41.Power Flow Control with Distributed facts AC Transmission System (D-FACTS) Devices

Abstract—Distributed Flexible AC Transmission System (D-

FACTS) devices offer many potential benefits to power system

operations. This paper illustrates the flexibility of control that is

achievable with D-FACTS devices. The impact of installing D-

FACTS devices is examined by studying the sensitivities of power

system quantities such as voltage magnitude, voltage angle, bus

power injections, line power flows, and real power losses with

respect to line impedance. Sensitivities enable us to quantify the

amount of control D-FACTS devices offer to the system.

Independently controllable lines are selected for power flow

control, and appropriate locations to install D-FACTS devices for

line flow control are determined. Then, D-FACTS device settings

are selected to achieve desired line flow objectives.

Index Terms— power flow control, distributed flexible AC

transmission systems, controllability, linear sensitivity analysis

I. INTRODUCTION

PPROXIMATELY two decades ago, Flexible AC

Transmission Systems (FACTS) were introduced. A

Flexible AC Transmission System incorporates power

electronics and controllers to enhance controllability and

increase transfer capability [1]. FACTS devices can improve

power system operation are by providing a means to control

power flow, to improve stability, and to better utilize the

existing transmission infrastructure. The benefits associated

with the use of FACTS devices have been demonstrated in

successful applications. One such application is in west Texas

where dynamic reactive compensation systems (DRCS)

correct abnormal voltages caused by the rapid changes in wind

production which is prevalent in the area [2].

Over the past two decades, technology in many areas of

electrical engineering has become faster, less expensive,

smaller, and more reliable. Advances in computing, wireless

communications, microprocessors, electronic devices, and

other technology advances have affected all aspects of life.

Improvements in electrical technology allow a revisit of

FACTS concepts from a fresh perspective.

Recently, Distributed Flexible AC Transmission System

(D-FACTS) devices [3], [4], [5] were introduced. D-FACTS

devices are power flow control devices which are small, light-

The authors would like to acknowledge the support of the support of NSF

through its grant CNS-0524695, the Power System Engineering Research

Center (PSERC), City Water Light and Power (CWLP) in Springfield, IL, and

the Grainger Foundation.

The authors are with the University of Illinois Urbana-Champaign, Urbana, IL 61801 (e-mail: [email protected]; [email protected]).

weight, and made of easily purchased mass-produced parts.

The achievement of flexible line flow control through the use

of effectively placed and configured D-FACTS devices is

explored in this paper.

II. POWER FLOW CONTROL

Control of power flow requires the ability to maintain or

change line impedances, bus voltage magnitudes, and phase

angle differences. Power controller devices such as FACTS

devices [6], [7], [8] affect some or all of these parameters.

D-FACTS devices, the Distributed Static Series

Compensator (DSSC) in particular, are series power flow

control devices which change the effective impedance of

transmission lines through the use of a synchronous voltage

source (SVS) [9]. A D-FACTS device changes the effective

line impedance by producing a voltage drop across the line

which is in quadrature with the line current. Thus, a D-

FACTS device provides either purely reactive or purely

capacitive compensation.

D-FACTS devices can be made to communicate wirelessly,

allowing them to receive commands for impedance injection

changes. In addition, D-FACTS devices can be configured to

operate autonomously during certain conditions [4]. D-FACTS

devices on different lines working together can be coordinated

in such a way that they can achieve some control objective.

Consider the simple 4-bus example below. D-FACTS

devices are first placed on line (1, 3) and then added to line

(2,4). The range of possible line flows due to D-FACTS

control is plotted in Figure 2 for each line in the system. It is

assumed that D-FACTS devices can change the line

impedance by +/-20% [4] of the uncompensated value.

Figure 1. 4-Bus System

Power Flow Control with Distributed Flexible

AC Transmission System (D-FACTS) Devices

Katherine M. Rogers, Student Member, IEEE, Thomas J. Overbye, Fellow, IEEE

A

Authorized licensed use limited to: Bharat University. Downloaded on June 17,2010 at 07:33:34 UTC from IEEE Xplore. Restrictions apply.

Figure 2. Line Flow Control of 4-bus System

A certain range of flows is achievable by D-FACTS control

of line (1,3). When D-FACTS are added to a second line, line

(2, 4), more control is possible. However, lines do not have

equal potential to provide control or to be controlled.

D-FACTS devices control of one line affects the flows on

all lines. The impact that the control of one line flow has on

other line flows is specific to the system. If a system has only

one loop, the flows are completely coupled and cannot be

controlled independently. The extent to which line flows can

be controlled independently is one measure of D-FACTS

potential. A method for identification of independently

controllable line flows is presented in Section IV. D-FACTS

devices should be placed at effective locations throughout the

system and set to achieve the desired control; this control is

described in Section V.

III. LINE IMPEDANCE SENSITIVITIES

Sensitivities are linearized relationships often used in power

systems analysis [10]. Sensitivities reveal the impact of a

small change in a variable on the rest of the system. Linear

approximations in nonlinear systems provide insight into

relationships which may otherwise be difficult to characterize.

Line impedance sensitivities are fundamental to the analysis of

placing and setting D-FACTS devices for line flow control.

The equations from which the sensitivities are derived in are

given in Appendix A.

A. Power Injection and State Variable Sensitivities

The negative inverse of the power flow Jacobian, J,

describes the way the state variables change in a solution of

the power flow due to bus power injection mismatch.

∆𝒔(𝜃 ,𝑉) = −𝑱 −1 ∙ 𝒇(𝑝 ,𝑞) (1)

The power injection to impedance sensitivity matrix 𝚼 is the

derivative of the entries of f(p,q) with respect to impedance:

∆𝒇(𝑝 ,𝑞) = 𝚼 ∙ ∆𝒙 (2)

The state to impedance sensitivity matrix Φ describes how

the state variables change in a solution of the power flow due

to a small impedance change:

𝚽 = −𝑱−1 ∙ 𝚼 (3)

∆𝒔(𝜃 ,𝑉) = 𝚽 ∙ ∆𝒙 (4)

The matrix Φ is the only full matrix needed, and its

computation involves 𝚼 and the inverse of J. The dimension

of the columns of 𝚼 is the number of lines equipped with D-

FACTS devices, k. The rows of 𝚼 are sparse since not every

bus is connected to each of the k lines. Thus, each column of

𝚼 is a sparse vector, and sparse vector methods may be used to

compute Φ using the fast-forward and full back schemes as

described in [11].

B. Power Flow Sensitivities

The relationships between state variables and real power

flows are represented by the power flow to state sensitivity

matrix Σ:

∆𝑷𝒇𝒍𝒐𝒘 = 𝚺 ∙ ∆𝒔(𝜃 ,𝑉) (5)

Each row in the power flow to impedance sensitivity matrix

Γ has one non-zero element corresponding to the line

impedance on the same line. Since a line has a sending end

and a receiving end power flow, each column in Γ has two

non-zero elements.

∆𝑷𝒇𝒍𝒐𝒘 = 𝚪 ∙ ∆𝒙 (6)

C. Loss Sensitivities

The sensitivity of losses (24) to real power line flows Τ is a

row vector of all ones with dimension of twice the number of

lines in the system, also the dimension of Pflow.

∆𝑃𝑙𝑜𝑠𝑠 = 𝚻 ∙ ∆𝑷𝒇𝒍𝒐𝒘 (7)

The aforementioned sensitivities define the complete

relationship between system losses and the reactive part of line

impedance. The total sensitivity of losses to line impedance Κ

has dimension equal to the number of lines with D-FACTS

devices:

𝚱 = 𝚻 𝚺 ∙ 𝚽 + 𝚪 (8)

∆𝑃𝑙𝑜𝑠𝑠 = 𝐊 ∙ ∆𝒙 (9)

The elements in Κ give the change in system losses due to a

small change in x and a solution of the power flow. Solution

of the power flow equations is important to consider.

Otherwise, our ability to analyze the impact of impedance-

changing devices would be limited to the direct sensitivities of

real power losses to line impedance found from Τ·Γ.

Including indirect sensitivities Τ·Σ·Φ in the analysis allows

representation of the impact of lines on other lines. The total

sensitivity representation (8) allows us to consider the use of a

D-FACTS device to provide control not only for the line on

which it is placed but also for other lines in the system.

IV. CONTROL POTENTIAL OF D-FACTS DEVICES

For any power system, it is useful to be able to determine

the control potential available from D-FACTS devices.


Analysis of the control of power systems with FACTS devices

[12], [13], [14] has been examined, but primarily with respect

to transient stability, where FACTS devices can be used for

control of certain modes of the system.

In this work, we are interested in the ability of D-FACTS

devices to provide control over line flows throughout the

system. When effective line impedances change, power flows

redistribute in the system. Our perspective is to show through

steady-state analysis the ability of D-FACTS devices to

control the way power flows distribute throughout the system.

A. Identification of Independently Controllable Line Flows

In some scenarios, it may be clear which lines need to be

targeted for control. The need to operate the system securely

is costly but crucial. D-FACTS devices can be used to relieve

a known overloaded element such as a line or transformer.

The ability to relieve an overloaded element through the use of

D-FACTS control is by itself a strong advantage. Since an

overloaded line or transformer can prevent many power

transfers from being able to take place, reducing the flow

through the overloaded element by even a few percent

improves the operation of the power grid.

From a broader perspective, D-FACTS devices can be used

throughout the system to provide the most comprehensive

control. In order to provide the most complete and effective

control for the entire system, it is necessary to identify how

the control of line flows are related to each other.

The coupling of the control of line flows is important to

understand so that money and control effort are not wasted in

attempts to independently control line flows which are highly

coupled. The following matrices show trivial cases where

control of line flows are completely decoupled (a) and

decoupled (b):

𝐚. 𝑥1 𝑥2 𝑥3

𝑃𝑓𝑙 ,1

𝑃𝑓𝑙 ,2

𝑃𝑓𝑙 ,3

1 0 00 2 00 0 1

𝐛. 𝑥1 𝑥2 𝑥3

𝑃𝑓𝑙 ,1

𝑃𝑓𝑙 ,2

𝑃𝑓𝑙 ,3

1 1 12 2 21 1 1

(10)

In the completely decoupled case, the vectors are orthogonal

and the angle between them is exactly 90 degrees. In the

completely coupled case, the row vectors are perfectly aligned

and the angle between them is exactly zero degrees. When the

row vectors are perfectly aligned but point in opposite

directions, the angle between them is 180 degrees, but they are

still completely coupled. Thus, coupling can be determined by

comparing the cosine of angles of vectors [15].

The cosine of the angle between two row vectors v1 and v2,

𝑐𝑜𝑠𝜃𝒗𝟏𝒗𝟐=

𝒗𝟏 ∙ 𝒗𝟐

𝒗𝟏 𝒗𝟐 (11)

of the total power flow to impedance sensitivity matrix

𝚺 ∙ 𝚽 + 𝚪 will be called the coupling index. The coupling

index has values between -1 and 1. When the coupling index

has an absolute value of 1, there is complete correlation, either

positive or negative, between the ways the two line flows

respond to D-FACTS control. When the coupling index is

zero, the line flows have the ability to be controlled

independently.

Consider the 4-bus example. The sensitivity matrix

𝚺 ∙ 𝚽 + 𝚪 for the system is given as follows,

𝑥(1,2) 𝑥(1,3) 𝑥(2,3) 𝑥(2,4) 𝑥(3,4)

𝑃𝑓𝑙(1,2)

𝑃𝑓𝑙(1,3)

𝑃𝑓𝑙(2,3)

𝑃𝑓𝑙(2,4)

𝑃𝑓𝑙(3,4)

1.23 1.43 −1.29 −1.84 −0.13

−1.22 −1.42 1.24 1.88 0.13

0.63 0.73 −3.77 3.67 0.27

0.63 0.74 2.46 −5.55 −0.41

−0.60 −0.71 −2.37 5.35 0.39

(12)

where the sending and receiving end line flows of a line

cannot be independently controlled, so they are not both

shown. The coupling indices are given by the following

symmetric matrix:

𝑃𝑓𝑙(1,2) 𝑃𝑓𝑙(1,3) 𝑃𝑓𝑙(2,3) 𝑃𝑓𝑙(2,4) 𝑃𝑓𝑙 (3,4)

𝑃𝑓𝑙(1,2)

𝑃𝑓𝑙(1,3)

𝑃𝑓𝑙(2,3)

𝑃𝑓𝑙(2,4)

𝑃𝑓𝑙(3,4)

1.00 −1.00 −0.01 0.50 −0.50

−1.00 1.00 0.03 −0.51 0.51

−0.01 0.03 1.00 −0.87 0.87

0.50 −0.51 −0.87 1.00 −1.00

−0.50 0.51 0.87 −1.00 1.00

(13)

The coupling indices indicate that the flow on lines (1,2) and

(1,3) cannot be independently controlled. Similarly, the flows

on (2,4) and (3,4) cannot be independently controlled. On the

other extreme, the coupling index between lines (1,2) and

(2,3) is very small which indicates a high ability to be

independently controlled.

The coupling indices provide insight into which line flows

should be targeted for control. For the 4-bus system, we will

choose to target lines (1,3) and (2,3) for control since they

have a small coupling index. The results of this control are

given in Section V.

B. Identification of Effective D-FACTS Locations

D-FACTS devices are unique because they are well-suited

to be placed at multiple locations in the system where their use

could be the most beneficial. Comparatively, if only one

FACTS device is used, all support goes to the same place.

However, reactive power support is most effective locally.

Sensitivities can be used to identify lines with a high impact

for particular applications. Lines with higher sensitivities are

able to provide more control, whereas lines with sensitivities

of zero have no impact. The locations for D-FACTS devices

are found by determining the lines with the highest

sensitivities for the objective.

Consider again the 4-bus system and its line flow

sensitivities given by (12). If the objective is to control the

flow of line (1,3), the relevant sensitivities are the second row

of (12):

𝑥 1,2 𝑥 1,3 𝑥 2,3 𝑥 2,4 𝑥 3,4

𝑃𝑓𝑙 1,3 −1.22 −1.42 1.24 1.88 0.13

(14)

In (14), the greatest impact on flow (1,3) comes from line

impedance (2,4); the next most effective line impedance is line

(1,3), etc. For controlling multiple line flows, D-FACTS

locations are similarly determined by finding the highest

sensitivities of the control objective to line impedance.


V. TRANSMISSION LINE POWER FLOW CONTROL

Once appropriate lines are targeted for control and effective

locations for D-FACTS devices are selected, the problem of

power flow control needs to be solved. The goal of the

problem can be stated as a desire to attain specified line flows

on any number of independently controllable lines through the

control of line impedance settings of D-FACTS devices on a

specified number of lines.

It is not always possible to achieve a specified power flow

on a line, so the line flow control equation, Pflow,calc(x) =

Pflow,spec(x) does not always have a solution. This is acceptable

because line flow control is merely an additional benefit. The

level of importance of a solution of the power balance

equations is much higher than the line flow control equations.

For any power system application, the power balance

equations f(p,q) must always be satisfied, but if some control

over the power flow on a line can be achieved, that can be

done as well.

Optimization methods are useful for problems that do not

have a solution [16]. The line flow control problem can be

examined in an optimization framework which reflects the

intuition behind what is being accomplished with D-FACTS

devices. The objective is to choose D-FACTS line impedance

settings to minimize the differences between the actual power

flows and the desired power flows. The objective function is

f0, where L is the number of line flows to be controlled:

𝑓0= 𝑷𝒇𝒍𝒐𝒘,𝑐𝑎𝑙𝑐 𝒙 -𝑷𝒇𝒍𝒐𝒘,𝑠𝑝𝑒𝑐 𝒙 𝑖

2𝑳

𝑖=1

(15)

The line flow control problem may be stated as follows:

min 𝑓0

𝑠𝑡 𝒇 𝑝 ,𝑞 (𝒔 𝜃 ,𝑉 ) = 0

𝒙 ≤ 𝒙𝑚𝑎𝑥 𝒙 ≥ 𝒙𝑚𝑖𝑛

(16)

The first constraint of (16) represents the AC power balance

equations. The next two constraints are constraints on how

much D-FACTS devices are able to change the line

impedances. The gradient of f0 is given by the following,

∇𝑓0 = 2𝝆(𝒙)𝑨′′ (17)

where the matrix A’’ is formed from elements of the power

flow to impedance total sensitivity matrix, 𝚺 ∙ 𝚽 + 𝚪. Thus,

D-FACTS devices are able to control line flows on any lines

with high enough sensitivities, not just their own line.

Important connections exist between sensitivities and

optimization theory [17], [18]. The sensitivities which

determine independently controllable line flows and effective

D-FACTS locations also exactly provide the gradient needed

to solve (15) using steepest descent. Steepest descent steps are

given by the following, where α is a positive, scalar step size:

𝒙𝑣+1 = 𝒙𝑣 − 𝛼∇𝑓0 (18)

Knowledge of the total sensitivity of an equation to the control

variables is enough to know how to minimize that function.

Minimizing the objective function is equivalent to controlling

real power line flows with D-FACTS devices.

A. 4-bus Test System

The results of control in the 4-bus case are presented here.

The approach described above is used to determine the D-

FACTS device settings. Lines (1,3) and (2,3) are first targeted

for control; their flow coupling index is 0.03. D-FACTS

devices are placed on lines (1,3), (2,3), and (2,4). The results

for the four possible control scenarios are shown in Table I. In

all cases, the control objective is achieved.

Table I. Decoupled Line Flow Control Control Case 1

Objective: Lower Flows on (1,3) and (2,3)

Original Flows

Line (1,3): 40.69 MW

Line(2,3): 80.17 MW

New Flows

Line (1,3): 37.82 MW

Line (2,3): 66.88 MW Control Case 2

Objective: Raise Flows on (1,3) and (2,3)

Original Flows

Line (1,3): 40.69 MW

Line(2,3): 80.17 MW

New Flows

Line (1,3): 42.53 MW

Line (2,3): 93.97 MW

Control Case 3

Objective: Raise Flow on (1,3) and Lower Flow on (2,3)

Original Flows

Line (1,3): 40.69 MW Line(2,3): 80.17 MW

New Flows

Line (1,3): 43.60 MW Line (2,3): 67.32 MW

Control Case 4

Objective: Lower Flow on (1,3) and Raise Flow on (2,3)

Original Flows

Line (1,3): 40.69 MW

Line(2,3): 80.17 MW

New Flows

Line (1,3): 36.58 MW

Line (2,3): 94.51 MW

Lines (1,2) and (2,3), with a flow coupling index of -1.0, are

also targeted for control. The results are given in Table II.

Table II. Coupled Line Flow Control Control Case 1

Objective: Lower Flows on (1,2) and (1,3)

Original Flows

Line (1,2): -40.57 MW

Line(1,3): 40.69 MW

New Flows

Line (1,2): -32.44 MW

Line (1,3): 32.61 MW

Control Case 2

Objective: Raise Flows on (1,2) and (1,3)

Original Flows

Line (1,2): -40.57 MW

Line(1,3): 40.69 MW

New Flows

Line (1,2): -48.71 MW

Line (1,3): 48.80 MW

Control Case 3

Objective: Raise Flow on (1,2) and Lower Flow on (1,3)

Original Flows

Line (1,2): -40.57 MW

Line(1,3): 40.69 MW

New Flows

Line (1,2): -42.37 MW

Line (1,3): 42.73 MW

Control Case 4

Objective: Lower Flow on (1,2) and Raise Flow on (1,3)

Original Flows

Line (1,2): -40.57 MW Line(1,3): 40.69 MW

New Flows

Line (1,2): -42.33 MW Line (1,3): 42.36 MW

The negative sign on the flows of line (1,2) indicates that the

flow is in the other direction. The control objectives in Table

II can not all be accomplished because lowering (raising) one

flow always lowers (raises) the other flow. In Cases 3 and 4,

however, the control objective can still be accomplished.

Thus, the decoupled line flows (1,3) and (2,3) in Table I are


shown to be independently controllable whereas the coupled

line flows (1,2) and (1,3) are not independently controllable.

B. Utility Test System

The ability of D-FACTS devices to improve power system

operation is evaluated for a system based upon a small North

American utility system provided by a US Midwest utility

who is closely involved in this project. The system of Figure

3 has 48 buses and 65 lines, although not all are in the utility’s

area. Buses 9 through 48 are in the utility’s area. The total

load is 571.44 MW.

Figure 3. Utility 48-bus Test System

The overloaded transformer between buses 7 and 8 is

targeted for control. It is assumed that the only lines which

may be equipped with D-FACTS devices are lines in the

utility’s area. Results for different numbers of lines with D-

FACTS devices are shown in Figure 4.

Figure 4. Utility Case, Line Flow Control

Significant flow reduction is achieved through the use of D-

FACTS devices on the five most sensitive lines. The flow

through the overloaded transformer is reduced from 284.91

MW to 270.90 MW (4.92%). The addition of D-FACTS

devices to increasingly less sensitive lines is not beneficial, as

shown by the flat portions of Figure 4.

Table III summarizes the results of the scenario where D-

FACTS devices are placed on the five most sensitive lines. For

the utility test system for this scenario, the D-FACTS settings

are all at their limits. When settings are at their limits, the

benefit of having D-FACTS devices will increase if the

amount of possible line impedance change increases. Also, if

line impedances are often set to their limits, settings can

potentially be chosen without completely solving the problem

for the optimal line impedance settings.

Table III. Utility System D-FACTS Line Impedance Settings

From

Bus #

To Bus

#

Original

x

High Limit

(+20%)

Low

Limit

(-20%)

D-FACTS

Setting

2 23 0.1498 0.17976 0.11984 Low Limit

6 15 0.00034 0.000408 0.000272 High Limit 8 13 0.0559 0.06708 0.04472 High Limit

10 46 0.02316 0.027792 0.018528 High Limit

13 15 0.01308 0.015696 0.010464 High Limit

VI. D-FACTS CONTROL FOR A GENERAL PROBLEM

The same control approach is extended to other power

system problems as follows

min 𝑓2(𝒔 𝜃 ,𝑉 , 𝒙)

𝑠𝑡 𝒇 𝑝 ,𝑞 (𝒔 𝜃 ,𝑉 ) = 0

𝒙 ≤ 𝒙𝑚𝑎𝑥 𝒙 ≥ 𝒙𝑚𝑖𝑛

(19)

where f2 is the objective function for the problem of interest

and D-FACTS devices are placed at locations in the system

determined by the sensitivities of the objective function f2 to

line impedance which are furthest from zero.

The direction of steepest descent is given by – 𝛻f2, where

𝛻f2 is the total derivative of the objective function with respect

to x. Line impedance settings to minimize f2 are

𝒙𝑣+1 = 𝒙𝑣 − 𝛼 ∙ 𝛻𝑓2 (20)

where α is a positive, scalar step size. D-FACTS devices may

then implement the final line impedance settings. This

approach can be used to implement D-FACTS applications

such as loss minimization and voltage control, briefly

described below.

A. Loss Minimization and Voltage Control

For loss minimization, f2 is the losses equation (24). The

total sensitivity of (24) to line impedances is given by 𝛻f2 = Κ,

where Κ comes from (8) and (9).

For voltage control including both raising and lowering

system voltages, f2 is the sum of the differences of the bus

voltages from specified values. The gradient 𝛻f2, is given by

𝛻f2, = 2𝜼(𝒙)𝚽V where ΦV, the sensitivities of voltages with

respect to line impedance, are the lower section of the state to

impedance sensitivity matrix, Φ= [𝚽θ , 𝚽V ]T .

B. Comments on Other Solution Methods

The general problem can potentially be solved using other

methods. The steepest descent optimization approach used in

this paper is a logical choice because it requires only

knowledge of the sensitivities and the ability to solve the

power flow, and it guarantees movement toward the optimum.

The ability to guarantee descent is important since the goal is

to determine the extent of D-FACTS abilities.

One approach, often using Newton’s method, treats the

effective reactances of D-FACTS devices as state variables

and solves the modified power flow equations for the line

impedances in addition to the other state variables. Problems

264

270

276

282

288

0 5 10 15 20 25 30

MW

Flo

w

Number of Lines with D-FACTS Devices


include that Newton’s method does not guarantee descent,

may not converge, and may not exhibit expected behavior if

started far from the solution. If second order sensitivities can

be calculated or approximated, the class of Newton-like

methods [18] may be worthwhile to investigate. Newton-like

methods also alleviate some of the problems with pure

Newton’s method.

VII. CONCLUSION

D-FACTS devices have the unique ability to be

incrementally installed on multiple lines throughout a system

to provide power flow control wherever needed. Effective D-

FACTS device locations and independently controllable flows

can be identified from sensitivities. After D-FACTS devices

are installed in certain fixed locations, their control objective

can easily be changed to target other lines flows. Thus, D-

FACTS devices can provide widespread, versatile control for

power systems.

In this paper, the successful control of line flows with D-

FACTS devices is presented for two test systems. A general

approach for line flow control with D-FACTS devices is

developed. The use of sensitivities in solving nonlinear

problems can be extrapolated to any application of interest and

to any system.

VIII. APPENDIX A

The AC power injection equations for real power P and

reactive power Q at a bus i are stated in (21) and (22),

𝑃𝑖 ,𝑐𝑎𝑙𝑐 =𝑉𝑖 𝑉𝑗 𝐺𝑖𝑗 𝑐𝑜𝑠 𝜃𝑖-𝜃𝑗 +𝐵𝑖𝑗 𝑠𝑖𝑛 𝜃𝑖-𝜃𝑗

𝑛

𝑗 =1

(21)

𝑄𝑖 ,𝑐𝑎𝑙𝑐 =𝑉𝑖 𝑉𝑗 𝐺𝑖𝑗 𝑠𝑖𝑛 𝜃𝑖-𝜃𝑗 -𝐵𝑖𝑗 𝑐𝑜𝑠 𝜃𝑖-𝜃𝑗

𝑛

𝑗 =1

(22)

where n is the number of buses. Power balance is expressed by

the vector f(p,q)(s(θ,V)) = ∆𝒑, ∆𝒒 𝑇 which must equal zero,

where s(θ,V) = 𝜽, 𝑽 𝑇 is a vector of bus voltage magnitudes

and angles, G+jB is the system admittance matrix,

∆𝑝𝑖=𝑃𝑖 ,𝑐𝑎𝑙𝑐 -(𝑃𝑖 ,𝑔𝑒𝑛 -𝑃𝑖 ,𝑙𝑜𝑎𝑑 ), and ∆𝑞𝑖=𝑃𝑖 ,𝑐𝑎𝑙𝑐 -(𝑄𝑖 ,𝑔𝑒𝑛 -𝑄𝑖 ,𝑙𝑜𝑎𝑑 ).

All real power line flows for the system comprise Pflow, and

system losses are the summation of all real power flows.

𝑃𝑓𝑙𝑜𝑤 ,𝑖𝑗 =-𝑉𝑖2𝐺𝑖𝑗 + 𝑉𝑖𝑉𝑗 𝐺𝑖𝑗 𝑐𝑜𝑠 𝜃𝑖-𝜃𝑗 +𝐵𝑖𝑗 𝑠𝑖𝑛 𝜃𝑖-𝜃𝑗 (23)

𝑃𝑙𝑜𝑠𝑠 = 𝑃𝑓𝑙𝑜𝑤 ,𝑖𝑗

𝑛

𝑗

𝑛

𝑖

𝑖 ≠ 𝑗 (24)

IX. REFERENCES

[1] FACTS Working Group, “Proposed Terms and Definitions for Flexible

AC Transmission System (FACTS)”, IEEE Transactions on Power

Delivery, Vol. 12, Issue 4, October 1997, p 1848-1853. [2] P. Hassink, D. Matthews, R. O'Keefe, F. Howell, S. Arabi, C. Edwards,

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[3] D. Divan, “Improving Power Line Utilization and Performance With D-

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2424.

[4] D. M. Divan, W. E. Brumsickle, R. S. Schneider, B. Kranz, R. W. Gascoigne, D. T. Bradshaw, M. R. Ingram, I. S. Grant, “A Distributed

Static Series Compensator System for Realizing Active Power Flow

Control on Existing Power Lines,” IEEE Transactions on Power Delivery, Vol. 22, No. 1, Jan 2007, p 642 - 649.

[5] H. Johal, D. Divan, “Design Considerations for Series-Connected

Distributed FACTS Converters,” IEEE Transactions on Industry Applications, Vol. 43, No. 6, Nov/Dec 2007, p 654 - 661.

[6] D. G. Ramey, R. J. Nelson, J. Bian, T. A. Lemak, “Use of FACTS

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on Power Systems, v 10, n 3, Aug. 1995, p 1401-7.

[14] X. R. Chen, N. C. Pahalawaththa, U. D. Annakkage, C. S. Kumble, “Output feedback TCSC controllers to improve damping of meshed

multi-machine power systems,” IEE Proceedings: Generation,

Transmission and Distribution, v 144, n 3, May, 1997, p 243-248. [15] A. M. A. Hamdan, A. M. Elabdalla, “Geometric Measures of Modal

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Power Systems Research, v 15, n 2, Oct 1988, p 147-155. [16] T. J. Overbye, “A Power Flow Measure for Unsolvable Cases,” IEEE

Transactions on Power Systems, Vol. 9, No. 3, Aug 1994, p 1359-1365.

[17] D. G. Luenberger, Linear and Nonlinear Programming, 2nd ed, Kluwer Academic Publishers Group, 2003.

[18] D. P. Bertsekas, Nonlinear Programming, 2nd ed, Massachusetts: Athena Scientific, 1999.

Katherine M. Rogers (S’05) received the B.S. degree in electrical engineering from the University of Texas at Austin in 2007 and the M.S.

degree from the University of Illinois Urbana-Champaign in 2009 and is

currently working toward the Ph.D. degree at the University of Illinois Urbana-Champaign. Her interests include sensitivity analysis, power system

analysis, and power system protection.

Thomas J. Overbye (S’87-M’92-SM’96-F’05) received the B.S., M.S. and

Ph.D. degrees in electrical engineering from the University of Wisconsin-

Madison. He is currently the Fox Family Professor of Electrical and Computer Engineering at the University of Illinois Urbana-Champaign. He

was with Madison Gas and Electric Company, Madison, WI, from 1983-1991.

His current research interests include power system visualization, power system analysis, and computer applications in power systems.


Documents

41.Power Flow Control with Distributed facts AC Transmission System (D-FACTS) Devices