Prentice Hall

New Jersey, 1996.

[39] A. Yazdani and R. Iravani, Voltage-sourced converters in power systems: modeling,

control, and applications. John Wiley & Sons, 2010.

[40] D. Jovcic, L. A. Lamont, and L. Xu, “VSC transmission model for analytical stud-

ies,” in Power Engineering Society General Meeting, 2003, IEEE, vol. 3, July 2003.

[41] A. J. Wood and B. F. Wollenberg, Power generation, operation, and control. John

Wiley & Sons, 2012.

[42] P. W. Sauer and M. Pai, Power system dynamics and stability. Prentice Hall New

Jersey, 1997.

[43] R. P. Stanley, “Acyclic orientations of graphs,” Discrete Mathematics, vol. 5, no. 2,

pp. 171 – 178, 1973. [Online]. Available: http://www.sciencedirect.com/science/

article/pii/0012365X73901088

http://www.sciencedirect.com/science/article/pii/0012365X73901088

http://www.sciencedirect.com/science/article/pii/0012365X73901088

Bibliography 64

[44] “IEEE 30 bus test system,” http://www.ee.washington.edu/research/pstca/pf30/

pg tca30bus.htm.

[45] B. C. Pal and A. K. Singh, “IEEE PES Task Force on Benchmark Systems for

Stability Controls–Report on the 68-Bus, 16-Machine, 5-Area System,” 2013.

[46] N. B. Negra, J. Todorovic, and T. Ackermann, “Loss evaluation of HVAC and HVDC

transmission solutions for large offshore wind farms,” Electric Power Systems Re-

search, vol. 76, no. 11, pp. 916 – 927, 2006.

[47] D. P. Bertsekas, Dynamic programming and optimal control. Athena Scientific

Belmont, MA, 1995, vol. 1, no. 2.

[48] P. H. Zipkin, Foundations of inventory management. McGraw Hill, New York,

2000.

[49] J. A. Taylor, D. S. Callaway, and K. Poolla, “Competitive energy storage in the

presence of renewables,” IEEE Transactions on Power Systems, vol. 28, no. 2, pp.

985–996, May 2013.

[50] J. H. Kim and W. B. Powell, “Optimal energy commitments with storage and in-

termittent supply,” Operations research, vol. 59, no. 6, pp. 1347–1360, 2011.

[51] J. Qin and R. Rajagopal, “Dynamic programming solution to distributed storage

operation and design,” in Power and Energy Society General Meeting (PES), 2013,

pp. 1–5.

http://www.ee.washington.edu/research/pstca/pf30/pg_tca30bus.htm

http://www.ee.washington.edu/research/pstca/pf30/pg_tca30bus.htm

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