DEGREE PROJECT, IN , SECOND LEVELELECTRIC POWER SYSTEMS

STOCKHOLM, SWEDEN 2015

On Accuracy of Conic Optimal PowerFlow

SHAN HUANG

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ELECTRICAL ENGINEERING

KTH ROYAL INSTITUTE OF TECHNOLOGY

Master Thesis

On Accuracy of Conic Optimal PowerFlow

Author:

Shan Huang

Supervisor:

Dr. Mohamadrez Baradar

Examiner:

Dr. Mohammad, Reza Hesamzadeh

Associate Professor

A thesis submitted in fulfilment of the requirements

for the degree of Master of Science

in the

Electricity Market Research Group(EMReG)

SCHOOL OF ELECTRICAL ENGINEERING

June 2015

Declaration of Authorship

I, Shan Huang, declare that this thesis titled, ’On Accuracy of Conic Optimal Power

Flow’ and the work presented in it are my own. I confirm that:

⌅ This work was done wholly or mainly while in candidature for a research degree

at this University.

⌅ Where any part of this thesis has previously been submitted for a degree or any

other qualification at this University or any other institution, this has been clearly

stated.

⌅ Where I have consulted the published work of others, this is always clearly at-

tributed.

⌅ Where I have quoted from the work of others, the source is always given. With

the exception of such quotations, this thesis is entirely my own work.

⌅ I have acknowledged all main sources of help.

⌅ Where the thesis is based on work done by myself jointly with others, I have made

clear exactly what was done by others and what I have contributed myself.

Signed:

Date:

i

KTH ROYAL INSTITUTE OF TECHNOLOGY

Abstract

Electricity Market Research Group(EMReG)

SCHOOL OF ELECTRICAL ENGINEERING

Master of Science

On Accuracy of Conic Optimal Power Flow

by Shan Huang

Nowadays, with the increasing need for security and economic operation of power sys-

tems, the optimal power flow (OPF) has been an essential and predominant tool for

economic and operation planning of power systems.

A developed conic formulation for OPF has been proposed [1]. This model is bases on

the line based flow equations and can be transformed into the form of second order cone

programming (SOCP). The SOCP formulation of OPF problem can be solved using

interior point methods (IPMs) [2].

In this thesis, a study on the performance of this developed conic formulation for OPF

is carried out. Firstly, the accuracy of SOCP formulation is studied. A more accurate

model is developed. The model is obtained by modifying the phase angle constrains

of SOCP formulation. The modified model can be solved using sequential conic pro-

gramming method. A comparison of results from these two models is made on di↵erent

test systems. Secondly, the SOCP formulation is applied to both small and large test

systems. The results of SOCP formulation is compared with the results from PSS/E

OPF. The performance of SOCP formulation has shown the accurate and e↵ectiveness

for solving OPF problems.

Acknowledgements

I would like to thank first Professor Mohammad Reza Hesamzadeh, for holding this

master project and allowing me to carry out with this project.

I also would like to thank my supervisor Dr. Mohamadreza Baradar, for his guidance

and patience during such long time throughout the whole project. Without his help and

advice, I could not finish this project.

Finally, I would like to thank my family for their love and support. Especially thanks

to my husband Ruoxi who encouraged me all the time.

iii

Contents

Declaration of Authorship i

Abstract ii

Acknowledgements iii

Contents iv

List of Figures vi

List of Tables vii

Abbreviations viii

1 Introduction 1

1.1 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Formulation of OPF problem . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 AC power flow equations . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 DC power flow equations . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 OPF methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Aim of the project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Second Order Cone Formulation for OPF 7

2.1 Line-flow based equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Second-order cone programming . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Optimal power flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Study on accuracy of SOCP OPF formulation 14

3.1 Modified phase angle di↵erence constrains . . . . . . . . . . . . . . . . . . 14

3.2 Optimal power flow and sequential conic programming . . . . . . . . . . . 15

3.3 Test results and comparison with SOCP OPF results . . . . . . . . . . . . 16

4 Simulation results and comparison with PSS/E 25

4.1 PSS/E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

iv

Contents v

5 Conclusion and further work 30

5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Bibliography 32

List of Figures

2.1 Equivalent circuit of a branch . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 Flow chart of solving the modified model . . . . . . . . . . . . . . . . . . 17

3.2 Results of voltage magnitudes of IEEE 30-bus under normal load oper-ation from MATPOWER (STRD), SOCP (SOCP) and modified model(SCP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Results of voltage magnitudes of IEEE 30-bus under high load operationfrom MATPOWER (STRD), SOCP(SOCP) and modified model(SCP) . . 18

3.4 Results of voltage magnitudes of IEEE 118-bus under normal load opera-tion from MATPOWER(STRD), SOCP(SOCP) and modified model(SCP) 19

3.5 Results of voltage magnitudes of IEEE 118-bus under high load operationfrom MATPOWER (STRD), SOCP (SOCP) and modified model (SCP) . 19

3.6 Results of voltage magnitudes of IEEE 2383-bus under normal load oper-ation from MATPOWER(STRD), SOCP(SOCP) and modified model(SCP) 23

4.1 PSS/E OPF data structure . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 IEEE 30-bus power flow diagram . . . . . . . . . . . . . . . . . . . . . . . 28

vi

List of Tables

3.1 OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER (MAT) of IEEE 118-bus under normal load operation . . . . . . 17

3.2 OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER (MAT) of IEEE 14-bus under normal load operation . . . . . . . 20

3.3 OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER of IEEE 14-bus under high load operation . . . . . . . . . . . . . 21

3.4 OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER of IEEE 30-bus under normal load operation A . . . . . . . . . . 21

3.5 OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER of IEEE 30-bus under normal load operation B . . . . . . . . . . 22

3.6 OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER of IEEE 30-bus under high load operation (A) . . . . . . . . . . 22

3.7 OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER of IEEE 30-bus under high load operation (B) . . . . . . . . . . 22

3.8 OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER (MAT) of IEEE 118-bus under high load operation . . . . . . . 23

3.9 OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER (MAT) of IEEE 2383-bus under normal operation . . . . . . . . 24

4.1 OPF results from SOCP OPF (SOCP) and PSS/E OPF of IEEE 30-bus(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 OPF results from SOCP OPF (SOCP) and PSS/E OPF of IEEE 30-bus(B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 OPF results from SOCP OPF (SOCP) and PSS/E OPF of IEEE 2383-bus 29

vii

Abbreviations

A Bus incidence matrix

AA Modified bus incidence matrix

C Loop incidence matrix

P

G

Vector of bus real power generation

Q

G

Vector of bus reactive power generation

P

f

Vector of real line flow

P

f

Vector of reactive line flow

P

loss

Vector of real losses of braches

Q

loss

Vector of reactive losses of braches

V

2 Vector of square of bus voltage

B Diagonal matrix with the diagonal elements are shunt of buses

R Diagonal matrix with the diagonal elements are resistance of branches

X Diagonal matrix with the diagonal elements are reactance of branches

� Phase angle of buses

Y

nk

Bus admittance matrix whose nkth element

V

i

n

Imaginary part of bus voltage

V

r

k

Real part of bus voltage

G

nk

Real part of Ynk

B

nk

Imaginary part of Ynk

viii

Chapter 1

Introduction

1.1 Historical background

Worldwide, to achieve the security and economic operation of power system is of great

significance to the development of national economy. Over the past decades, with the

growing of consumer demand, development of renewable power plants and some other

factors, the need for finding e�cient and reliable optimization models which can solve

both security and economic issues of system operation is increasing [3]. In the 1960s,

the node voltage constraints, branch flow constraints and other operation constraints

were first introduced into the economic dispatch problem by Carpentier [4]. Although

the problem not successfully solved, this nonlinear programming formulation introduced

by Capentier was the first mathematical model of optimal power flow (OPF) problem.

Optimal power flow (OPF) seeks to find an optimal operational point of power system

to satisfy an economic, planning and reliable objective subject to the power flow con-

straints and system control limits. OPF is developed as a predominant tool for economic

and operation planning of power systems. OPF problem often is a kind of optimization

problem with the character of non-linear, non-convex and larger scale [3]. Di↵erent for-

mulations have been developed to obtain various objectives by using di↵erent control

variables and constraints. However the optimizations of power flow problems always

include the power flow equations as the constraints in the network. Moreover, di↵erent

1

Chapter 1. Introduction 2

types of optimization problems always have di↵erent names depending on the optimiza-

tion programming format of formulations. The formulations can be classified as linear

programming, nonlinear programming, quadratic programming and etc.

The solution methods for OPF problems also have been developed for decades. Many

mathematical programming methods can be utilized to OPF problems, and some soft-

ware are developed to solve OPF problems. But, due to the increasing size and com-

plexity of power system, it is di�cult to find a proper formulation and solution method

for all OPF problems.

1.2 Formulation of OPF problem

The standard form of formulation of OPF problem can be represented as follows [5]:

min f(u, x)

s.t.g(u, x) = 0

h(u, x) 0

(1.1)

Where f(u, x) represents the objective function which is the aim of optimization. Vec-

tor function g(u, x) and h(u, x) represent the equalities and inequalities of constraints.

The vector u and x represent the state variables and controllable variable separately.

The state variables can describe the state of power system and controllable variables

representing the control equipment of system. Di↵erent state and controllable variables

can be chosen according to the form of power flow equations and particular operation

situation of the system.

The equalities of constrains g(u, x) are composed of power flow equations and other

operation balance constrains. The form of power flow equations are various. The power

flow equations is classified into two kinds: alternating current (AC) power flow equations

and direct current (DC) power flow equations. The inequalities of constrains h(u, x)

often contain upper and lower bounds of variables, such as voltage magnitude and phase

angles.

Chapter 1. Introduction 3

1.2.1 AC power flow equations

AC power flow equations model the power system completely. Many kinds of AC power

flow equations are present in literature. The normal form of AC power flow equations

can be written as:

P

n

= Re(Vn

NX

k=1

Y

⇤nk

· V ⇤nk

) (1.2)

Q

n

= Im(Vn

NX

k=1

Y

⇤nk

· V ⇤nk

) (1.3)

The polar form of AC power flow equations are often used in OPF formulation. The

variables are set as the voltage magnitudes and voltage phase angle.

P

n

=NX

k=1

|Vn

| |Vk

| |Ynk

| cos(�n

� �

k

� �

nk

) (1.4)

Q

n

=NX

k=1

|Vn

| |Vk

| |Ynk

| sin(�n

� �

k

� �

nk

) (1.5)

Another form called rectangular form of power flow equations are proposed in recent

papers [6]:

P

n

=NX

k=1

G

nk

(V r

n

V

r

k

+ V

i

n

V

i

k

) +B

nk

(V i

n

V

r

k

� V

r

n

V

i

k

) (1.6)

Q

n

=NX

k=1

G

nk

(V r

n

V

r

k

� V

i

n

V

i

k

)�B

nk

(V i

n

V

r

k

+ V

r

n

V

i

k

) (1.7)

As can be seen from the equations, the variables are set as the real and imaginary part

of voltages.

There are some other power flow equations whose variables are branch flow variables. [7]

introduced a new set of power flow equations named branch flow model, which concerns

the balances of branch flow. The variables of this model are current magnitude and

angle, voltage magnitude and angle and real and reactive line flow of the system.

Chapter 1. Introduction 4

The feasible region of the OPF formulations including the AC power flow equations are

nonlinear and non-convex. Due to such reason, it is di�cult to find a global solution.

Many researches into convexification of OPF formulations has been active for a long

time. Paper [8] shows that the AC power flow equations can be relaxed using semidef-

inite programming (SDP). Some limitations of SDP relaxation for OPF problems are

also analyzed for meshed networks in [8]. [9] proposed an approach for relaxing the

branch flow equations for radial networks, i.e., relaxing the equalities to inequalities and

obtaining a second order cone programming. In [10], two relaxations are used for solving

the OPF problems for both meshed and radial networks. The voltage and current angle

constrains are eliminated and the conic relaxation is used. These two relaxations are

proven exact for radial networks, but for meshed networks, in order to obtain the global

solution, some phase shifters should be used in the network.

1.2.2 DC power flow equations

The DC power flow equations also represent the AC power system, but it is simplified

by using a linear form. To derive the DC power equations, some assumptions are made:

• The di↵erences of voltage angles are small. sin(�n

��

k

) = �

n

��

k

, cos(�n

��

k

) = 1.

• All the voltage magnitude are set to 1 p.u.

• Only line reactance are taken into consideration.

The DC power flow equations are written as follows:

P

n

=NX

k=1

B

nk

(�n

� �

k

) (1.8)

Many OPF formulations use DC power flow equations due to the linear constraints which

can be solved with fast speed and reliability. However, the DC power flow equations

neglect the losses of lines and the solution may result in large errors when applied to

large system [11].

Chapter 1. Introduction 5

1.3 OPF methods

The most popular methods for solving OPF problems are Newton method [12], sequen-

tial linear programming (SLP) [13], sequential quadratic programming (SQP) [14] and

interior point methods (IPMs) [15]. Newton method, SLP, SQP are active set methods

for solving OPF problems. The active set methods su↵ering the di�culty of identifica-

tion of the active set of inequality constrains may have low e�ciency when solving the

OPF problem. IPMs do not have such question. IPMs reach the optimal solution by

following a declined path of feasible region. IPMs is widely used and can be seen as

the most e�cient and fastest algorithm for solving OPF problems. Some literature have

studied the enhancements over IPMs, Primal-Dual Interior Point Methods (PDIPMs)

[16], Mehrotra’s predictor-corrector techniques [17], and trust region techniques [18].

1.4 Aim of the project

A conic formulation for OPF problems are developed in [19], [1]. The formulation is

based on line flow equations and formed in a second order programming. We make full

study on the SOCP formulation for OPF. To achieve this aim, the accuracy and e�ciency

are analyzed through applying this model to several systems, including large systems.

Also, a comparison is made between results from SOCP formulation implemented in

GAMS and di↵erent software.

1.5 Outline of thesis

In chapter 1, the historical background of OPF is introduced. The research on OPF

formulation and solution methods are also described.

In chapter 2, the conic formulation of OPF based on line flow equations is illustrated.

The non-linear equalities of line flow based equations can be replaced by inequalities

and the formulation for OPF can be formed in SOCP.

Chapter 1. Introduction 6

In chapter 3, a study on accuracy of the conic formulation for OPF is carried out. A

modified model which change the phase angle constrains of conic formulation is de-

scribed. The modified model can obtain a more accurate solution compared with conic

formulation. The comparison is analyzed between these two models

In chapter 4, the performances of SOCP formulation is studied On both small and large

test systems. The comparison of the results from SOCP formulation and the results

from PSS/E OPF is studied.

Chapter 2

Second Order Cone Formulation

for OPF

As the development of the optimal power flow formulation, a second order cone program-

ming formulation for OPF in a meshed network has been developed [19]. The developed

conic programming formulation is based on line flow based equations which has been

first described in [20]. Compared to conventional power flow models, the line flow based

model can directly reflect the line power flows of power system. The line flow based

equations have been studied in many papers. Since the nonlinearity of the equations of

branch losses, [21] has neglected the equations representing the line losses. [22] used the

approximated methods to linear the line losses. But in [19], all the nonlinear equations

of real and reactive losses are taken into consideration.

In order to develop the nonconvex OPF problem to convex optimization problem, some

approximations are used to linearize the angle phase constrains. Besides, the nonlinear

equations of power losses are replaced by relaxing the equalities into inequalities which

can be formed as rotated quadratic cones. By using these methods, the problem which

is formulated by using the line flow variables is formed as a second order programming

problem. The problem can be e�ciently solved through polynomial time interior point

methods [2].

Also, the author has proven the accuracy of the proposed OPF problem by comparing

the results of SOCP formulation with the results obtained from MATPOWER. The

tightness of the convex relaxation was also examined and proved.

7

Chapter 2. Second Order Cone Formulation for OPF 8

In this section, the SOCP formulation based on line flow based equations will be de-

scribed in details.

2.1 Line-flow based equations

Chapter 2 Second order cone formulation for OPF

13

2.1 Line-flow based equations

Figure 1: equivalent circuit of a branch

First, the interrelation of voltages and phase angles of a branch can be derived. An

equivalent circuit of a branch of system is shown in figure 1 and we define the line flow

direction is from bus i to bus j. The voltage drop from each branch can be written as 𝑉𝑖∠𝛿𝑖 = 𝑉𝑗∠𝛿𝑗 +

𝑃𝑓 − 𝑗𝑄𝑓

𝑉𝑗∠ − 𝛿𝑗(𝑅𝑓 + 𝑗𝑋𝑓) (9)

Then calculate the square of the magnitude of both sides and the following equation

can be obtained 𝑉𝑖

2𝑉𝑗2 = 𝑉𝑗

4 + 2𝑉𝑗2(𝑅𝑓𝑃𝑓 + 𝑋𝑓𝑄𝑓) + (𝑃𝑓

2 + 𝑄𝑓2)(𝑅𝑓

2 + 𝑋𝑓2) (10)

Diving (10) by 𝑉𝑗2, (11) is obtained as

𝑉𝑖2 − 𝑉𝑗

2 = 2(𝑅𝑃𝑓 + 𝑋𝑄𝑓) + 𝑃𝑙𝑜𝑠𝑠𝑅𝑓 + 𝑄𝑙𝑜𝑠𝑠𝑋𝑓 (11)

Where

𝑃𝑙𝑜𝑠𝑠,𝑓 =𝑃𝑓

2 + 𝑄𝑓2

𝑉𝑗2 𝑅𝑓 (12)

𝑄𝑙𝑜𝑠𝑠,𝑓 =

𝑃𝑓2 + 𝑄𝑓

2

𝑉𝑗2 𝑋𝑓

(13)

The interrelation of phase angle across a branch can be written as

𝑅𝑓 + 𝑗𝑋𝑓

Bus i Bus j

jB

𝑃𝑓, 𝑄𝑓

jB

Figure 2.1: Equivalent circuit of a branch

First, the interrelation of voltages and phase angles of a branch can be derived. An

equivalent circuit of a branch of system is shown in figure 1 and we define the line flow

direction is from bus i to bus j. The voltage drop from each branch can be written as

V

i

\�i

= V

j

\�j

+P

f

� jQ

f

V

j

\� �

j

(Rf

+ jX

f

) (2.1)

Then calculate the square of the magnitude of both sides and the following equation can

be obtained

V

2i

V

2j

= V

4j

+ 2V 2j

(Rf

P

f

+X

f

Q

f

) +�P

2f

+Q

2f

� �R

2f

+X

2f

�(2.2)

Diving 2.2 by V

2j

, 2.3 is obtained as

V

2i

� V

2j

= 2 (RP

f

+XQ

f

) + P

loss

R

f

+Q

loss

X

f

(2.3)

Where

P

loss,f

=P

2f

+Q

2f

V

2j

R

f

(2.4)

Q

loss,f

=P

2f

+Q

2f

V

2j

X

f

(2.5)

The interrelation of phase angle across a branch can be written as

V

i

V

j

sin (�i

� �

j

) = X

f

P

f

�R

f

Q

f

(2.6)

Chapter 2. Second Order Cone Formulation for OPF 9

Consider a connected graph W which can represent a meshed power system with n nodes

which represent the buses and l links which represent the lines in the network. We regard

the graph W as a directed graph and define the orientation as the smaller number node

points to the bigger number node of a link, i.e. a link f is connected by (i, j), if i < j,

the direction of link is from node i to node j. So the direction of line flow is the same

with the directed link. Choose a spinning tree T of graph, the spinning tree includes

(n-1) links and the number of fundamental cycles of graph is (l-n+1). According to

graph theory, the line-flow based equations can be derived.

The active and reactive power balance at each bus can be written as follows: for all

i 2 n, f 2 l,

P

Gi

� P

Li

=lX

f=1

A (i, f) · Pf

+lX

f=1

A

0(i, f) · P

loss,f

(2.7)

Q

Gi

�Q

Li

=lX

f=1

A(i, f) ·Qf

+lX

f=1

A

0(i, f) ·Q

loss,f

�B(i, i)V 2i

(2.8)

The elemets of matrix A and AA are defined as

A

il

=

8>>><

>>>:

1, if line l leaves bus i

�1, if line goes into bus i

0, otherwise

(2.9)

A

0il

=

8<

:1, if line l leaves bus i

0, otherwise

(2.10)

Since 2.6 is a trigonometric function, some assumptions are made to linearized it. As-

suming sin (�i

� �

j

) ⇡ �

i

� �

j

and V

i

V

j

⇡ 1, 2.6 can be written as

�

i

� �

j

= X (f, f)Pf

�R (f, f)Qf

(2.11)

In a meshed network, the sum of phase angle drops around fundamental cycles is zero.

So the equations of cycle phase angle can be obtained by neglecting the approximations

used in 2.11.

0 = CXP � CRQ (2.12)

A fundamental cycle is consisted of a line that does not belong to the spinning tree

together with the path in T. We define the direction of each fundamental cycle is the

Chapter 2. Second Order Cone Formulation for OPF 10

same with the direction of the line outside T. So C can be partitioned into:

C = [CT

C?] (2.13)

where C

T

is(n � 1) ⇥ (l � n + 1)corresponding to the lines included in T and C? is

(l � n + 1) ⇥ (l � n + 1) corresponding to the lines which do not belong to T. C? is a

diagonal matrix with all the diagonal elements equal to 1. C? can be calculated by

C

T

=��A

⇤�1T

A

⇤?�T

(2.14)

A

⇤ is a reduced incidence matrix which is derived by removing one row of A.

Therefore, a meshed network is described by 2.3, 2.4, 2.5, 2.7, 2.8 and 2.12.

2.2 Second-order cone programming

The second order cone programming problem is a kind of convex programming prob-

lems, which minimize a linear function subject to a set of linear constrains and nonlinear

constrains which are in form of convex cones. The SOCP problems can be solved e�-

ciently by using polynomial time interior point methods. The standard form of SOCP

is as follows [2]:

Min c

T

x

subject to Ax = b

x

min

x x

max

x 2 C

(2.15)

where x 2 R

n, C is to be a convex cone, c and b are the constant number.

Put the variables x into di↵erent sets S

p, p = 1, ..., k, and each variable x is only

contained in one set Sp. Define

C = {x 2 R

n

, : xs

p 2 C

p

, p = 1, ..., k} (2.16)

where xs

p is defined as the variables which are include in set Sp. And C

p

must be fulfilled

the one of these two following forms:

Chapter 2. Second Order Cone Formulation for OPF 11

Quadratic cone:

C

p

= {x 2 R

np, : x1 �

vuutnpX

i=2

x

2j

} (2.17)

Rotated quadratic cone:

C

p

= {x 2 R

np, : 2x1x2 �

npX

i=3

x

2j

, x1, x2 � 0} (2.18)

If the problem is constrained by linear equalities and quadratic inequalities which can

be formed as these two types of cones, the problem can be formed in SOCP and solved

by using interior point methods [2].

2.3 Optimal power flow

The line flow based equations are used to formulate the optimal power flow. The objec-

tive function is set to minimize the fuel cost of all the generators in the system. Since

the objective function should be formed in a linear form, the piecewise linear cost model

of generators is considered. The objective function is written as

Min c0PGi

+ c1 (2.19)

where, the constant c0, c1 are the coe�cients with the piecewise linear cost model of

generators. The vector of variables of the proposed line flow based model is defined as

X =

2

6666666666666664

V

2T

P

T

Q

T

P

T

G

Q

T

G

P

T

loss

Q

T

loss

3

7777777777777775

(2.20)

Chapter 2. Second Order Cone Formulation for OPF 12

But the feasible set of the OPF problem is nonconvex due to the quadratic equation of

2.4 and 2.5. And can be deduced from by using the following equation.

Q

loss,f

=X(f, f)

R(f, f)P

loss,f

(2.21)

Because V

2j

� 0, Ploss,f

� 0,relax 2.4 into inequalities

P

loss,f

�P

2f

+Q

2f

V

2j

R(f, f) (2.22)

Form 2.4 as a rotated quadratic cone

2V 2j

P

⇤losss,f

� P

2f

+Q

2f

(2.23)

P

⇤loss,f

=P

loss,f

2R(f, f)(2.24)

By transforming 2.4 to 2.23, the feasible set of this problem is convex.

Therefore, the equality and inequality constrains of the problem can be written as

P

G

� P

L

�A · P �AA · Ploss

= 0 (2.25)

Q

G

�Q

L

�A ·Q�AA ·Qloss

�B · Y = 0 (2.26)

CXP � CRQ = 0 (2.27)

A

T · Y � 2RP + 2XQ+RP

loss

+XQ

loss

= 0 (2.28)

Q

loss

·R� P

loss

·X = 0 (2.29)

2P ⇤loss

·R = P

l

oss (2.30)

2V 2j

P

⇤loss,f

� P

2f

+Q

2f

(2.31)

V

2i

V

2i

V

2i

(2.32)

P

Gi

P

Gi

P

Gi

(2.33)

Q

Gi

Q

Gi

Q

Gi

(2.34)

P

i

P

i

P

i

(2.35)

Chapter 2. Second Order Cone Formulation for OPF 13

Q

i

Q

i

Q

i

(2.36)

where the underline and overline represent the upper and lower bound of these line flow

variables.

Chapter 3

Study on accuracy of SOCP OPF

formulation

In the proposed SOCP OPF formulation, in order to linearize the phase angle drops

across a line, some approximations are used. However, this will a↵ect the accuracy of the

solution results. Can we find a proper way to linearize the phase angle constrains more

accurately. To increase the accuracy of the OPF results, a modified model is illustrated

in this section. The model is obtained by modifying the phase angle constraints in the

SOCP OPF formulation. In this modified model, the interrelations of phase angles are

described as functions of line flow variables and linearize the functions by using Taylor’s

series. The modified model can be solved using sequential conic programming method as

used in [23]. This means that an external loop is added to the previous model. Although

some approximations are still used because of truncating the high order terms of Taylor’s

series, the solution will be more accurate and closer to the exact operation point through

the iterative loops.

3.1 Modified phase angle di↵erence constrains

We can obtain the imaginary part and real part of both sides in 2.1 as follows

V

i

V

j

sin(�i

� �

j

) = X

l

P

f

�R

l

Q

f

(3.1)

14

Chapter 3. Study on accuracy of SOCP OPF formulation 15

V

i

V

j

cos(�i

� �

j

) = R

l

P

f

+X

l

Q

f

+ V

2j

(3.2)

Then the phase angle di↵erence through a branch can be obtained

�

i

� �

j

= tan�1(X

l

P

f

�R

l

Q

f

R

l

P

f

+X

l

Q

f

+ V

2j

) (3.3)

The trigonometric functions are not compatible in the conic quadratic format, Equation

3.3 should be linearized. Define

T

ij

= X

l

P

f

�R

l

Q

f

(3.4)

S

ij

= R

l

P

f

+X

l

Q

f

+ V

2j

(3.5)

By following Taylor’s series and neglect the higher terms, Equation 3.3 can be expressed

at an estimate point (S(q)in

, T (q)in

).

�

i

� �

j

+T

(q)ij

S

(q)2

ij

+ T

(q)2

ij

S

ij

�S

(q)ij

S

(q)2

ij

+ T

(q)2

ij

T

ij

= tan�1S

(q)ij

T

(q)ij

(3.6)

Equation 3.6 is a linearization of Equation 3.3, which shows the interrelations between

phase angles in a more accurate way instead of using the approximations in 2.11.

3.2 Optimal power flow and sequential conic programming

The objective function is the same with the one of SOCP OPF formulation. The variables

of modified model are defined as

Chapter 3. Study on accuracy of SOCP OPF formulation 16

X =

2

66666666666666666666666664

V

2T

P

T

G

Q

T

G

P

T

loss

Q

T

loss

P

T

Q

T

�

T

T

T

S

T

3

77777777777777777777777775

(3.7)

Because of including the linearized angle constraints (50), an iterative procedure of

solving the problem is necessary. At each iteration, the formulation can be formed as

a SOCP programming problem. The conic programming OPF formulation is formed as

follows.

Minimize 2.19

Subject to: 2.25, 2.26, 2.28, 2.29, 2.30, 2.31, 2.32, 2.33, 2.34, 2.35, 2.36, 3.4, 3.5, 3.6.

The process can be done by: first set q equal to 0 and give initial values to S

0ij

and T

0ij

.

Second, solve the conic formulation defined above, increase q by 1, the results can be

obtained and denote the results Sq

ij

, T q

ij

. If������Sq

ij

� S

(q�1)ij

������ < ✏ and

������T q

ij

� T

(q�1)ij

������ < ✏,

the final results of problem are derived, otherwise, repeat the step two.

Figure 3.1 shows the flow chart of the process solving the modified model.

3.3 Test results and comparison with SOCP OPF results

The accuracy comparison between the results of modified model and SOCP model is car-

ried out by comparing with the results of the standard OPF formulation. The the mod-

ified model and SOCP model is programmed in the platform GAMS and the MOSEK

[24] solver is used. The standard OPF formulation is carried out in MATPOWE. The

di↵erent OPF models are applied to three test systems IEEE14-bus, IEEE 30-bus, IEEE

118-bus and IEEE 2383-bus. In order to investigate the di↵erences of results from these

Chapter 3. Study on accuracy of SOCP OPF formulation 17

Start

Set q = 0, S0ij

= 1,

T

0ij

= 0, ✏ = 10�6

Solve the conic formula-tion q = q + 1, Sq

ij

,T q

ij

������Sq

ij

� S

(q�1)ij

������

and������T q

ij

� T

(q�1)ij

������ <

✏?

Display the results

End

yes

no

Figure 3.1: Flow chart of solving the modified model

two models, we modify the small test systems by increasing the load to maximum value

which a feasible solution can be obtained (since the load has a great impact on the

voltage and phase angle).

Table 3.1: OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER (MAT) of IEEE 118-bus under normal load operation

MAT SOCP MM

Total cost [$] 86331.14 86334.08 86331.14Total P losses [MW] 74.557 74.704 74.557Total Q losses [MVAR] 438.62 439.555 438.634Total P gen [MW] 4316.56 4316.704 4316.557Total Q gen [MVAR] 301.91 302.625 301.917Elapsed Time [s] 0.89 0.543 2.729

For IEEE 14-bus, the results from three models are shown in table 3.2 and 3.3. The

Chapter 3. Study on accuracy of SOCP OPF formulation 18

Figure 3.2: Results of voltage magnitudes of IEEE 30-bus under normal load opera-tion from MATPOWER (STRD), SOCP (SOCP) and modified model (SCP)

Figure 3.3: Results of voltage magnitudes of IEEE 30-bus under high load operationfrom MATPOWER(STRD), SOCP(SOCP) and modified model(SCP)

Chapter 3. Study on accuracy of SOCP OPF formulation 19

Figure 3.4: Results of voltage magnitudes of IEEE 118-bus under normal load oper-ation from MATPOWER(STRD), SOCP(SOCP) and modified model(SCP)

Figure 3.5: Results of voltage magnitudes of IEEE 118-bus under high load operationfrom MATPOWER(STRD), SOCP(SOCP) and modified model(SCP)

Chapter 3. Study on accuracy of SOCP OPF formulation 20

Table 3.2: OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER (MAT) of IEEE 14-bus under normal load operation

Bus Voltage P

G

Q

G

[p.u] [MW ] [MVAR]

MAT SOCP MM MAT SOCP MM MAT SOCP MM

1 1.06 1.06 1.06 128.648 128.654 128.649 0 0 02 1.052 1.052 1.052 140 140 140 34 34.2 343 1.019 1.019 1.019 0 0 0 34.74 34.50 34.624 1.017 1.017 1.017 0 0 0 0 0 05 1.022 1.022 1.022 0 0 0 0 0 06 1.018 1.018 1.018 0 0 0 24 24 247 1.02 1.02 1.02 0 0 0 0 0 08 1.06 1.06 1.06 0 0 0 24 24 249 0.998 0.998 0.998 0 0 0 0 0 010 0.994 0.994 0.994 0 0 0 0 0 011 1.002 1.002 1.002 0 0 0 0 0 012 1.002 1.002 1.002 0 0 0 0 0 013 0.997 0.997 0.997 0 0 0 0 0 014 0.978 0.978 0.978 0 0 0 0 0 0

Elapsed time [s] MAT SOCP MM0.61 0.512 1.394

results which are from modified model is closer to the standard results than those which

are from the SOCP formulation. But the di↵erences from modified model and SOCP

formulation are quite small. For IEEE 30-bus system under both normal load and high

operation, from table 3.4, 3.5, 3.6 and 3.7, it can be seen that the results from SOCP

and modified model are still close to each other, compared to the standard results from

MATPOWER, the results from modified model are more accurate. Furthermore, the

modified model has reduced the voltage magnitude errors, figure 3.2 shows that the

voltage magnitudes of bus node number 25 and 29 are corrected by the modified model

when network is under normal load operation. Similarly, when the network under high

load operation, figure 3.3 shows that the voltage magnitudes of bus node number 10 and

22 are corrected.

The results from di↵erent models of IEEE 118-bus are shown in table 3.1 and 3.8. The

modified model still gets better results. And the accuracy of voltage magnitudes is

increased by the modified model. Since it is hard to display all the bus nodes in one

figure, only 55 bus nodes are chosen to show in figure 3.4 and 3.5. The maximum voltage

Chapter 3. Study on accuracy of SOCP OPF formulation 21

Table 3.3: OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER of IEEE 14-bus under high load operation

Bus Voltage P

G

Q

G

[p.u] [MW ] [MVAR]

MAT SOCP MM MAT SOCP MM MAT SOCP MM

1 1.06 1.06 1.06 54.02 54.153 54.017 0 0 02 1.06 1.06 1.06 129.65 128.44 129.646 46.17 46 46.1713 1.039 1.039 1.039 92.85 94.503 92.85 40 40 404 1.024 1.024 1.024 0 0 0 0 0 05 1.031 1.031 1.031 0 0 0 0 0 06 1.009 1.009 1.009 81.49 80.474 81.493 24 24 247 1.002 1.002 1.002 0 0 0 0 0 08 1.041 1.041 1.041 36.15 36.53 36.161 24 24 249 0.968 0.968 0.968 0 0 0 0 0 010 0.962 0.962 0.962 0 0 0 0 0 011 0.979 0.979 0.979 0 0 0 0 0 012 0.984 0.984 0.984 0 0 0 0 0 013 0.974 0.974 0.974 0 0 0 0 0 014 0.94 0.94 0.94 0 0 0 0 0 0

Elapsed time [s] MAT SOCP MM0.62 0.538 1.853

Table 3.4: OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER of IEEE 30-bus under normal load operation A

Gen P

G

Q

G

[MW ] [MVAR]

MAT SOCP M MAT SOCP M

1 63.19 63.18 63.19 -0.88 -1.112 -0.932 80 80 80 27.49 27.776 27.50613 0 0 0 15.41 15.382 15.40422 50 50 50 15.64 15.621 15.59723 0 0 0 6.09 6.127 6.0927 0 0 0 16.32 16.265 16.401

magnitude errors compared to MATPOWER of modified model and SOCP formulation

are 10�5 and 10�3 respectively. The accuracy of voltage magnitudes of 22 bus nodes

under normal load operation and 36 bus nodes under high operation has been improved

about 0.1%.

For the larger test case IEEE 2383-bus system, from table 3.9, the same with the other

three cases, the accuracy of results is increased by the modified model. In this case, the

Chapter 3. Study on accuracy of SOCP OPF formulation 22

Table 3.5: OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER of IEEE 30-bus under normal load operation B

MAT SOCP MM

Total cost [$] 316.38 316.39 316.38Total P losses [MW] 3.992 3.988 3.992Total Q losses [MVAR] 15.06 15.051 15.058Total P gen [MW] 193.19 193.188 193.192Total Q gen [MVAR] 80.07 80.059 80.067

Elapsed Time [s] MAT SOCP MM0.72 0.526 2.065

Table 3.6: OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER of IEEE 30-bus under high load operation (A)

Gen P

G

Q

G

[MW ] [MVAR]

MAT SOCP M MAT SOCP M

1 80 80 80 5.83 5.742 51.5822 80 80 80 54.5 54.601 54.72913 40 40 40 22.41 22.404 22.33822 50 50 50 39.63 39.624 39.62623 30 30 30 13.41 13.418 13.50727 38.557 38.56 38.55 21.85 21.842 21.834

Table 3.7: OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER of IEEE 30-bus under high load operation (B)

MAT SOCP MM

Total cost [$] 685.31 685.29 685.31Total P losses [MW] 6.377 6.377 6.37Total Q losses [MVAR] 21.88 21.875 21.871Total P gen [MW] 318.56 318.56 318.55Total Q gen [MVAR] 157.63 157.632 157.616

Elapsed Time [s] MAT SOCP MM0.74 0.542 2.153

accuracy of voltage magnitudes has gotten greater increase as shown in figure 3.6 (The

figure only shows a small part of buses.). The results of voltage magnitudes for modified

model is much closer to the standard results. The maximum increase of accuracy is

about 1.7%.

Also, it should be noted that the elapsed time of modified model is almost three times of

that of SOCP formulation, which is because the external iterative procedure of modified

Chapter 3. Study on accuracy of SOCP OPF formulation 23

Figure 3.6: Results of voltage magnitudes of IEEE 2383-bus under normal load op-eration from MATPOWER(STRD), SOCP(SOCP) and modified model(SCP)

Table 3.8: OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER (MAT) of IEEE 118-bus under high load operation

MAT SOCP MM

Total cost [$] 200160 200146 200150Total P losses [MW] 176.943 177.323 176.948Total Q losses [MVAR] 1020.02 1019.611 1019.255Total P gen [MW] 8236.76 8237.123 8236.748Total Q gen [MVAR] 2199.72 2199.128 2199.043Elapsed Time [s] 0.95 0.582 2.874

model. Moreover, the modified model has introduced more variables which may cause

the complexity of computation.

In summary, in case of using old method, the results are still accurate in a shorter execute

time. However, the new method has increased the accuracy of simulation results with

longer time. When the system becoming large or under high load operation, the e↵ect

of accuracy of voltage magnitude is getting larger when using the old method. The

new method can reduce such e↵ect, but the consuming time is much longer. There are

advantages and drawbacks of both methods, according to di↵erent situation the more

appropriate one should be chosen.

Chapter 3. Study on accuracy of SOCP OPF formulation 24

Table 3.9: OPF results from modified model (MM), SOCP OPF (SOCP) and MAT-POWER (MAT) of IEEE 2383-bus under normal operation

MAT SOCP MM

Total cost [$] 1.8784+e6 1.8775+e6 1.8782e+6Total P losses [MW] 700.304 690 699.857Total Q losses [Mvar] 3864.11 3840.6 3861.5Total P gen [MW] 25258.7 25248.39 25258.3Total Q gen [Mvar] 6675.2 6651.7 6672.6Elapsed Time [s] 7.1 6.4 16.1

Chapter 4

Simulation results and

comparison with PSS/E

In order to investigate the performance of SOCP formulation, we also compare the

simulation results from SOCP formulation with the other commercial software. As is

known to all, PSS/E is the best performing commercial software in the market, so PSS/E

is used as the compared software.

In this chapter, a small test system (IEEE 30-bus) and a large test power system (IEEE

2383-bus) are both modeled with SOCP formulation using the GAMS MOSEK solver

and in PSS/E OPF package. After the implementation of modeling with these two

methods, the comparison of results is made.

4.1 PSS/E

PSS/E is widely used commercial software after it was introduced in 1976. This software

is a set of computer programs which can deal with the basic function of simulation work

of the power system. PSS/E can handle the simulation work as follows [25]:

• Power Flow

• Optimal Power Flow

• Fault Analysis

25

Chapter 4. Simulation results and comparison with PSS/E 26

• Dynamic Simulations

• Open network Access and Price calculation

• Equivalent Construction

PSS/E is composed of some modules which can handle di↵erent simulations of small or

large systems. In this project, we only use the power flow and optimal power modules

of PSS/E.

PSS/E power flow module is the basis of PSS/E module and it can be used to simulate

and analyze power flow network.

PSS/E optimal power flow (PSS/E OPF) module is an advanced PSS/E module. The

OPF module can solve the power system with some constrains and limitations to achieve

some economical goals. PSS/E OPF can achieve the optimization by minimizing di↵er-

ent objective functions,

To solve the OPF problems in PSS/E OPF, the OPF model data must be set correctly.

The OPF model data is consisted of transmission part and generation part of network.

The transmission part modeling in PSS/E can be implemented by inputting all the

information of buses and connection links elements (branches, shunts, etc.). After that

the network can be represented by the PSS/E model.

The generation part modeling in PSS/E is to model all generators respectively with

proper data sets that can represent the characteristics of generators. The data sets are

composed of three sections:

• Generation dispatching data

• Generation cost curve

The generation dispatching data is represented by using the dispatch tables in PSS/E.

All the data in each table can describe the state of each generator in the network. The

data sets are shown as follows:

• Maximum of generation Max

• Minimum of generation Min

Chapter 4. Simulation results and comparison with PSS/E 27

OPFraw data

Rawdata-loadflow data

OPF dataeditor

OPFand loadflow data

Figure 4.1: PSS/E OPF data structure

• Coe�cient of fuel cost

• Cost curve type

• Cost table

• In service or not

The cost curves of generators in PSS/E are the fuel cost curves. The types of cost

curves contain piece-wise linear functions, piece-wise quadratic functions and polynomial

functions.

4.2 Simulation results

As can be seen from the simulation results from SOCP formulation and PSS/E, the

results for IEEE 30-bus from SOCP formulation and PSS/E are almost identical, but

the elapsed time of SOCP formulation coded in platform GAMS is longer than that of

PSS/E. But when the SOCP formulation is executed in the GAMS software, it takes

time to compile the data files and execute the commands, which account for significant

portion of the elapsed time. The execute time of MOSEK optimizer is only 0.06 seconds.

Figure 4.2 shows the power flow diagram of IEEE 30-bus after implementation modelling

in PSS/E.

Chapter 4. Simulation results and comparison with PSS/E 28

Cha

pter

4 S

imul

atio

n re

sults

and

com

paris

on w

ith P

SS/E

36

Fi

gure

8: I

EEE

30-b

us p

ower

flow

dia

gram

1 1 1.0

141.7

163.2

-0.9R

2 2 1.0

141.3

30.9

-5.3

-30.7

2.5

121.7

12.7

180.0

27.5R

3 3 1.0

138.5

32.3 4.4

-31.8

-4.7

12.4

1.2

4 4 1.0138.0

31.7 3.4

-31.1

-4.0

29.4

3.5

-29.4

-3.2

17.6

1.6

5 5 1.0

138.821.0 3.6

-20.8

-4.9

6 6 1.0

137.2

36.3 5.2

-35.6

-5.1

28.0 7.7

-27.9

-7.4

7 7 1.0

136.6

20.8 4.9

-20.6

-5.4

2.2

4.4

-2.2

-5.5

122.8

10.9

8 8 1.0135.5

29.0

24.2

-28.8

-23.7

130.0

30.0

9 9 1.0139.4

10.5

-7.7

-10.5

8.1

10 10 1.0

140.5

6.0

-4.4

-6.0

4.7

10.5

-8.1

-10.58.3

15.8

2.0

1-20.6

11 11 1.0139.4

-0.0

-0.0

-0.0

-0.0

12 12 1.0

139.0

24.9

-2.2

-24.93.8

111.2 7.5

13 13 1.0

141.7

0.0-15.1

0.0

15.4

10.0

15.4R

14 14 1.0

137.6

5.41.4

-5.4

-1.3

16.2

1.6

15 15 1.0

137.9

9.41.0

-9.3

-0.9

-0.8

-0.3

0.8

0.3

18.2

2.5

16 16 1.0

138.7

-1.1 1.4

1.1

-1.4

13.5

1.8

17 17 1.0

139.3

13.7

6.4

-13.6

-6.3

-4.6

-0.4

4.6

0.5

19.0

5.8

18 18 1.0

137.2

1.7

1.6

-1.6

-1.6

13.2

0.9

19 19 1.0

137.2

-1.6 0.7

1.6

-0.7

19.5

3.4

20 20 1.0

137.9

13.5

3.9

-13.3

-3.5

-11.1

-2.7

11.1

2.8

12.2

0.7

21 21 1.0141.1

-8.8

-2.0

8.82.1

117.5

11.2

22 22

-7.6

-2.7

7.62.8

-26.3

-13.3

26.4

13.4

1

23 23 1.0139.0

-1.3

-3.5

1.33.5

13.21.6

10.06.1R

24 24

16.0-0.6

-15.7

1.0

-4.5 1.0

4.6

-0.9

16.7

1-4.6

25 25 1.0139.8

2.4

-2.2

-2.42.2

26 26 1.0

137.5

3.52.4

-3.5

-2.3

13.52.3

27 27 1.0141.2

-1.1

-4.6

1.24.6

10.0

16.4R

28 28 1.0137.1

15.7

-4.0

-15.6

3.1

-1.2

-6.3

1.2

4.4

-14.4

8.5

14.4

-7.4

29 29 1.0138.6

6.21.7

-6.1

-1.5

12.4

0.9

30 30 1.0137.1

7.11.6

-7.0

-1.4

3.70.6

-3.6

-0.5

110.6

1.9

0.0

8.7

1.0

139.5

15.6R

1.0141.7

50.0

0.0

Figure 4.2: IEEE 30-bus power flow diagram

Chapter 4. Simulation results and comparison with PSS/E 29

For large system IEEE 2383-bus system, the results from SOCP formulation are also

close to the results from PSS/E, the execute time of MOSEK optimizer is 0.63 seconds

which is longer than the elapsed time of PSS/E, which is a little longer that the execute

time of PSS/E . Compared with PSS/E OPF, the SOCP formulation still get high

accuracy and e�ciency for both small and large test systems.

Table 4.1: OPF results from SOCP OPF (SOCP) and PSS/E OPF of IEEE 30-bus(A)

Gen P

G

[MW ] Q

G

[MVAR]PSS/E SOCP PSS/E SOCP

1 63.19 63.18 -0.88 -1.1122 80 80 27.47 27.77613 0 0 15.41 15.38222 50 50 15.62 15.62123 0 0 6.08 6.12727 0 0 16.37 16.265

Table 4.2: OPF results from SOCP OPF (SOCP) and PSS/E OPF of IEEE 30-bus(B)

PSS/E SOCP

Total cost [$] 316.38 316.39Total P losses [MW] 3.992 3.988Total Q losses [Mvar] 15.057 15.051Total P gen [MW] 193.19 193.188Total Q gen [Mvar] 80.07 80.059Elapsed Time [s] 0.065 0.526

Table 4.3: OPF results from SOCP OPF (SOCP) and PSS/E OPF of IEEE 2383-bus

PSS/E SOCP

Total cost [$] 1.8785e+6 1.8782e+6Total P losses [MW] 700.7 690Total Q losses [Mvar] 3864.1 3840.6Total P gen [MW] 25258.7 25248.9Total Q gen [Mvar] 6675.1 6651.7Elapsed Time [s] 0.336 6.409

Chapter 5

Conclusion and further work

5.1 Conclusion

This thesis has studied the developed second order cone programing formulation for

OPF. The accuracy and computation time are considered as the two key aspects of

studying.

The developed conic formulation of OPF is introduced in details firstly, approximated

method and conic relaxation are used to transform constrains to second order program-

ming format.

A modified model is developed to study on accuracy of the SOCP formulation and tested

on four test systems including small and large systems under normal load and high load

operation.

The modified model is also based on the line flow equations, only the phase angle con-

strains are modified to obtain a more accurate result. The di↵erences of phase angle

are descried as functions of line flow variables and linearized in a Taylor’s series. The

modified model can be solved by using sequential conic programming method.

The modified model has improved the accuracy of solution results. Especially, For

large system or system under high load operation, the accuracy of voltage magnitude is

improved a lot compared with the SOCP formulation. But the sequential method applied

in modified model causes the algorithm complicated and slow speed of computation.

30

Chapter 5. Conclusion and further work 31

A comparison is carried out between the results of SOCP formulation and PSS/E soft-

ware to investigate the performance of SOCP formulation. After implementation

the SOCP formulation and PSS/E modelling on one small system and one large scale

system, it can be found that the SOCP formulation is quite accurate even for larger

system and the elapsed time is more than that of PSS/E, but it is because the time of

compiling data and executing commands in GAMs platform, for the small system the

MOSEK optimizer time is less than the elapsed time of PSS/E.

After studying the performances of SOCP formulation for OPF problem, we can say

that the SOCP formulation of OPF can be seen as an accurate and reliable formulation

for OPF problems and can be solved e�ciently by using IPMs.

5.2 Further work

In this project, we only consider minimizing the total fuel cost as the objective function,

some other objective functions such as minimizing the active losses can be studied to

analyze the performances of SOCP formulation. Since the application of FACTS in the

power systems is increasing, to model the FACTS in the power systems in the SOCP

formulation is an important work for future.

The fundamental loops are not used in modified model, the modified model only use

linearized equations to represent the di↵erences of phase angles, the modified model can

be used to analyze the optimal transmission switching problems.

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