Multi-stage fuzzy load frequency control using PSO

Energy Conversion and Management 49 (2008) 2570–2580

Contents lists available at ScienceDirect

Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Multi-stage fuzzy load frequency control using PSO H. Shayeghi a,*, A. Jalili b, H.A. Shayanfar c a b c

Technical Engineering Department, University of Mohaghegh Ardabili, Ardabil, Iran Islamic Azad University, Ardabil Branch, Ardabil, Iran Center of Excellence for Power Automation and Operation, Electrical Engineering Department, Iran University of Science and Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 15 September 2007 Accepted 18 May 2008 Available online 3 July 2008 Keywords: LFC MSF Restructured power system Power system control PSO

a b s t r a c t In this paper, a particle swarm optimization (PSO) based multi-stage fuzzy (PSOMSF) controller is proposed for solution of the load frequency control (LFC) problem in a restructured power system that operate under deregulation based on the bilateral policy scheme. In this strategy the control is tuned on line from the knowledge base and fuzzy inference, which request fewer sources and has two rule base sets. In the proposed method, for achieving the desired level of robust performance, exact tuning of membership functions is very important. Thus, to reduce the design effort and find a better fuzzy system control, membership functions are designed automatically by PSO algorithm, that has a strong ability to find the most optimistic results. The motivation for using the PSO technique is to reduce fuzzy system effort and take large parametric uncertainties into account. This newly developed control strategy combines the advantage of PSO and fuzzy system control techniques and leads to a flexible controller with simple stricture that is easy to implement. The proposed PSO based MSF (PSOMSF) controller is tested on a three-area restructured power system under different operating conditions and contract variations. The results of the proposed PSOMSF controller are compared with genetic algorithm based multi-stage fuzzy (GAMSF) control through some performance indices to illustrate its robust performance for a wide range of system parameters and load changes. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Global analysis of the power system markets shows that the frequency control is one of the most profitable ancillary services at these systems. This service is related to the short-term balance of energy and frequency of the power systems. The most common methods used to accomplish frequency control are generator governor response (primary frequency regulation) and load frequency control (LFC). The goal of LFC is to reestablish primary frequency regulation capacity, return the frequency to its nominal value and minimize unscheduled tie-line power flows between neighboring control areas. From the mechanisms used to manage the provision this service in ancillary markets, the bilateral contracts or competitive offers stand out [1]. During the past decade, several proposed LFC scenarios have been attempted to adapt traditional LFC schemes to the change of environment in the power systems under deregulation [2–4]. In a power system, each control area contains different kinds of uncertainties and various disturbances due to increased complexity, system modeling errors and changing power system structure. As a result, a fixed controller based on classical theory is not certainly suitable for the LFC problem. It is desirable that a flexible * Corresponding author. Tel.: +98 451 5517374; fax: +98 451 5512904. E-mail address: [email protected] (H. Shayeghi). 0196-8904/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2008.05.015

controller be developed. Efforts have been made to design load frequency controllers with better performance to cope with parameter changes, using various adaptive neural networks and robust methods [5–10]. The proposed methods show good dynamical responses, but robustness in the presence of model dynamical uncertainties and system nonlinearities were not considered. Also, some of them suggest complex state feedback or high order dynamical controllers, which are not practical for industry practices. Recently, some authors proposed fuzzy PID methods to improve performance of the LFC problem [11–13]. It should be pointed out that they require a three-dimensional rule base. This problem makes the design process is more difficult. To overcome this drawback, in author’s pervious papers [14,15] a improved control strategy based on fuzzy theory and GA technique have been proposed. The resulting structure is a multi-stage fuzzy (MSF) controller using two-dimensional inference engines (rule base) to perform reasonably the task of a three-dimensional controller. The proposed method requires fewer resources to operate and its role in the system response is more apparent, i.e. it is easier to understand the effect of a two-dimensional controller than a three-dimensional one [14]. In order for a fuzzy rule based control system to perform well, the fuzzy sets must be carefully designed. A major problem plaguing the effective use of this method is the difficulty of accurately constructing the membership functions. Because, it is a computationally expensive combinatorial optimization and also

H. Shayeghi et al. / Energy Conversion and Management 49 (2008) 2570–2580

extraction of an appropriate set of membership function from the expert may be tedious, time consuming and process specific. Thus, to reduce fuzzy system effort cost, in our pervious paper [15] a GA technique based on the hill climbing method have been proposed. Although, GA seems to be good methods to solve optimization problems, when applied to problems consisting of more number of local optima, the solution from GA are just near global optimum areas. Also, it takes long simulation time to obtain the solution. Moreover, when the number of parameter is more, optimization problem is complex and coding chromosomes with more gens for increasing algorithm accuracy is caused GA convergent speed will become very slow, so that convergent accuracy may be influenced by the slow convergent speed. Here, to overcome these drawbacks, a PSO based MSF (PSOMSF) controller is proposed. In this study, PSO technique is used for tuning membership functions of MSF controller. This method is proposed to improve optimization synthesis such that the global optima are guaranteed and the speed of algorithms convergence is extremely improved, too. PSO algorithm can be used to solve many of the same kinds of problems as GA and does not suffer from of GA’s difficulties [16–18]. PSO is a novel population based metaheuristic, which utilize the swarm intelligence generated by the cooperation and competition between the particle in a swarm and has emerged as a useful tool for engineering optimization. It has also been found to be robust in solving problems featuring nonlinearing, non-differentiability and high dimensionality. The proposed PSOMSF controller is tested on a three-area restructured power system under different operating conditions in comparison with the GAMSF [15] controller through some performance indices. Results evaluation show that the proposed method achieves good robust performance for wide range of system parameters and load changes in the presence of system nonlinearities and is superior to the other controllers.

2571

2. Generalized LFC model In the deregulated power systems, the vertically integrated utility no longer exists. However, the common LFC objectives, i.e. restoring the frequency and the net interchanges to their desired values for each control area, still remain. The deregulated power system consists of GENCOs, TRANSCOs and DISCOs with an open access policy. In the new structure, GENCOs may or may not participate in the LFC task and DISCOs have the liberty to contract with any available GENCOs in their own or other areas. Thus, various combinations of possible contracted scenarios between DISCOs and GENCOs are possible. All the transactions have to be cleared by the independent system operator (ISO) or other responsible organizations. In this new environment, it is desirable that a new model for LFC scheme be developed to account for the effects of possible load following contracts on system dynamics. Based on the idea presented in [19], the concept of an ‘ augmented generation participation matrix’ (AGPM) to express the possible contracts following is presented here. The AGPM shows the participation factor of a GENCO in the load following contract with a DISCO. The rows and columns of AGPM matrix equal the total number of GENCOs and DISCOs in the overall power system, respectively. Consider the number of GENCOs and DISCOs in area i be ni and mi in a large scale power system with N control areas. The structure of AGPM is given by

2

AGPM11 6 6 ... AGPM ¼ 4 AGPMN1

3 AGPM1N 7 7 ... 5    AGPMNN  .. .

where,

Fig. 1. The generalized LFC scheme in the restructured system.

ð1Þ

2572

H. Shayeghi et al. / Energy Conversion and Management 49 (2008) 2570–2580

2 6 6 AGPMij ¼ 6 4

gpf ðsi þ1Þðzj þ1Þ



gpf ðsi þ1Þðzj þmj Þ

.. .

..

.. .

.

   gpf ðsi þni Þðzj þmj Þ

gpf ðsi þni Þðzj þ1Þ

3

Table 2 GENCOs parameter

7 7 7 5

For i, j = 1, . . ., N and

si ¼

i1 X

ni ; zj ¼

j1 X

mj

and s1 ¼ z1 ¼ 0

In the above, gpfij refers to ‘generation participation factor’ and shows the participation factor of GENCO i in total load following requirement of DISCO j based on the contract. Sum of all entries in each column of AGPM is unity. The diagonal sub-matrices of AGPM correspond to local demands and off-diagonal sub-matrices correspond to demands of DISCOs in one area on GENCOs in another area. Block diagram of the generalized LFC scheme in a restructured system is shown in Fig. 1. The nomenclature used is given in Appendix A. Dashed lines show interfaces between areas and the demand signals based on the possible contracts. These new information signals are absent in the traditional LFC scheme. As there are many GENCOs in each area, ACE signal has to be distributed among them due to their ACE participation factor in the Pni LFC task and j¼1 aji ¼ 1. We can write [20]:

di ¼ DPLoc;j þ DPdi ; DPLoc;j ¼

mi X

DPLji ;

DPdi ¼

j¼1

gi ¼

N X

mi X

DPULji

ð2Þ

j¼1

T ij Dfj

fi ¼ DPtie;i;sch

ð3Þ

DPtie;ik;sch

ð4Þ

k¼1 and k6¼i nl X mk X j¼1



apf ðsl þjÞðzk þiÞ DPLðzk þiÞk

i¼1 nk X mi X i¼1

apf ðsk þiÞðzl þjÞ DPLðzl þjÞl

ð5Þ

j¼1

Area 1

G22

G12

D12

D11

G21

D21

G11

Area 3

D22

DPtie;ik;sch ¼

G23 D23

D13

G13

1-1

2-1

1-2

2-2

1-3

2-3

Rate (MW) TT (s) TH (s) R (Hz/pu)

1000 0.36 0.06 2.4 0.5

800 0.42 0.07 3.3 0.5

1100 0.44 0.06 2.5 0.5

900 0.4 0.08 2.4 0.5

1000 0.36 0.07 2.4 0.5

1020 0. 4 0.08 3.3 0.5

DPtie;ierror ¼ DPtie;iactual  fi

qi ¼ ½ q1i    qki    qni i ; qki ¼

"m j N X X j¼1

DPm;ki ¼ qki þ apf ki DPdi ;

k ¼ 1; 2; . . . ; ni

# gpf ðsi þkÞðzj þtÞ DP Ltj

ð6Þ ð7Þ

t¼1

ð8Þ

DPm,ki is the desired total power generation of a GENCO k in area i and must track the demand of the DISCOs in contract with it in the steady state. A three-area power system, shown in Fig. 2 is considered as a test system to illustrate the effectiveness of the proposed control strategy. It is assumed that each control area includes two GENCOs and DISCOs. The power system parameters are given in Tables 1 and 2. 3. PSO based MSF controller scheme

j¼1 and j6¼i N X

GENCOs (k in area i)

a

k¼1

k¼1

MVAbase (1000 MW) parameter

Gij : GENCO i-j Dij : DISCOi-j

Area 2 Fig. 2. A three-area restructured power system.

Table 1 Control area parameters Parameter

Area 1

Area 2

Area 3

KP (Hz/pu) TP (s) B (pu/Hz) Tij (pu/Hz)

120 20 0.8675 T12 = T13 = T23 = 0.545

72 14.3 0.785

91 10.6 0.87

Fuzzy logic control is one of the most successful areas in the application of fuzzy theory and is excellent alternatives to the conventional control methodology when the processes are too complex for analysis by conventional mathematical techniques [25,26]. Because of the complexity and multi-variable conditions of power systems, conventional control methods may not give satisfactory solutions. On the other hand, their robustness and reliability make fuzzy controllers useful for solving a wide range of control problems in power systems. In this paper, particle swarm optimization based multi-stage fuzzy controller is proposed for solution of the LFC problem. The motivation of using the proposed PSOMSF controller is to take large parametric uncertainties, system nonlinearities and minimize of area load disturbances into account. As MSF controller is elaborately explained in author’s pervious paper [14], here we points out only the salient features of it briefly. This control strategy combines fuzzy PD controller and integral controller with a fuzzy switch. The fuzzy PD stage is employed to penalize fast change and large overshoots in the control input due to corresponding practical constraints. The integral stage is also used in order to get disturbance rejection and zero steady state error. It should be noted that the exact tuning of membership functions in MSF control strategy is very important to achieve the desired level of system robust performance. Because it is a computationally expensive combinatorial optimization problem and also extraction of an appropriate set of membership function from the expert may be tedious, time consuming and process specific. In order to overcome this drawback and reduce fuzzy system effort and cost, PSO algorithm is being used to optimal tune of membership functions in the proposed MSF controller. Particle swarm optimization (PSO) algorithm, which is tailored for optimizing difficult numerical functions and based on metaphor of human social interaction, is capable of mimicking the ability of human societies to process knowledge [20]. It has roots in two main component methodologies: artificial life (such as bird flocking, fish schooling and swarming); and evolutionary computation. Although, the PSO algorithm is initially developed as a tool for modeling social behavior, it has been applied in different areas.

2573

H. Shayeghi et al. / Energy Conversion and Management 49 (2008) 2570–2580

Moreover, it has been recognized as a computational intelligence technique intimately related to evolutionary algorithms. Details of the original PSO algorithm can be found in Kennedy et al. [21– 23]. PSO is a populated search method for optimization of continuous nonlinear functions resembling the movement of organisms in a bird flock or fish school. Its key concept is that potential solutions are flown through hyperspace and are accelerated towards better or more optimum solutions. Its paradigm can be implemented in simple form of computer codes and is computationally inexpensive in terms of both memory requirements and speed. It lies somewhere between evolutionary programming and genetic algorithms. As in evolutionary computation paradigms, the concept of fitness is employed and candidate solutions to the problem are termed particles or sometimes individuals, each of which adjusts its flying based on the flying experiences of both itself and its companions. It keeps track of its coordinates in hyperspace which are associated with its previous best fitness solution, and also of its counterpart corresponding to the overall best value acquired thus far by any other particle in the population. Vectors are taken as presentation of particles since most optimization problems are convenient for such variable presentations. In fact, the fundamental principles of swarm intelligence are adaptability, diverse response, proximity, quality, and stability. It is adaptive corresponding to the change of the best group value. The allocation of responses between the individual and group values ensures a diversity of response. The higher dimensional space calculations of the PSO concept are performed over a series of time steps. The population is responding to the quality factors of the previous best individual values and the previous best group values. The principle of stability is adhered to since the population changes its state if and only if the best group value changes [24]. Fig. 3 shows the structure of the proposed PSOMSF controller for solution of the LFC problem. In this structure, input values are converted to truth-value vectors and applied to their respective rule base. The output truth-value vectors are not defuzzified to crisp value as with a single stage fuzzy logic controller but are passed onto the next stage as a truth value vector input. The darkened lines in Fig. 3 indicate truth value vectors. In this effort, all membership functions are defined as triangular partitions with seven segments from 1 to 1. Zero (ZO) is the center membership function which is centered at zero. The partitions are also symmetric about the ZO membership function as shown in Fig. 4. The remaining parts of the partition are negative big (NB), negative medium (NM), negative small (NS), positive small (PS), positive medium (PM) and positive big (PB). There are two rule bases used in the MSF controller. The first is called the PD rule bases as it operates on truth vectors form the error (e) and change in error (De) inputs. A typical PD rule base for

NB

-1

NM

NS

-b

-a

ZO

0

PS

a

PM

PB

b

1

Fig. 4. Symmetric fuzzy partition.

the fuzzy logic controller is given in Table 3. This rule base responds to a negative input from either error (e) or change in error (De) with a negative value thus driving the system to ward the commanded value. Table 4 shows a PID switch rule base. This rule base is designed to pass through the PD input if the PD input is not in zero fuzzy set. If the PD input is in the zero fuzzy set, then the R PID switch rule base passes the integral error values ð eÞ. This rule base operates as the behavior switch, giving control to PD feedback when the system is in motion and reverting to integral feedback to remove steady state error when the system is no longer moving. The operation used to determine the consequence value at the intersection of two input fuzzy value is given as

ci;j ¼ Pðai bj Þ;

i; j ¼ 1; 2; . . . Nm

ð9Þ

where, ai is the membership value of ith fuzzy set for a given e input, and bj is likewise for a De input. The operator used to determine the membership value of the kth consequence set is

Ck ¼

X

C i;j ;

i; j ¼ 1; 2; . . . Nm

ð10Þ

The defuzzification uses the weighted average method where Ck is the peak point of the kth output fuzzy membership function.

d¼

X

C k ck =

X

C k k ¼ 1; . . . ; N ðsets in output pointÞ:

3.1. Membership functions tuning strategy by PSO In the proposed PSOMSF controller, we must tune the linguistic hedge combinations which are difficult to be contributed according to human experience and knowledge. To acquire an optimal combination, this paper employs PSO technique [27] to improve optimization synthesis and find the global optimum value of fitness function. PSO is a population based evolutionary search algorithms characterized as conceptually simple, easy to impalement and computationally efficient. As it is reported in [28], this optimization technique can be used to solve many of the same kinds of problems as GA, and does not suffer from some of GAs difficulties. PSO has also been found to be robust in solving problem featuring

PSO Technique

Kds yref

Fuzzyfy P

Ki/s

di

Fuzzyfy D PID Switch Rule Base

PD Rule Base

ð11Þ

Fuzzyfy I

Defuzzify

HF Controller

PSOMSF Controller

Fig. 3. Structure of the proposed PSOMSF control strategy.

ζi

ρi

Nominal Model of Area i

ACEi

2574

H. Shayeghi et al. / Energy Conversion and Management 49 (2008) 2570–2580

Table 3 PD rule base

De

e

NB NM NS ZO PS PM PB

NB

NM

NS

ZO

PS

PM

PB

NB NB NB NB NM NS ZO

NB NB NB NM NS ZO PS

NB NB NM NS ZO PS PM

NB NM NS ZO PS PM PB

NM NS ZO PS PM PB PB

NS ZO PS PM PB PB PB

ZO PS PM PB PB PB PB

Table 4 PID switch rule base PD values R

e

NB NM NS ZO PS PM PB

NB

NM

NS

ZO

PS

PM

PB

NB NB NB NB NB NB NB

NM NM NM NM M NM NM

NS NS NS NS NS NS NS

NB NM NS ZO PS PM PB

PS PS PS PS PS PS PS

PM PM PM PM PM PM PM

PB PB PB PB PB PB PB

nonlinearing, non-differentiability and high-dimensionality. It is as the search method to improve the speed of convergence and find the global optimum value of fitness function. In this work, the PSO module works offline. PSO searches the optimal linguistic hedge combination according to the controlled plants. According to Fig. 4 for exact tuning of used membership functions in the proposed method we must find the optimal value for a and b parameters, where 0 < a < b < 1. This new approach features many advantages; it is simple, fast and can be coded in few lines. Also, its storage requirement is minimal. Moreover, this approach is advantageous over evolutionary and genetic algorithms in many ways. First, PSO has memory. That is, every particle remembers its best solution (local best) as well as the group best solution (global best). Another advantage of PSO is that the initial population of the PSO is maintained, and so there is no need for applying operators to the population, a process that is time and memory-storage-consuming. In addition, PSO is based on ‘‘constructive cooperation” between particles, in contrast with the

genetic algorithms, which are based on ‘‘ the survival of the fittest”. PSO starts with a population of random solutions ‘‘particles” in a D-dimension space. The ith particle is represented by Xi = (xi1, xi2, . . . , xiD). Each particle keeps track of its coordinates in hyperspace, which are associated with the fittest solution it has achieved so far. The value of the fitness for particle i (pbest) is also stored as Pi = (pi1, pi2, . . . ,piD). The global version of PSO keeps track of the overall best value (gbest), and its location, obtained thus far by any particle in the population. PSO consists of, at each step, changing the velocity of each particle toward its pbest and gbest according to Eq. (12). The velocity of particle i is represented as Vi = (vi1, vi2, . . . ,viD). Acceleration is weighted by a random term, with separate random numbers being generated for acceleration toward pbest and gbest. The position of the ith particle is then updated according to Eq. (13) [27].

vid ¼ w  vid þ c1  randðÞ  ðPid  xid Þ þ c2  randðÞ  ðPgd  xid Þ ð12Þ xid ¼ xid þ vid

ð13Þ

where, pid = pbest and pgd = gbest. Several modifications have been proposed in the literature to improve PSO algorithm speed and convergence toward the global minimum. One modification is to introduce a local-oriented paradigm (lbest) with different neighborhoods. It is concluded that gbest version performs best in terms of median number of iterations to converge. However, lbest version with neighborhoods of two is most resistant to local minima. PSO algorithm is further improved via using a time decreasing inertia weight, which leads to a reduction in the number of iterations [27]. Fig. 5 shows the flowchart of the proposed PSO algorithm. It should be noted that choice properly fitness function is very important in synthesis procedure. Because different fitness functions promote different PSO behaviors, which generate fitness value providing a performance measure of the problem considered [14]. For our optimization problem, the flowing fitness function is introduced:

f ðITAEÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P3 i¼1 ITAEi 3

;

ITAEi ¼

Z

t

Select parameters of PSO: N, C1, C2, C and w Generate the randomly positions and velocities of particles

Update pbest and gbest Initialize, pbest with a copy of the position for particle, determine gbest Satisfying stopping criterion

ð14Þ

Here, PSO procedure is applied to exact tune of membership functions of the proposed MSF controller for the solution of LFC problem. Results of membership function set values are listed in Table 5 and Fig. 6 shows the minimum fitness value evolving pro-

Start

Evaluate the fitness of each particle

tjACEi jdt

0

No Update velocities and positions according to Eqs. (12) and (13)

Yes Optimal value of member ship functions hedge End Fig. 5. Flowchart of the proposed PSO technique for exact tuning of membership functions.

2575

H. Shayeghi et al. / Energy Conversion and Management 49 (2008) 2570–2580 Table 5 Optimal values of parameters a and b Membership function

ACEi

DACEi

R

ACE

Output

PSO Result

0.0563 0.1384

0.0288 0.4014

0.0362 0.4091

0.1232 0.8016

a b

Fitness Valuw

0.95

0.9

0.85

4.1. Case 1 In this scenario, DISCOs have the freedom to have a contract with any GENCO in their or another areas. Consider that all the DISCOs contract with the available GENCOs for power as per following AGPM. All GENCOs participate in the LFC task.

0.8

0.75

it). This is to take GRC into account, i.e. the practical limit on the rate of the change in the generating power of each GENCO. The results in Refs. [20,26] indicated that GRC would influence the dynamic responses of the system significantly and lead to larger overshot and longer settling time. The proposed PSOMSF controller is applied for each control area of the restructured power system as shown in Fig. 2. To illustrate robustness of the proposed control strategy against parametric uncertainties and contract variations, simulations are carried out for four cases of possible contracts under various operating conditions and large load demands. Performance of the proposed PSOMSF controller is compared with the GAMSF [15] controller.

0

50

100 150 200 250 0

350 400 450 500

Iteration

2

Fig. 6. Minimum fitness value convergence.

cess. In order to acquire better performance, number of particle, particle size, number of iteration, C1, C2, C is chosen as 20, 8, 500, 2, 2 and 1, respectively. Also, the inertia weight, w, is linearly decreasing from 0.9 to 0.4. 4. Simulation results In the simulation study, the linear model of turbine DPVki/DPTki in Fig. 1 is replaced by a nonlinear model of Fig. 7 (with ±0.05 lim-

ΔPVki

1 TTki

-

d -d

1 s

ΔPT ki

0:25

0

0:5 0

6 0:5 0:25 0 0:25 0 6 6 6 0 0:5 0:25 0 0 6 AGPM ¼ 6 6 0:25 0 0:5 0:75 0 6 6 0:25 0 0 0:5 4 0 0

0

0

0

0

3

07 7 7 07 7 7 07 7 7 05 1

It is assume that a large step load 0.1 pu MW is demanded by each DISCOs in all areas. Power system responses with 25% decrease in system parameters are shown in Fig. 8. Using the proposed method, the frequency deviation of the all areas are quickly driven back to zero and has small settling time. Also the tie-line power flow properly converges to the specified value, of Eq. (5), in the steady state case (Fig. 8), i.e.; DPtie21,sch = 0 and DPtie23,sch = +0.025 pu MW.

In this case, It may happen that a DISCO violates a contract by demanding more power than that specified in the contract. This excess power must be reflected as a local load of the area but not

0.2 0

ΔPtie 1-2

ΔF 1

0.06

-1

0

10

20

0

30

0.8

ΔF2

0

4.2. Case 2

Fig. 7. Nonlinear turbine model with GRC.

-0.04

0

10

0

10

20

30

20

30

0

0.1

-1

0

10

20

30

ΔPtie 2-3

0

ΔF 3

0:25

-0.4

0.04 0 -0.02

-1

0

10

20

Time (Sec)

30

Time (Sec)

Fig. 8. Deviation of frequency and tie-lines power flows; solid (PSOMSF) and dashed (GAMSF).

2576

H. Shayeghi et al. / Energy Conversion and Management 49 (2008) 2570–2580

as the contracted demand and taken up by GENCOs in the same area. Consider case 1 again with modification that DISCO of areas 1, 2 and 3 demands 0.07, 0.04 and 0.02 pu MW of excess power, respectively. The total local load in areas is computed as

DPLoc;1 ¼ 0:1 þ 0:1 þ 0:07 ¼ 0:27; DPLoc;2 ¼ 0:1 þ 0:1 þ 0:04 ¼ 0:24; DPLoc;3 ¼ 0:1 þ 0:1 þ 0:01 ¼ 0:21 pu MW

4.3. Case 3 Consider case 2 again. Assume, in addition to the specified contracted loead demands 0.1 pu MW, a bounded random step load change as a large uncontracted demand as shown in Fig. 10 appears in each control area, where

0:05 6 DPdi 6 þ0:05 pu MW

Power system responses in this scenario with 25% increase in system parameters are shown in Fig. 9. Using the proposed method, the frequency deviation of all areas and tie-line power flows are quickly driven back to zero and have small settling time. Also, the tie-line power flows properly converges to the specified value, Eq. (5), in the steady state, i.e.: DPtie21,sch = 0 and DPtie31,sch = + 0.025 pu MW.

The propose of this scenario is to test the robustness of the proposed controller against uncertainties and random large load disturbances. The power system responses in this case with 25% decrease in system parameters are shown in Figs. 11–13. The simulation results demonstrate that the proposed control strategy track the load fluctuations and meet robustness for a wide range of load disturbances and possible contract scenarios under plant parameter changes.

0.5

0.04

0

ΔPtie 1-2

ΔF1

0

-1.4 0

10

30

20

-0.08

ΔF 2

1

0

10

20

30

0

10

20

30

0

-1.5 0

10

20

0.09

30

ΔPtie 2-3

0.5

ΔF 3

0

0

-0.07

-1.5 0

10

20

30

Fig. 9. Deviation of frequency and tile lines power flows; solid (PSOMSF) and dashed (GAMSF).

ΔPd 1

0.24 0.2 0.16 0

40

80

120

160

200

0

40

80

120

160

200

0

40

80

120

160

200

ΔPd 2

0.24 0.2 0.16

ΔPd 3

0.24 0.2 0.16

Fig. 10. Random disturbance load patterns in areas 1, 2 and 3.

2577

H. Shayeghi et al. / Energy Conversion and Management 49 (2008) 2570–2580

0.6

ΔF1

0

-0.6

-1.2 0

40

80

120

160

200

0

40

80

120

160

200

0

40

80

120

160

200

ΔF 2

1.5

0

-1.5 0.4

ΔF3

0

-0.6

-1.4 Fig. 11. Deviation of frequency, solid (PSOMSF) and dashed (GAMSF).

ΔF 1

0.15

0

-0.1 20

60

100

140

ΔF2

0.15

0

20

40

60

80

100

120

140

20

40

60

80

100

120

140

ΔF 3

0.08

0

-0.12 Fig. 12. Scales of deviation of frequency, solid (PSOMSF) and dashed (GAMSF).

To demonstrate performance robustness of the proposed method, the integral of the time multiplied absolute value of the error

(ITAE) and figure of demerit (FD) based on the system performance characteristics are being used as

2578

H. Shayeghi et al. / Energy Conversion and Management 49 (2008) 2570–2580

ΔPtie 1-2

0.08

0

-0.08 0

40

80

120

160

200

0

40

80

120

160

200

ΔPtie 2-3

0.12

0.06

0

Fig. 13. Deviation tile lines power flows; solid (PSOMSF), dashed (GAMSF).

Table 6 Value of ITAE Test no.

PPC (%)

1 2 3 4 5 0 (Nominal) 6 7 8 9 10

ITAE ¼

Table 7 Value of FD

Z

25 20 15 10 5 0 +5 +10 +15 +20 +25

Scenario 1

Scenario 2

Test no.

PSOMSF

GAMSF

PSOMSF

GAMSF

764.9 787.0 803.4 824.6 869.4 918.7 983.1 1059.1 1128.6 1187.5 1232.0

866.0 899.0 942.2 981.8 1046.9 1128.9 1269.6 1469.8 1631.1 1738.2 1920.3

1526.6 1561.2 1593.4 1616.0 1641.2 1733.6 1846.9 2002.1 2150.3 2281.1 2406.5

2030.2 2076.6 2289.4 2518.1 2755.6 2989.9 3263.3 3439.5 3520.1 3645.4 3678.2

10

tðjACE1 ðtÞj þ jACE2 ðtÞj þ jACE3 ðtÞjÞdt

ð15Þ

0

FD ¼ ðOS  7Þ2 þ ðUS  4Þ2 þ ðTs  0:1Þ2

1 2 3 4 5 0 (Nominal) 6 7 8 9 10

PPC (%)

25 20 15 10 5 0 +5 +10 +15 +20 +25

Scenario 1

Scenario 2

PSOMSF

GAMSF

PSOMSF

GAMSF

295.7 271.3 212.84 203.7 201.9 202.2 204.3 206.2 207.6 209.1 209.2

584.9 551.7 534.3 514.0 511.7 541.1 614.4 757.6 869.7 874.9 1010.1

1155.7 991.5 831.4 686.4 580.1 557.5 554.8 558.5 552.3 545.3 558.2

2219.3 2103.3 2191.9 2278.5 2313.7 2260.6 23156.0 2183.1 1859.8 1806.7 1991.1

Remark 4.1. Examination of Figs. 14 and 15 reveals that the proposed control strategy achieves good robust performance against parametric uncertainties and is superior to other controllers.

ð16Þ

where, overshoot (OS), undershoot (US) and settling time (for 3% band of the total load demand in area 1) of frequency deviation area 1 is considered for evaluation of the FD. Numerical results of performance robustness for cases 1 and 2 are listed in Tables 6 and 7, whereas the system parameters are varied from 25% to 25% of the nominal values. Figs. 14 and 15 shows the maximum variation percent of ITAE and FD from theirs values in the nominal conditions.

Remark 4.2. The worst case, as seen from Tables 6 and 7, occurs when the all system parameters have been had +25% increasing from their nominal values. Remark 4.3. we have considered different cases for LFC control of a three-area power system. The simulation results show that in comparison with GAMSF controller, the system performance is significantly improved by the PSOMSF controller designed in this paper for a wide range of load disturbances and possible contract

Fig. 14. Values of performance indices under cases 1: (a) ITAE (b) FD.

H. Shayeghi et al. / Energy Conversion and Management 49 (2008) 2570–2580

2579

Fig. 15. Values of performance indices under cases 2: (a) ITAE (b) FD.

scenarios under different plant parameter changes even in the presence of GRC.

5. Conclusions A new PSO based multi-stage fuzzy controller is proposed for solution of the LFC problem in a restructured power system in this paper. This control strategy was chosen because of increasing the complexity and changing structure of power systems. This newly developed control strategy combines advantage a fuzzy PD and integral controllers with a fuzzy switch to achieve the desired level of robust performance, such as frequency regulation, tracking of load demand and disturbance attenuation under load fluctuation for a wide range of the plant parameters changes and system nonlinearities. In order to reduce design effort and find better fuzzy system control, a PSO based algorithm has been used to tune membership functions automatically. The PSO algorithm proposed in this paper is easy to implement without additional computational complexity. Thereby experiments this algorithm gives quite promising results. The ability to jump out the local optima, the convergence precision and speed are remarkably enhanced and thus the high precision and efficiency are achieved. The effectiveness of the proposed method is tested on a threearea restructured power system for a wide range of load demands and disturbances under different operating conditions. Compared with GAs in term of ITAE and FD, the PSO demonstrates its superiority in computational complexity, success rate and solution quality and following conclusions can be drawn about the proposed method. 1. It is characterized as a simple heuristic of well balanced mechanism with flexibility to enhance and due to non-model base, it can be used to control a wide range of complex and nonlinear systems. 2. It does not depend on the nature of the function it minimizes and is insensitive to the initial searching point, there by ensuring quality solution for different trial runs. 3. It is effective and ensures robust performance for a wide range operating conditions under plant parameter changes and system nonlinearities. Also, it dose not require an accurate model of LFC problem, has simple structure and easy to implement. Moreover, the design process is less demanding than that of other controllers. 4. The system performance characteristics in terms of ‘ITAE’ and ‘FD’ indices reveal that this control strategy is a promising control scheme for the AGC problem in the deregulated power systems. Appendix A. Nomenclature F Ptie

area frequency net tie-line power flow turbine power

PT PV PC ACE apf D KP TP TT TH R B Tij Pd PLji PULji Pm,ji PLoc

g f

turbine power governor valve position governor set point area control error ACE participation factor deviation from nominal value subsystem equivalent gain subsystem equivalent time constant turbine time constant governor time constant droop characteristic frequency bias tie line synchronizing coefficient between areas i and j area load disturbance contracted demand of Disco j in area i un-contracted demand of Disco j in area i power generation of GENCO j in area i total local demand area interface scheduled power tie-line power flow deviation (DPtie,sch.)

References [1] Raineri R, Rios S, Schiele D. Technical and economic aspects of ancillary services markets in the electric power industry: an international comparison. Energy Policy 2006;34(13):1540–55. [2] Christie RD, Bose A. Load frequency control issues in power system operations after deregulation. IEEE Trans Power Syst 1996;11(3):1191–200. [3] Kumar J, Hoe NG, Sheble G. LFC simulator for price-based operation. Part I: Modeling. IEEE Trans Power Syst 1997;12(2):527–32. [4] Donde V, Pai A, Hiskens IA. Simulation and optimization in a LFC system after deregulation. IEEE Trans Power Syst 2001;16(3):481–9. [5] Kazemi M, Karrari M, Menhajm M. Decentralized robust adaptive-output feedback controller for power system load frequency control. Elect Eng 2002;84(2):75–83. [6] Zeynelgil HL, Demiroren A, Sengor NS. The Application of ANN technique to automatic generation control for multi-area power system. Elect Power Energy Syst 2002;24:354345–545354. [7] Bevrani H, Mitani Y, Tsuji K. Robust decentralized LFC in a restructured power system. Energy Convers Manage 2004;45:2297–312. [8] Lim KY, Wang Y, Zhou R. Robust decentralized load frequency control of multiarea power system. IEE Proc Gener Transm Distrib 1996;43(5):377–86. [9] Chang C, Fu W. Area load frequency control using fuzzy gain scheduling of PI controllers. Elect Power Syst Res 1997;42:145–52. [10] Feliachi A. On load frequency control in a restructured environment. In: Proceeding of the IEEE international conference on control application, 15–18 September 1996. p. 437–41. [11] Shayeghi H, Shayanfar HA, Jalili A, Khazaraee M. Area load frequency control using fuzzy PID type controller in a restructured power system. In: Proceedings of the international conference on artificial intelligence Las Vegas Nevada, USA; June 27–30, 2005. p. 344–50. [12] Yesil E, Guzelkaya M, Eksin I. Self tuning fuzzy PID type load and frequency controller. Energy Convers Manage 2004;45:377–90. [13] Petrov M, Ganchev I, Taneva A. Fuzzy PID control of nonlinear plants. In: Proceedings of the first international IEEE symposium on intelligence systems, September 2002. p. 30–5. [14] Shayeghi H, Shayanfar HA, Jalili A. Multi stage fuzzy PID power system automatic generation controller in the deregulated environment. Energy Convers Manage 2006;47(18):2829–45. [15] Shayeghi H, Jalili A, Shayanfar HA. Robust modified GA based multi-stage fuzzy LFC. J Energy Convers Manage 2007;48(5):1656–70. [16] Zahiria SH, Seyedin SA. Swarm intelligence based classifiers. J Franklin Inst 2007;344:362–76.

2580

H. Shayeghi et al. / Energy Conversion and Management 49 (2008) 2570–2580

[17] Higashi N, Iba H. Particle swarm optimization with Gaussian mutation. In: Proceedings of the IEEE swarm intelligence symposium (SIS’03), 2003. p. 72–9. [18] Yang S, Wang M, Jiao L. A quantum particle swarm optimization. In: Proceedings of the congress on evolutionary computation. vol. 1, 2004. p. 320–24. [19] Shayeghi H, Shayanfar HA, Malik OP. Robust decentralized neural networks based LFC in a deregulated power system. Elect Power Syst Res 2007;77(3):241–51. [20] Kennedy J. The particle swarm: social adaptation of knowledge. In: Proceedings of the international conference on evolutionary and computation, Indianapolis; 1997. p. 303–08. [21] Kennedy J, Eberhart R, Shi Y. Swarm intelligence. San Francisco: Morgan Kaufman Publishers; 2001. [22] Clerc M, Kennedy J. The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE Trans Evolut Comput 2002;6(1):58–73. [23] Chau KW. Particle swarm optimization training algorithm for ANNs in stage prediction of Shing Mun River. J Hydrol 2006;329(3–4):363–7. [24] Chau KW. Application of a PSO-based neural network in analysis of outcomes of construction claims. Automat Constr 2007;16:642–6. [25] Chen G, Ogmen H. Fuzzy PID controller: design, performance evaluation and stability analysis. Inform Sci 2000;123:249–70. [26] Visioli A. Tuning of PID controllers with fuzzy logic. In: Proceeding of the IEEE international conference on control theory and applications. vol. 4, No. 1, January 2001, p. 1–8. [27] Al-Awami Ali T, Abdel-Magid YL, Abido MA. A particle-swarm-based approach of power system stability enhancement with unified power flow controller. Elect Power Energy Syst 2007;29:251–9. [28] Yu XM, Xiong XY, Wu YW. A PSO-based approach to optimal 28 capacitor placement with harmonic distortion consideration. Elect Power Syst Res 2004;71:27–33.

Hossein Shayeghi received the B.S. and M.S. degrees in Electrical and Control Engineering in 1996 and 1998, respectively. He received his Ph.D. degree in Electrical Engineering from Iran University of Science and Technology, Tehran, Iran in 2006. Currently, he is an Assistance Professor in Technical Eng. Department of University of Mohaghegh Ardabili, Ardabil, Iran. His research interests are in the Application of Robust Control, Artificial Intelligence to Power System Control Design, Operation, Planning and Power System Restructuring. He is a member of Iranian Association of Electrical and Electronic Engineers and IEEE.

Aref Jalili received the B.S. and M.S. degrees in Electrical Engineering from Azad University, Ardabil and South Tehran Branches, Iran in 2003 and 2005, respectively. Currently, he is a Ph.D. student in Electrical Engineering at Science and Technology Research Branch of the Azad University, Tehran, Iran. His areas of interest in research are Application of Fuzzy Logic and Genetic Algorithm to Power System Control and Restructuring.

Heidarali Shayanfar received the B.S. and M.S. degrees in Electrical Engineering in 1973 and 1979, respectively. He received his Ph.D. degree in Electrical Engineering from Michigan State University, U.S.A., in 1981. Currently, he is a Full Professor in Electrical Engineering Department of Iran University of Science and Technology, Tehran, Iran. His research interests are in the Application of Artificial Intelligence to Power System Control Design, Dynamic Load Modeling, Power System Observability Studies and Voltage Collapse. He is a member of Iranian Association of Electrical and Electronic Engineers and IEEE.