Sensitivity and Particle Swarm Optimization-based ...

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Sensitivity and Particle Swarm Optimization-based Congestion Management a

K. S. Pandya & S. K. Joshi

b

a

Electrical Engineering Department, Charotar Institute of Technology, Gujarat, India b

Electrical Engineering Department, Maharaja Sayajirao University of Baroda, Gujarat, India Version of record first published: 30 Jan 2013.

To cite this article: K. S. Pandya & S. K. Joshi (2013): Sensitivity and Particle Swarm Optimizationbased Congestion Management, Electric Power Components and Systems, 41:4, 465-484 To link to this article: http://dx.doi.org/10.1080/15325008.2012.749555

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Electric Power Components and Systems, 41:465–484, 2013 Copyright © Taylor & Francis Group, LLC ISSN: 1532-5008 print/1532-5016 online DOI: 10.1080/15325008.2012.749555

Sensitivity and Particle Swarm Optimization-based Congestion Management K. S. PANDYA 1 and S. K. JOSHI 2 Downloaded by [Charotar Institute of Applied Sciences] at 21:08 25 February 2013

1

Electrical Engineering Department, Charotar Institute of Technology, Gujarat, India 2 Electrical Engineering Department, Maharaja Sayajirao University of Baroda, Gujarat, India Abstract In the competitive electricity market, it is not always possible to discharge all contracted power transactions due to congestion in one or more transmission lines. In most cases, the independent system operator tries to remove congestion by rescheduling output power of the generators. But all generators do not have the same effect (sensitivity) on the power flow of the congested lines, so it is not an economical way to reschedule output power of all generators for managing congestion. Thus, in this article, active and reactive power generator sensitivity factors of the generators to the congested lines have been used to determine the number of generators participating in congestion management. Second, a particle swarm optimization based algorithm has been suggested to minimize the deviations of rescheduled values of active power and reactive power of generators from scheduled values, considering voltage stability enhancement and voltage profile improvement criteria. Thus, rescheduling costs of active power and reactive power were minimized by particle swarm optimization. The effectiveness and feasibility of the proposed algorithm have been tested on IEEE 30bus and New England 39-bus systems, and the obtained results have been compared with previous literature in terms of solution quality. Keywords congestion management, particle swarm optimization, rescheduling, sensitivity analysis, voltage stability

1. Introduction Congestion is defined as the overloading of one or more transmission lines and/or transformers in the power system. In the deregulated electricity market, congestion occurs when the transmission system is unable to accommodate all of their desired transactions due to violation of MVA limits of transmission lines. In such a market, most of the time, the transmission lines operate near their stability limits, as all market players try to maximize their profits from various transactions by fully utilizing transmission systems. Congestion may also occur due to various factors, such as lack of coordination between generation companies (GENCOs) and transmission companies (TRANSCOs), contingency like generator/line outage, sudden change in load demand, and failure of various equipment. Congestion may lead to a rise in cost of electricity, tripping of overloaded lines, and consequential tripping of other healthy lines. It may also create voltage stability Received 1 June 2012; accepted 10 November 2012. Address correspondence to Mr. Kartik S. Pandya, Electrical Engineering Department, Charotar Institute of Technology, Changa, Gujarat, Changa, 388 421, India. E-mail: kartikpandya.ee@ ecchanga.ac.in, [email protected]

465

Downloaded by [Charotar Institute of Applied Sciences] at 21:08 25 February 2013

466

K. S. Pandya and S. K. Joshi

related problems. Congestion should be relieved to maintain power system stability and security; otherwise, it may result in system blackout with heavy loss of revenue. So, congestion management is given the highest priority, followed by cost recovery, etc., by the Federal Energy Regulatory Commission (FERC) [1] and many other utilities. Various algorithms and methods for congestion management have been proposed to date. In [2], congestion costs of pool and bilateral models were minimized considering active power redispatching of generators, but the effect of reactive power rescheduling of generators in managing congestion was ignored. Fang and David [3] proposed a method of willingness to pay to avoid curtailment of bilateral transaction to manage congestion. In [4], an analytical tool—the transmission dispatch and congestion management system—was developed to help the independent system operator (ISO) to manage congestion. In [5], applications of flexible AC transmission systems (FACTS) devices, for example, a thyristor-controlled series compensator (TCSC) and thyristorcontrolled phase angle regulator (TCPAR) were proposed to manage congestion rapidly and efficiently. Padhy [6] proposed a hybrid fuzzy model to determine optimal transactions and their corresponding load curtailment for managing congestion. Kumar et al. [7] suggested real and reactive power flow sensitivity indexes to identify different congested zones. Output powers of generators and the reactive power of optimally placed capacitor were rescheduled to minimize rescheduling cost. Talukdar et al. [8] proposed a method for selecting loads to be curtailed out and participating generators, based upon their sensitivities to the overloaded lines. However, reactive power was not rescheduled to remove congestion. Acharya and Mithulananthan [9] established a locational marginal price (LMP) difference and a congestion rent based concept to locate TCSC to reduce congestion. Yesuratnam and Thukaram [10] proposed a relative electrical distance (RED) based concept for real power rescheduling to manage congestion. But the generators with same RED contributed the same power to the congested line. In this case, the rescheduling cost was not optimized if generators had different cost functions. In [11], multi-objective particle swarm optimization (PSO) was used to alleviate congestion from the maximum number of lines and to minimize the total cost of generation. In [12], PSO was used to minimize the rescheduling cost of active power. However, the effect of rescheduling of reactive power output of generators and voltage stability constraints were ignored. The purpose of this article is to use an efficient method for selecting the number of participating generators and optimum rescheduling of active and reactive power output of generators for managing congestion at a minimum rescheduling cost. Generally, all generators do not have the same effect (sensitivity) on the power flow of a congested line. Some generators are a little sensitive to the congested line, so they do not affect the power flow of a congested line. So in a practical situation, only a few generators take part in removing congestion. Active power and reactive power sensitivity factors of generators to the congested line are learned first. The number of participating generators is selected from sensitivity factors. Second, active and reactive power rescheduling of participating generators is optimally done in such a way that the total active and reactive power rescheduling costs get minimized. Sometimes, congestion alleviation results in large voltage deviations or very low voltage profile at load buses, which may invite voltage collapse. So, output voltages of generators (reactive power output of generators) have been rescheduled to keep load bus voltages within the permissible limits. Also, voltage stability indicators (L-index [13]) of all load buses have been determined to check voltage stability in the post-rescheduling state. In this article, the congestion management problem has been formulated as a nonlinear optimal power flow (OPF) problem and has been solved by the PSO method,


Sensitivity and PSO-based Congestion Management

467

because classical methods have many limitations in solving such a problem. Conventional classical optimization methods, such as the gradient method, lambda iteration, linear programming, etc., rely on the convexity assumption of objective function. They fail to capture discontinuities of the objective function and may get trapped into local minima or diverge at all. The choice of initial starting point also affects the quality of the solution. Also, they could find only a single optimized solution in a single simulation run. PSO has shown its superiority over other classical and artificial intelligence methods in terms of solution quality and computational efficiency in solving many OPF problems, including the economic load dispatch problem [14, 15], the optimal reactive power dispatch (loss minimization) problem [16, 17], the unit commitment problem [18], artificial neural network training [19], price and load forecasting [20, 21], fault classification [22], and power quality related problems [23]. The effectiveness of the PSO-based algorithm has been tested on IEEE 30-bus and New England 39-bus systems. This article is organized as follows. Section 2 describes an overview of the PSO method, and Section 3 explains the theory for formulating generator sensitivity factors. Section 4 explains the formulation of an OPF problem and the proposed PSO-based algorithm for congestion management. Section 5 discusses the simulation results, and in Section 6, conclusive remarks are given.

2. Overview of PSO PSO is a fast, simple, and efficient population-based optimization method that was proposed by Kennedy and Eberhart [24]. Each particle updates its position based upon its own best position, global best position among particles, and its previous velocity vector according to the following equations: vikC1 D w vik C c1 r1 pbesti xik C c2 r2 gbest xik ; (1) xikC1 D xik C vikC1 ;

(2)

where vikC1 is the velocity of the i th particle at the .k C 1/th iteration; w is the inertia weight of the particle; vik is the velocity of the i th particle at the kth iteration; c1 and c2 are the positive constants having values between Œ0; 2:5; r1 and r2 are randomly generated numbers between Œ0; 1; pbesti is the best position of the i th particle obtained based upon its own experience; gbest is the global best position of the particle in the population; xikC1 is the position of the i th particle at the .k C 1/th iteration; xik is the position of the i th particle at the kth iteration; is the constriction factor, which may help ensure convergence. Suitable selection of inertia weight w provides good balance between global and local explorations: wmax wmin iter; (3) w D wmax itermax where wmax is the value of inertia weight at the beginning of iterations, wmin is the value of inertia weight at the end of iterations, iter is the current iteration number, and itermax is the maximum number of iterations.

468


3. Formulation of Generator Sensitivity Factors All generators have different sensitivities to the power flow of a congested line. A change in active power flow .Pij / in a transmission line k connected between bus i and bus j due to unit change in active power injection .P Ggn / at bus n by generator g can be defined as an active power generator sensitivity factor .GSPk gn /. Mathematically, it can be written for line k as


GSPk gn D

.Pij / : .P Ggn /

(4)

The detailed theory of Eq. (4) was given in [12]. Similarly, the reactive power generator sensitivity factor [7] for line k can be written as k GSQgn D

.Qij / : .QGgn /

(5)

The reactive power flow equation of line k can be written as Qij D Vi2 Bij C Vi Vj Gij sin.i

j /

Vi Vj Bij cos.i

j /;

(6)

where Vi and i are voltage magnitude and voltage angle of bus i , respectively. Gij and Bij are conductance and susceptance of line k connected between buses i and j , respectively. Neglecting Q-ı coupling, Eq. (5) can be written as

k GSQgn D

.@Qij / .@Vi / .@Qij / .@Vj / C : .@Vi / .@QGgn / .@Vj / .@QGgn /

(7)

The first and third terms of Eq. (7) can be obtained by differentiating Eq. (6) as follows: .@Qij / D 2Vi Bij C Vj Gij sin.i .@Vi /

j /

Vj Bij cos.i

j /

(8)

and .@Qij / D Vi Gij sin.i .@Vj /

j /

Vi Bij cos.i

j /:

(9)

The reactive power injected at bus i can be expressed as Qi D QGi

QDi ;

(10)


469

where QGi and QDi are reactive power generation and demand at bus i , respectively. Qi can be written as n X ˚ .Gij sin.i

Qi D jVi j2 Bi i C jVi j

j //jVj j ;

Bij cos.i

j /

j D1;j ¤i


∴

∴

@Qi D 2Bi i Vi C @Vi

n X ˚ .Gij sin.i

j /

j //jVj j ;

Bij cos.i

j D1;j ¤i

n X ˚ @Qi D jVi j .Gij sin.i @Vj

j /

j // :

Bij cos.i

j D1;j ¤i

(11)

(12)

(13)

Equations (12) and (13) are the matrices of partial derivatives of bus injected reactive powers at bus i and j with respect to bus voltage magnitudes at buses i and j , respectively. Taking the inverse of Eqs. (12) and (13) gives @Vi @Qi D @QGgn @Vi

1

@Qi @Vj D @QGgn @Vj

1

;

(14)

:

(15)

Equations (14) and (15) are second and fourth terms of Eq. (7), respectively. Thus, reactive power generator sensitivity factors could be obtained by using Eq. (7).

4. Optimization Problem Formulation and Proposed Algorithm The active and reactive power redispatching costs of generators for congestion management in a pool model is formulated as a non-linear OPF problem and has been solved by PSO:

Min

Ng X

CP g .Pg /Pg C

g

Ng X

CQg .Qg /Qg C k1 Lmax C k2

g

Nd X

j1

Vi j C PF; (16)

i D1

subject to the power flow balance equations (equality constraints), 8 ˆ ˆ ˆ ˆ PGi ˆ ˆ
> > ij / D 0 > > > = > > > > ij / D 0 > > ;

;

(17)

470


and various operating constraints (inequality constraints), Pg

Pgmin D Pgmin Pg Pgmax D Pgmax

Qg

Qgmin D Qgmin Qg Qgmax D Qgmax jSk j Skmax ;


Vi

Pg ; Qg ;

g8Ng ; g8Ng ;

(18) (19) (20)

k8Nl ;

Vimin D Vimin Vi Vimax D Vimax

Vi ;

i 8Nb :

(21)

Generator sensitivity factors are considered as an inequality constraint as given in Eq. (22): 0 12 0 12 Ng Ng X X k @ GSPk gn Pg C Pij A C @ GSQgn Qg C Qij A g

g

2 Sijmax ;

(22)

ij 2 Nl ;

where CP g is the cost of the active power rescheduling corresponding to the incremental/decremented price bids submitted by generator g participating in congestion management (these are the prices at which the generators are willing to adjust their real power output); Pg is the active power rescheduling of the generator g; Qg is the reactive power rescheduling of the generator g; CQg .Qg / is the cost of reactive power rescheduling (opportunity cost) of generator g participating in congestion management, expressed as q CQg .Qg / D CgP .SG max / CgP SG2 max Qg2 ; (23) where CgP is the cost of active power generation of generator g and is expressed as a quadratic function as 2 CgP .P Ggn / D an P Ggn C bn .P Ggn / C cn ; (24) where an , bn ; and cn are predetermined cost coefficients of the gth generator, SG max is the maximum apparent power limit of generator g, and is the profit rate of active power generation, taken between 5% to 10% (taken here as 7.5%); k1 is a constant D 10,000; as the L-index varies between 0 and 1, its value is multiplied by a large constant (e.g., 10,000) so that its effect becomes significant in the objective function; Lmax is the maximum value of voltage stability indicator (L-index, which gives a scalar value to each load bus and lay in the range from zero [no-load case] to unity [voltage collapse point]); k2 is a constant D 1000; Nd X i D1

j1

Vi j

(25)



471

is the voltage profile improvement (summation of load bus voltage deviations from 1.0 p.u.) PF is the penalty function; PGi and QGi are the active and reactive power generations at bus i ; PDi and QDi are the active and reactive power demands at bus i ; jVi j∠ıi is the complex voltage at bus i ; jYij j∠ij is the ij th element of bus admittance matrix; Pgmin and Pgmax are the real power generation limits of generator g; Pgmin and Pgmax are the minimum and maximum limits of the change in generator active power output, respectively; Qgmin and Qgmax are the reactive power generation limits of generator g; Qgmin and Qgmax are the minimum and maximum limits of the change in generator reactive power output, respectively; Sk is the power flow in transmission line k caused by all contracts requesting the transmission service; Skmax D Sijmax is the MVA flow limit of the kth transmission line connected between bus i and bus j ; Vimin and Vimax is the voltage magnitude limits at bus i ; Vimin and Vimax is the minimum and maximum limits of the change in bus voltage magnitude at bus i , respectively; Liindex is the voltage stability indicator (L-index) of bus i ; Nl is the total number of transmission lines; Nb is the total number of buses; Ng is the total number of generator buses; Nd is the total number of load buses; and Pij and Qij are the original active power and reactive power flow in line k (between bus i and bus j / caused by all transactions requesting the transmission service. The square penalty function is used to handle inequality constraints of active power output of the slack bus generator, reactive power output of generator buses, voltage magnitude of all buses, and MVA limit of all transmission lines, respectively, as shown in Eqs. (26) and (27): PF D k3 f .P1 / C k4

Ng X i D1

8 0 ˆ ˆ < f .x/ D .x x max /2 ˆ ˆ : min .x x/2

f .Qgi / C k5

Nb X i D1

9 if x min x x max > > = max if x > x ; > > ; min if x < x

f .Vi / C k6

Nl X

f .Sk /;

(26)

kD1

(27)

where k3 , k4 , k5 , and k6 are the penalty coefficients equal to 1000; and x min and x max are the minimum and maximum limits of variable x. The proposed PSO-based algorithm for congestion management is given next. Step 1: Run full AC Newton–Raphson load flow to identify the overloaded lines. Step 2: Find the sensitivity of all generators to the congested lines; i.e., find active power generator sensitivity factors and reactive power generator sensitivity factors of all generators corresponding to each congested line.

472

K. S. Pandya and S. K. Joshi Table 1 Representation of a particle

Particle no.


1 2 i

Generator active power rescheduling Pg1;1 Pg1;2 Pg1;i

PgNG;1 PgNG;2 PgNG;i

Generator voltage rescheduling (reactive power rescheduling) Vg1;1 Vg1;2 Vg1;i

VgNG;1 VgNG;2 VgNG;i

Step 3: Based upon obtained sensitivity factors, identify the generators that will take part in managing congestion. Step 4: Apply PSO. Initialize particles with their position and velocity. Each particle is made of continuous variables. The values of these variables are the amount of active power rescheduling (PgNG;i ) and amount of generator voltage rescheduling (VgNG;i ) required by the generators to manage congestion. As the reactive power output of a generator is a function of generator voltage, any rescheduling in generator output voltage will reschedule its reactive power. These variables are generated randomly within their permissible minimum and maximum limits. The particles can be presented in matrix form as shown in Table 1, where Pg1;i and PgNG;i are the active power rescheduling of participating generators of the i th particle, and Vg1;i and VgNG;i are the voltage rescheduling (reactive power rescheduling) of participating generators of the i th particle. If there are a total of i particles, and if each particle consists of j control variables, then the dimension of a population becomes i j . Step 5: Run full AC Newton–Raphson load flow to get line flows, active power rescheduling, reactive power rescheduling, line losses, and voltage magnitude of all buses. Step 6: Find constraint violation and calculate penalty function of each particle using Eq. (26). Step 7: Calculate the fitness function of each particle using Eq. (16). Step 8: Find out the “global best” .gbest / particle having the minimum value of fitness function in the whole population and the “personal best” .pbesti / of all particles. Step 9: Generate new population using Eqs. (1) and (2). Step 10: Go to Step 5 until convergence criterion is satisfied. Step 11: Stop the simulation.

5. Simulation Results and Discussion The simulation studies were carried out on Intel CORE 2 Duo processor (2 GB RAM, 2.20-GHz system, MATLAB 7.6 platform [The MathWorks, Natick, Massachusetts, USA]). The IEEE 30-bus system [25] and New England 39-bus system [26] have been used to test the effectiveness of the proposed algorithm. The performance of the proposed algorithm on IEEE 30-bus system has been compared with [8, 12], whereas the


473


Table 2 Details of power flow of the congested lines of IEEE 30-bus system Congested line

Power flow (MW)

Line flow limit (MW)

1-2 2-9

170.30 68.75

130 65

results obtained by the proposed method on the 39-bus system have been compared with [10, 12]. 5.1.

IEEE 30-bus Test System

The IEEE 30-bus test system consists of 6 generator buses, 24 load buses, and 41 transmission lines. The slack bus generator is assigned number 1; the remaining generators are assigned numbers 2, 3, 4, 5, and 6, respectively. Load buses are numbered from 7 to 30. Here, two lines, i.e., line 1 (between buses 1 and 2) and line 6 (between buses 2 and 9), are found to be congested. Details of power flow of the congested lines are given in Table 2. The values of active power generators sensitivity factors (GSPk gn ) and reactive k power generator sensitivity factors (GSQgn ) computed for lines 1-2 and 2-9 are given in Table 3. A negative value of sensitivity factor of a generator indicates that an increase in generation for that generator decreases the power flow in the congested line; a positive sensitivity factor of a generator indicates that an increase in generation increases power flow in the congested line. From Table 3, it is seen that generators 1, 2, 3, 4, and 6 have negative sensitivity factors, while generator 5 has positive active power sensitivity factors for both congested lines. Therefore, only generators 1, 2, 3, 4, and 6 were selected for removing congestion from the congested lines. Generator 5 did not take part in removing congestion. PSO was then applied to optimally reschedule active and reactive power output of generators to manage congestion. Table 4 shows the comparison of results obtained by PSO with those of other published studies. It can be seen that the cost of active power rescheduling and total

Table 3 Generator sensitivity factors of congested lines of IEEE 30-bus system Generator number

Congested lines

1

2

3

4

5

6

Line 1 (bus 1-2)

GSP1 gn 1 GSQgn

0.0 0.779

0.077 0.867

0.127 0.744

0.105 0.788

C0.17 0.761

0.420 0.775

Line 6 (bus 2-9)

GSP6 gn 6 GSQgn

0.0 0.356

0.014 0.351

0.029 0.327

0.029 0.368

C0.326 0.354

0.116 0.357

474

K. S. Pandya and S. K. Joshi Table 4 Comparison of results obtained by PSO for congestion management of IEEE 30-bus system Results given in [12]


Proposed method Cost of active power rescheduling ($/day) Cost of reactive power rescheduling ($/day) Resultant power flow (MW) Line 1 Line 6 Active power rescheduling (MW) P1 P2 P3 P4 P5 P6 Total active power rescheduling (MW) Reactive power rescheduling (MVAR) Q1 Q2 Q3 Q4 Q5 Q6 Total reactive power rescheduling (MVAR) Active power losses (MW)

Results given in [8]

31,286

50,466

50,700

7641

Not reported

Not reported

128.16 63.24

129 60

130 60

43.20 C16.67 C10.06 C14.20 Not participated C2.75 86.88

59 C19.9 C13 C6 C6.5 C7 111.4

58 C20.5 C14.5 C8 C9.2 — 110.2

29.99 C80.00 C00.94 00.00 Not participated 31.42 142.35

Not reported

Not reported

11.39

15

Not reported

active power rescheduling obtained by the proposed method is less than those of [8, 12]. Also, it is interesting to note that the total rescheduling cost (active power rescheduling cost C reactive power rescheduling cost) obtained by the proposed method is still less than those of [8, 12]. It is therefore preferable to reschedule the reactive power output of generators for removing congestion. Overloading of both congested lines was sufficiently removed by the proposed method. Also, obtained active power losses by the proposed method were fewer than that of [12].



475

Figure 1. L-index values of some load buses before and after rescheduling for IEEE 30-bus system.

Furthermore, reactive power rescheduling helped in improving voltage stability of the load buses, and it took the system far away from the voltage collapse point. It is clear from Figure 1 that voltage stability has increased because the L-index values of the load buses have considerably decreased in the post-rescheduling state. Reactive power rescheduling also decreased the deviation in voltage of load buses from the rated 1.0-p.u. value, thus improving the voltage profile of the load buses. The results are given in Table 5. 5.2.

New England 39-bus System

The New England 39-bus system is a simplified representation of the 345-kV transmission system in the New England region having 10 generator buses, 29 load buses, and 46 transmission lines. In this system, line 11 (between buses 16 and 17) is found to be congested. Its power flow details are given in Table 6. The values of generators sensitivity factors computed for the congested line are given in Table 7. It can be seen that generators 4, 5, and 9 have almost zero value of active power sensitivity factors, so any changes in the active power output of those generators do not reduce power flow of the congested line. The active power output of generators 4, 5, and 9 were therefore not rescheduled for managing congestion. From remaining

Table 5 Voltage stability and voltage deviation indicators in pre-rescheduling and post-rescheduling states of 30-bus system

Lmax PVoltage deviation

Pre-rescheduling

Post-rescheduling

0.1007

0.0815

1.205

0.659

476



Table 6 Details of power flow of a congested line of New England 39-bus system Congested line

Power flow (MW)

Line flow limit (MW)

Line 11 (bus 16-17)

531.25

425

generators, the generators with higher negative active power generator sensitivity factors were selected for congestion management, because any increase in active power output of those generators had decreased the power flow of the congested line. Hence, active power output of generators 1, 2, 3, 6, and 10 was rescheduled to alleviate congestion, as they had relatively higher negative active power sensitivity factors, whereas all generators had positive reactive power sensitivity factors to the congested line. Generally, reactive power output of many generators is decreased to reduce losses and power flow of a congested line; therefore, generators with higher positive sensitivity factors were selected for removing congestion, because a small decrease in reactive power output of those generators significantly reduced power flow of the congested line. As the reactive power sensitivity factors of generators 1, 3, 8, 9, and 10 had higher values than those of the remaining generators, their reactive powers had been rescheduled to manage congestion. It is worth noting that sometimes a generator that has maximum real power sensitivity may not have maximum reactive power sensitivity to the congested line. In such a case, if the generator has positive real power sensitivity, it will not be selected for congestion management because any rise in its real output power will increase the power flow of the congested line; if its output power is decreased, it will reduce the profit of the GENCO because it can sell a smaller amount of real power. Hence, generators with relatively higher negative real power sensitivity will be selected for congestion management. If the same generator has a negative reactive power sensitivity factor to the congested line, then any increase in its reactive output power will decrease the power flow of the congested line but will increase the power losses of the generator and, thus, reduce the efficiency of the generator. So the generator with a negative reactive power sensitivity should not be used for congestion management. Generators with relatively higher positive reactive power sensitivity are selected to manage congestion, and their output reactive powers are decreased to reduce the power flow of the congested line. From Table 8, it is clear that the total active power rescheduling required by the proposed method is greater than that of [10], but active power rescheduling cost obtained by the proposed method is lesser than that of [10]. This is because, in the proposed algorithm, active and reactive power rescheduling costs have been simultaneously optimized by PSO, whereas in [10], optimization of the active power rescheduling cost has not been conducted. As a result, it could not find the optimal solution. In addition, it can be seen that only five generators were sufficient to manage congestion without violating any system constraint, whereas in [10], the output power of all generators were rescheduled to manage congestion. It can be seen from Tables 9 and 10 that the proposed method could effectively remove congestion from the congested line as compared to other reported methods. It decreased the L-index values of the load buses and line losses. Reactive power rescheduling also decreased the deviation in voltage of load buses from the rated 1.0-p.u. value. Thus, it improved the voltage profile of the load buses.

477

0.0356

C0.0870

GSP11gn

11 GSQgn

1

C0.0764

0.0122

2

C0.0801

0.0126

3

C0.0700

0.0001

4

C0.0681

0.0002

5

C0.0732

0.0102

6

Generator number

C0.0747

0.0087

7

C0.0803

0.0072

8

Table 7 Generator sensitivity factors of a congested line 11 (bus 16-17) of New England 39-bus system


C0.0826

0.000

9

C0.0891

0.0239

10

478

K. S. Pandya and S. K. Joshi Table 8 Comparison of results obtained by PSO for congestion management of New England 39-bus system


Proposed method Cost of active power rescheduling ($/day) Cost of reactive power rescheduling ($/day) Active power rescheduling (MW) P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 Total active power rescheduling (MW) Reactive power rescheduling (MVAR) Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Total reactive power rescheduling (MVAR)



22,028

Not reported

35,911

711

Not reported

Not reported

756.62 C260.79 C211.97 Not participating Not participating C108.87 Not participating Not participating Not participating C192.37 1530.62

149.1 C65.6 129 Not participating Not participating Not participating Not participating C75.4 C52.1 C83 554.2

99.59 C98.75 159.64 C12.34 C24.69 C24.69 C12.34 C24.69 C12.34 C49.38 518.45

17.88 Not participating 120.74 Not participating Not participating Not participating Not participating 49.48 85.21 C113.75 387.06

Not reported

Not reported

The tunable parameters of PSO have great influence on its convergence characteristics, so tuning of parameters is quite essential. Thus, the effect of different values of various parameters (i.e., population size, acceleration constants, constriction factor, inertia weight, and velocity of particles) on the convergence of the algorithm was studied for 50 different trials. Finally, those values of parameters that gave the best rescheduling costs were selected, and they are given in Table 11 for both systems.

479

Line 11 — —

System component

Active power flow (MW) Vmin (p.u.) Active power loss (MW)

System parameter 531.25 0.929 59.35

Pre-rescheduling state

359.45 0.958 55.20

Proposed method

416.5 0.945 57.31


Post-rescheduling state

Table 9 Comparison of some results in pre-rescheduling and post-rescheduling states of 39-bus system


433.5 0.932 58.00


480

K. S. Pandya and S. K. Joshi Table 10 Voltage stability and voltage deviation indicators in pre-rescheduling and post-rescheduling states of 39-bus system


Lmax PVoltage deviation

5.3.

Pre-rescheduling

Post-rescheduling

0.1662 0.0444

0.1080 0.0371

Statistical Results and Convergence Characteristics of PSO

As PSO is a stochastic optimization method, it randomly generates the population of particles in each new simulation, so different results may be obtained in each simulation. Thus, simulations were carried out 50 times, and statistical results—namely, the worst, average, and best rescheduling costs—were obtained for both systems, as given in Table 12. Figures 2 and 3 show the graph of convergence characteristics of PSO for the 30-bus and 39-bus systems, respectively. From both figures, it is observed that total active power rescheduling costs gradually decrease with the number of iterations and finally obtain their minimum values. The nature of both graphs also indicates that the selected parameters of PSO are proper. It is seen that the PSO-based algorithm can remove congestion from the overloaded lines within 120 iterations, which justifies the fact that it is a fast method. An average simulation time required by the proposed method for the 30-bus system was approximately 4.5 min. Figure 4 shows the convergence characteristic of PSO for reactive power rescheduling cost of both systems. In this case also, reactive power rescheduling costs gradually decrease with the number of iterations and finally obtain their optimum (minimum) values.

Table 11 Selected parameters of PSO for IEEE 30-bus and New England 39-bus systems PSO parameters Population size (number of particles) Acceleration constants (C1 and C2 ) Constriction factor () Maximum and minimum inertia weights (wmax; wmin ) Maximum and minimum velocity of particles (vmax ; vmax ) Convergence criterion

IEEE 30-bus system

New England 39-bus system

50

75

2.1 and 2.0

1.4 and 1.4

0.729 1 and 0.2

0.30 0.7 and 0.3

0.45 and

0.45

0.25 and

Minimum 35 iterations

0.25


481

Table 12 Statistical results of rescheduling costs for IEEE 30-bus and New England 39-bus systems New England 39-bus system


IEEE 30-bus system Rescheduling cost ($/day)

Worst cost

Best cost

Average cost

Worst cost

Best cost

Average cost

Active power rescheduling cost ($/day) Reactive power rescheduling cost ($/day)

41,000

28,130

31,286

26,000

21,610

22,028

8200

6051

7641

1350

640

711

Figure 2. Convergence characteristic of PSO for active power rescheduling cost of 30-bus system.

6. Conclusions In this article, a PSO-based algorithm has been suggested for minimizing active power rescheduling cost and reactive power rescheduling cost of generators to alleviate congestion in IEEE 30-bus and New England 39-bus systems. The contribution of this article to the available literature can be concluded as follows. (1) Instead of using all generators for managing congestion, only a few generators can be used to manage congestion, and the generators that take part in congestion management may be selected based upon their sensitivities to the congested lines.


482


Figure 3. Convergence characteristic of PSO for active power rescheduling cost of 39-bus system.

Figure 4. Convergence characteristic of PSO for reactive power rescheduling cost.

(2) The effect of generator reactive powers should be considered in managing congestion. Rescheduling of reactive power of generators along with their active power rescheduling reduce overall rescheduling cost to manage congestion. (3) Reactive power rescheduling helps in improving the voltage profile of the load buses, and it enhances voltage stability of the system in the post-rescheduling state. (4) Losses obtained by the proposed method are significantly lower than those of other reported methods.


483

The proposed algorithm improves the performance of the system in the post-rescheduling state; this experiment shows encouraging results, suggesting that the proposed approach is capable of efficiently determining higher quality solutions addressing congestion management. In the future, the same algorithm will be extended to solve a dynamic congestion management problem.


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