Electric Power Systems Research 70 (2004) 115–127

Application of GA/GA-SA based fuzzy automatic generationcontrol of a multi-area thermal generating system

S.P. Ghoshal

Department of Electrical Engineering, National Institute of Technology,Durgapur 713209 West Bengal, India

Received 24 July 2003; received in revised form 26 September 2003; accepted 13 November 2003

Abstract

Optimal integral gains (for integral gain control) and proportional-integral-derivative gains (for PID control) are computed by geneticalgorithm (GA) and then hybrid genetic algorithm-simulated annealing (GA-SA) techniques for nominal values of area input parameters andoptimal transient responses of area frequency deviations in terms of settling times, undershoots, overshoots and df/dt as output with incrementalincrease of area load for interconnected three equal generating areas. Though it is well known that the normal PID control is usually superiorto integral control because of the advantages of each of the three individual control actions (proportional, integral and derivative), the author’scontribution in the paper is optimizing these individual PID gains through GA or GA-SA methods to obtain an optimal PID controller, whichwould be further better than an optimal integral controller. These optimal PID gains are tested by plotting transient responses analytically byMATLAB based software program and then by “SIMULINK of MATLAB software.” Both methods yield same results and prove that optimalPID gains are superior to suboptimal, arbitrary PID gains and optimal integral gains as well with respect to transient responses. The author’snext contribution is to show optimal PID gains as determined by hybrid GA-SA technique to be more globally optimal than those determinedby GA method. For off-nominal input parameters, transient responses as determined by fast acting Sugeno fuzzy logic technique reflect thesame superiority of GA-SA based optimized gains, specially for PID control, the same has also been verified by “MATLAB–SIMULINK”software.© 2004 Elsevier B.V. All rights reserved.

Keywords:Thermal generating systems; Automatic generation control; Genetic algorithm; Simulated annealing; Optimal PID gains; Sugeno fuzzy control

1. Introduction

Area loads and system operating parameters are alwayschanging in a large scale interconnected power systems.So, effective megawatt-frequency control under these ever-changing conditions will always try to satisfactorily main-tain area frequency and mutual tie-line exchanges betweenareas very close to specified nominal values for stable oper-ation of the system.

Various load frequency control strategies[1,4] have beendeveloped in order to have better dynamic transient re-sponse over the last decades. Fixed integral gain controllersfor nominal operating conditions fail to provide best controlperformance over a wide range of off-nominal operatingconditions. So, to achieve optimal performance, various

E-mail address:spghoshalrecdgp@yahoo.com (S.P. Ghoshal).

state feedback[5,6], state adaptive optimal controllers[7],ANN, fuzzy based controllers[3,8] have been proposed,all of which in general suffer from complicated compu-tational burden and memory. Recently, Matrix–Riccatibased integral gains for adaptive fuzzy load frequencycontrol [9] and genetic algorithm (GA)[2] based inte-gral gains[10,11] have been developed to cater for theuncertainties in the operating input parameters. In thepresent paper, nominal optimal integral gains and nom-inal optimal proportional-integral-derivative (PID) gainsare determined by GA technique and then by hybrid ge-netic algorithm-simulated annealing (GA-SA) techniquefor nominal input parameters for studying comparativeperformance between GA and GA-SA based optimiza-tions. These gains are then utilized by Sugeno fuzzy logic[9–11] to provide optimal performance for widely varyingoff-nominal input parameters. It is shown that optimal PID

0378-7796/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.epsr.2003.11.013

116 S.P. Ghoshal / Electric Power Systems Research 70 (2004) 115–127

control is much superior to conventional integral controland arbitrary, suboptimal PID control. Also, GA-SA basedintegral gains and PID gains are more globally optimal thancorresponding GA based gains.

2. System models

Closed loop integral controlled/PID controlled ringconnected three-area thermal generating system and theassumed directions of tie-line power flows in automaticgeneration control are shown inFigs. 1, 2 and 3(a), respec-tively. Block diagram equations are shown inAppendix A.Active power-frequency control of these closed loop controlsystems means minimizing the area control errors (ACEi)to zero so that area frequency and tie-line interchangesare maintained at their scheduled values, respectively, forwidely varying system input parameters and load changes.

ACEi =∑

j

�Ptie,i,j + bi�fi

Fig. 1. Closed loop INTEGRAL (Gi) or PID (PIDi) controlled ring connected three-area reheat system. Inputs to the PID controllers’ are ACE1(s),ACE2(s), ACE3(s), respectively. Inputs to Summ1, Summ2, and Summ3 are{b1�f1(s), �Ptie,1,2(s) as +ve, �Ptie,1,3(s) as +ve}, {b2�f2(s) as +ve,�Ptie,1,2(s) as−ve, �Ptie,2,3(s) as+ve}, {b3�f3(s) as+ve, �Ptie,1,3(s) as−ve, �Ptie,2,3(s) as−ve} respectively. Inputs to Summ4, Summ5, and Summ6are (G1 × ACE1(s) as +ve, and (1/R1)�f1(s) as −ve}, {G2 × ACE2(s) as +ve, and (1/R2)ACE2(s) as −ve}, (G3 × ACE3(s) as +ve, (1/R1)�f1(s) as−ve} respectively. Inputs to Summ7, Summ8, and Summ9 are{�PD1(s) as−ve, �PG1(s) as+ve �Ptie,1,2(s) as−ve, �Ptie,1,3(s) as+ve}, {�PG2(s)as+ve, �Ptie,1,2(s) as+ve, �Ptie,2,3(s) as−ve}, {�PG3(s) as+ve, �Ptie,1,3(s) as+ve, �Ptie,2,3(s) as+ve} respectively. Inputs to Summ10, Summ11,and Summ12 are{�f1(s) as+ve, �f2(s) as−ve}, {�f2(s) as+ve, �f3(s) as−ve}, {�f1(s) as+ve, �f3(s) as−ve} respectively.

Fig. 2. Assumed tie-line power flow directions.

where ACEi is the area control error ofith area;bi, fre-quency bias coefficient ofith area;�fi, frequency error ofith area;�Ptie,i,j, tie-line interchange error between theitharea andjth area. All are shown inFigs. 1, 2 and 3(a). Theintegral of ACEi over a given time intervalτ in Laplace do-main is defined by:(−Gi)[

∑j �Ptie,i,j(s)+bi�fi(s)]. Gi is

either−Ki/s (integral control) or PIDi (−Kp −Ki/s −Kds)(for PID control). The varying systems input parametersare: (1) area time constant (tp) (2) tie-line synchronizingcoefficient (t12) and (3) frequency bias coefficient (b).

S.P. Ghoshal / Electric Power Systems Research 70 (2004) 115–127 117

3. Optimization of gains by GA method in off-line

The base of GA is rooted in natural evolutionary and bio-logical process that ensures the “survival of the fittest.” Thatis, the species, which can adopt external environment moreefficiently, will survive. Hence, GA’s are essentially searchalgorithm based on the mechanics of nature and natural ge-netics. They combine solution evaluation with randomized,structured exchanges of information between solutions toobtain optimality. The power of this algorithm stems fromits ability to exploit historical information structures fromprevious solution guesses in an attempt to increase perfor-mance of future solutions. GA’s are considered to be robustin nature because restrictions on the solution space are notmade during the process.

A population size is chosen, containing several solutionchromosomes, each being a fixed length string, consistingof some genes (some binary bits). Evaluation of each chro-mosome solution string is accomplished by decoding thebinary bits and calculating fitness function of the objectiveusing the decoded value of the solution. In the present study,evaluation of figure of demerit is an alternative of conven-

Fig. 3. (a) “MATLAB–SIMULINK” software based three-area automatic generation control block diagram. Inputs to the PID controllers’ are ACE1(s),ACE2(s), ACE3(s), respectively, Gains 1, 3, and 5 are allb gain, gain 2, and gain 4 are 1/R, inputs to sum, Sum3 and 6 are{b�f1(s), �Ptie,1,2(s), �Ptie,1,3(s)all as +ve}, {+b�f2(s), �Ptie,1,2(s) as −ve, �Ptie,2,3(s) as +ve}, {+b�f3(s), �Ptie,1,3(s) as −ve, �Ptie,2,3(s) as −ve} respectively, Inputs to Sum2,Sum5, and Sum8 are{step�PD1/s as−ve, �PG1(s) as+ve, �Ptie,1,2(s) as –ve,�Ptie,1,3(s) as−ve}, {�PG2(s) as+ve, �Ptie,1,2(s) as+ve, �Ptie,2,3(s)as –ve,{�PG3(s) as+ve, �Ptie,1,3(s) as+ve, �Ptie,2,3(s) as+ve}, respectively. Each transfer Fcn 1, 5, and 8 is ((1+ scTr)/(1+ sTg)(1+ sTt)(1+ sTr)),each transfer Fcn 3, 6, and 9 isKp/(1+stp), den(s) is (1+sTg)(1+sTt)(1+sTr). As per 11th row ofTable 2, nominaltp1, t12, andb1 are 20.0, 0.145, and0.275, respectively, optimal PID gainsKp, Ki, andKd are−1.7842,−0.5190,−1.1122, respectively, as per 11th row ofTable 2(negative sign indicatesnegative feedback), arbitrary PID gain values chosen are−1.0, −1.0, −1.0, other fixed parameters likeTg1 , Tt1 , Tr1 , c and R are as given inSection 6.(b) “MATLAB–SIMULINK” software based plot of optimal response of�f1(t) for optimal PID gainsKp, Ki, and Kd as −1.7842,−0.5190,−1.1122,respectively, as per 11th row ofTable 2undershoot is−0.0172 p.u., overshoot is zero, settling time is 5 s. (c) “MATLAB–SIMULINK” software basedplot of suboptimal response of�f1(t) for arbitrary PID gainsKp, Ki, andKd as−1.0, −1.0, −1.0, respectively, undershoot is−0.0185 p.u., overshoot is3 × 10−3 > optimal overshoot of (b), settling time is 9 s� optimal settling time of (b).

tional maximization of fitness function. The strings with lessfigure of demerit will only survive for the next generation,the strings with more figure of demerit will die. The pro-cess is essentially selection, elitism and copying the elitestrings in place of dying strings to regenerate the same pop-ulation size. The newly generated strings now produce newoff-springs by crossover depending on the crossover rateand some off-springs undergo mutation operation dependingupon the mutation probability to avoid premature conver-gence to sub-optimal solution. In this way, a new populationdifferent from the old one is formed in each genetic itera-tion cycle. The whole process is repeated for several geneticiteration cycles till required optimal or near optimal solu-tion is obtained for which the figure of demerit is the grandminimum.

In our problem, integral gain solution is represented bya string of sixteen binary bits and PID gain solution by astring of forty eight binary bits, each consecutive sixteenbits representing proportional, integral and derivative gain.The binary to decimal decoded value of each sixteen bitsgives corresponding gain value. Because the problem is op-timization of system transient performance by minimizing

118 S.P. Ghoshal / Electric Power Systems Research 70 (2004) 115–127

Fig. 3. (Continued).

undershoot, overshoot, settling time and df/dt of transient re-sponse of deviations in area 1 frequency, a newly designedperformance function (which is sum of squares of all theabove mentioned system performance characteristics suit-ably weighted) named as “figure of demerit”(as shown be-low) is minimized in each genetic iteration cycle. Popula-tion size chosen is 50. The figure of demerit is computedfor each solution string of the population, the less the fig-ure of demerit, the better the solution string is. The betterstring survives in the next population. In this way, 50% ofthe strings are selected (elite strings) in order of increasingvalues of figure of demerit and the same are copied on therest dying strings having higher values of figure of demerit tomake the population size of the new generation same as 50.

Maximum allowable genetic iteration cycles is selectedas 100 because minimum limits of figure of demerit, i.e.optimal transient performance are found to be reachedconsistently within 100 genetic iteration cycles for allnominal system parameters, even with various changes inpopulation size, crossover rate and mutation probability.Execution times lie within a few minutes of the order of3–6.6 min.

figure of demerit= (overshoot× 1000)2

+ (undershoot× 100)2+(settling time)2

+[(

df

dt

)× 100

]2

S.P. Ghoshal / Electric Power Systems Research 70 (2004) 115–127 119

Undershoot, overshoot, df/dt and settling time of area 1frequency deviation�f1(t) are considered for evaluation offigure of demerit.

These values are computed by first sampling the transientresponse curve of�f1(t). Number of samples chosen as 400has become sufficient for good accuracy because maximumsettling time as seen from the curve rarely exceeds 100 s.After sampling, the magnitudes of the samples are checkedto be more than or equal to 0.0001 p.u., reckoning from thelast sample where the magnitude is very much less than0.0001 p.u. The instant of time when the above condition issatisfied is in our case called to be settling time. Undershootand overshoot are nothing but the minimum or the maximumvalues of the samples respectively. The maximum rate ofchange of frequency (df/dt) is computed by determining themaximum of the differences in consecutive sample valuesdivided by the sampling interval.

Because overshoot, undershoot and df/dt values are verysmall compared to settling time, these are weighted by largemultipliers so that the reduction of these values may com-pete with reduction of settling time during GA optimizationprocess.

4. Optimization of gains by hybrid GA-SA methodin off-line

Some well-known disadvantages in GA are poor con-vergence of GA near global optimum and the prematureconvergence to the suboptimum. The author improves thepremature convergence of GA by adopting the basic ex-ponential acceptance probability of SA as the criterion oftesting and accepting or rejecting the SA-induced perturbedGA cycle generated candidate solutions. The effect of SAinduced small perturbation on the GA generated solutionis to inject new member solution in the GA population andhelp escaping the local suboptimal minimum of figure ofdemerit of our problem and thus preventing the prematureconvergence of the GA. The mixed use of GA and SA thusdoes not hurt the advantages of each individual algorithm butshows a reasonable combination of local and global search.

Annealing physically refers to the process of heating upa solid to a high temperature followed by slow coolingachieved by decreasing the temperature of the environmentin steps. At each step the temperature is maintained con-stant for a period of time sufficient for the solid to reachthermal equilibrium. In our study, post applying the simu-lated annealing method after GA method is firstly to get alittle perturbed neighbor to every GA based solution of in-tegral gain string or PID gain strings of the population bychanging some of the lesser weighted binary bits of the so-lution and then calculating the figure of demerit. A moveto the neighbor string is selected if either it has lower fig-ure of demerit than that corresponding pre-calculated oneby GA or, in case the neighbor solution has higher figure ofdemerit, i.e. poorer, if exp(−�E/Cp) ≥ U(0, 1), where�E

is the increase in figure of demerit if we move to the poorerneighbor solution,Cp is control temperature, equals toC0

× (β)k, where 0< β < 1, k is the genetic iteration cycle;U(0,1), random number uniformly distributed over [0,1] andC0, initial temperature. Proper choice ofC0 and β valuesdepends on numerical experimentation in our study. A toolow β or too smallk results in imperfect annealing, thus fail-ing to yield globally optimal solutions. A wastefully highC0 also misses optimal solution and unnecessarily requiresmore number of iterations. BestC0 is selected as 75◦C andbestβ is selected as 0.995 after performing a lot of computerbased computations.

5. Sugeno fuzzy logic [9–11] as applied to gainscheduling on-line

The whole process involves essentially three steps as (a)fuzzification of input and output variables and formation offuzzy rule base table (off-line) but fast computing, on-line(b) fuzzy inference and finally (c) defuzzification to obtainthe crisp, non-fuzzy output.

5.1. Fuzzification of the inputs[11]

The numerical system input parameters chosen areon-line system information of power system time constanttp, tie-line constantt12 and frequency bias constantb. Theseinputs are firstly fuzzified by defining fuzzy subsets like“Small,” “Medium” and “Large” and triangular member-ship functions. The subsets for each input are:{Small,Medium, Large}. The limits of the subsets are defined asfollow:

1. “Small” of tp is 0 < tp ≤ 20; “Medium” of tp is 10 ≤tp ≤ 30; “Large” of tp is 20≤ tp. Central values oftp inthese subsets are, respectively 10 (S), 20 (M), 30 (L).

2. “Small” of t12 is 0 < t12 ≤ 0.345; “Medium” of t12 is0.145 ≤ t12 ≤ 0.545; “Large” of t12 is 0.545 ≤ t12.Central values oft12 in these subsets are, respectively,0.145 (S), 0.345 (M), 0.545 (L).

3. “Small” of b is 0 < b ≤ 0.275, “Medium” ofb is 0.125≤b ≤ 0.425, “Large” ofb is 0.275≤ b. Central values ofb in these subsets are, respectively, 0.125 (S), 0.275 (M),0.425 (L).

All the central values are the nominal system parameters.The output solution chosen is GA/GA-SA based optimalvalue of integral gain for integral control and proportional,integral and derivative gains for PID control. Sugeno fuzzyrule base tables formed are: (1)Table 1contains GA basedoptimized nominal integral gains and GA-SA based opti-mized nominal integral gains for nominal input parametersand (2)Table 2contains GA based optimized nominal PIDgains and GA-SA based optimized nominal PID gains fornominal input parameters.

120 S.P. Ghoshal / Electric Power Systems Research 70 (2004) 115–127

Table 1Nominal input parameters vs. various performance parameters for integral gain control (different rows), first sub-row and second sub-row of each rowof columns 4–10 indicate values for GA optimization and GA-SA optimization, respectively

tp t12 b No. of geneticiterations

Ki

(−ve)USH (−ve)× 102

OSH× 103

Settlingtime (s)

df/dt× 102

Figure ofdemerit

10.0 0.145 0.125 65 0.6605 4.41 8.6 17.5 8.83 478.235114 0.6597 sub-row sub-row 17.5 8.83 478.0789

10.0 0.145 0.275 40 0.4534 4.41 11.7 20.0 8.82 634.654531 0.4297 do 11.2 do do 622.8292

10.0 0.145 0.425 32 0.1338 4.21 5.9 15.2 9.0 381.881639 0.1363 do 4.9 do 8.4 362.0301

10.0 0.345 0.125 85 0.5343 3.55 5.6 31.5 7.09 1086.450 0.4763 do 4.8 do 7.10 1077.9

10.0 0.345 0.275 75 0.2522 3.55 4.9 29.5 7.10 957.443344 0.2277 do 4.3 do do 951.7428

10.0 0.345 0.425 57 0.1412 3.55 4.0 31.0 7.10 1039.740 0.1873 do 5.8 29.5 do 966.243

10.0 0.545 0.125 68 0.4456 3.01 9.6 49.2 7.53 2578.140 0.4453 do do do do 2578.1

10.0 0.545 0.275 82 0.1724 3.04 8.0 48.4 7.54 2473.237 0.1723 do do do do 2473.2

10.0 0.545 0.425 47 0.1335 2.96 9.3 49.0 5.69 2529.018 0.1113 3.05 7.8 47.6 7.54 2392.9

20.0 0.145 0.125 54 0.5000 3.26 7.5 21.0 5.17 533.897933 0.5000 do do do do 533.8979

20.0 0.145 0.275 59 0.3311 3.25 9.8 23.5 5.17 685.449566 0.3309 do do do do 685.3263

20.0 0.145 0.425 35 0.1416 3.28 5.9 18.5 5.17 415.129042 0.1414 do do do do 414.9968

20.0 0.345 0.125 36 0.4142 2.45 5.0 32.4 5.11 1106.755 0.4142 do do do do 1106.7

20.0 0.345 0.275 51 0.2766 2.33 6.9 27.5 4.66 830.286553 0.2765 do do do do 830.2353

20.0 0.345 0.425 55 0.1646 2.33 6.0 29.5 4.66 933.763359 0.1475 do 5.4 do do 926.2009

20.0 0.545 0.125 11 0.4426 2.09 6.1 39.5 4.19 1619.153 0.3913 2.10 5.3 do do 1610.7

20.0 0.545 0.275 23 0.2189 2.10 6.4 37.5 4.19 1469.029 0.2106 do 6.1 do do 1465.8

20.0 0.545 0.425 48 0.1408 2.21 5.9 37.2 4.78 1446.232 0.1309 2.23 5.4 do do 1440.9

30.0 0.145 0.125 31 0.3906 2.71 6.1 30.5 3.63 987.409552 0.3867 do 6.0 do do 986.6123

30.0 0.145 0.275 35 0.2812 2.70 8.9 26.0 3.62 775.445231 0.2803 do do do do 774.9456

30.0 0.145 0.425 40 0.1250 2.71 5.5 29.5 3.63 921.234754 0.1238 do do do do 920.5045

30.0 0.345 0.125 36 0.3750 2.01 3.38 38.0 3.38 1477.162 0.3720 do 4.2 do do 1476.7

30.0 0.345 0.275 25 0.1876 2.01 4.3 34.5 3.38 1224.636 0.1822 do 4.2 do do 1223.2

30.0 0.345 0.425 36 0.1261 2.07 4.7 34.4 3.60 1223.042 0.1151 do 4.2 do do 1218.2

30.0 0.545 0.125 39 0.3753 2.05 4.7 44.4 3.44 2009.456 0.3410 2.07 4.1 do do 2004.0

30.0 0.545 0.275 27 0.1896 1.93 4.2 39.5 3.16 1591.958 0.1832 do 4.0 do do 1590.1

30.0 0.545 0.425 26 0.1339 2.03 5.0 39.2 3.44 1577.755 0.1280 2.04 4.7 do do 1574.8

Total genetic cycles= 100. Results shown in bold form indicate more prominent effect of GA-SA method over the GA method regarding transientperformance characteristics.

S.P. Ghoshal / Electric Power Systems Research 70 (2004) 115–127 121

Table 2Nominal input parameters vs. various performance parameters for PID gain control (different rows), first sub-row and second sub-row of columns 4–10of each row indicate values for GA optimization and GA-SA optimization, respectively

tp t12 b GeneticIterations

Kp

(−ve)Ki

(−ve)Kd

(−ve)USH (−ve)× 102

OSH× 103

Settlingtime (s)

df/dt× 102

Figure ofdemerit

10.0 0.145 0.125 60 1.9001 1.9600 1.4607 2.46 2.3 7.0 4.91 84.30691.9993 1.9976 1.9247 2.11 2.1 7.0 4.22 75.4848

10.0 0.145 0.275 55 1.8749 0.5624 1.5000 1.57 0.1 4.5 3.14 32.60031.9443 0.5910 1.3225 1.75 0.1 4.0 3.38 30.2840

10.0 0.145 0.425 64 1.7500 0.4375 0.8750 1.61 0.1 4.0 3.23 29.00791.7668 0.4211 0.7350 1.77 0.0 4.5 3.55 35.9677

10.0 0.345 0.125 40 0.9420 1.1056 1.1056 1.92 0.7 7.5 3.84 75.24751.2462 1.9532 1.9947 1.19 2.2 7.5 2.38 68.1609

10.0 0.345 0.275 44 1.9355 1.8300 1.9062 0.69 1.6 7.0 1.38 53.9981.6533 0.6450 1.4900 0.95 0.1 4.5 1.89 24.7403

10.0 0.345 0.425 30 1.4868 1.7694 1.2299 0.79 2.1 8.5 1.59 79.79581.9122 0.5589 1.9990 0.58 0.1 5.0 1.16 26.6857

10.0 0.545 0.125 39 1.7495 1.8125 1.6247 1.66 2.9 11.5 3.11 153.25630.7603 1.8132 1.6129 1.71 3.5 7.5 2.58 78.1134

10.0 0.545 0.275 38 1.9062 0.7202 2.0000 0.68 0.1 5.0 0.77 26.05871.4429 0.5922 1.9947 0.76 0.1 5.5 0.90 31.6510

10.0 0.545 0.425 21 1.8017 1.7267 1.6875 0.38 1.4 6.5 0.73 44.82631.9473 1.8681 1.9045 0.35 1.3 6.5 0.71 44.6170

20.0 0.145 0.125 62 1.9938 1.5845 1.3700 1.98 2.8 10.5 3.95 137.46911.9964 1.9396 1.2497 2.01 3.9 9.0 4.02 116.2660

20.0 0.145 0.275 32 1.9757 1.7162 1.0363 1.72 3.3 6.5 3.44 67.78061.7842 0.5190 1.1122 1.72 0.1 4.0 3.44 30.7578

20.0 0.145 0.425 53 1.8750 1.8127 1.1250 1.39 3.1 6.0 2.78 55.06151.8711 0.4369 0.8724 1.56 0.0 4.0 3.12 28.1657

20.0 0.345 0.125 33 1.7899 1.3797 1.4759 1.56 1.5 8.0 3.12 78.55851.9973 1.1275 1.6815 1.48 0.7 7.5 2.96 67.6670

20.0 0.345 0.275 39 1.6225 0.5919 0.9424 1.50 0.1 4.0 3.00 27.26011.6923 0.6079 1.1174 1.40 0.1 4.0 2.81 25.8496

20.0 0.345 0.425 74 1.5885 0.4632 1.3396 1.06 0.0 5.0 2.13 30.66171.9093 0.5332 0.8121 1.31 0.0 4.0 2.62 24.6107

20.0 0.545 0.125 22 1.4918 1.0374 1.7498 1.31 0.9 8.3 5.75 104.49951.4994 1.5004 1.7384 1.27 2.1 8.0 2.30 75.2459

20.0 0.545 0.275 32 1.8959 0.6170 1.5586 0.92 0.0 6.0 1.83 40.19091.8950 0.6684 1.2498 1.05 0.1 4.5 2.10 25.7678

20.0 0.545 0.425 28 1.7802 0.5418 1.2133 0.87 0.0 5.0 1.75 28.80951.9461 0.6018 0.9224 1.00 0.0 4.5 2.00 25.2605

30.0 0.145 0.125 30 1.9331 1.5866 1.3521 1.61 4.1 11.5 3.05 160.67511.9965 1.4886 1.3746 1.60 3.7 11.5 3.04 157.5991

30.0 0.145 0.275 65 1.9914 0.5632 1.0000 1.40 0.7 6.5 2.81 52.58471.9228 0.5364 1.1234 1.37 0.5 6.5 2.75 51.8954

30.0 0.145 0.425 36 1.7392 0.3578 1.3615 1.18 0.0 6.4 3.46 54.32111.7487 0.3776 0.9996 1.26 0.0 5.0 2.53 32.9745

30.0 0.345 0.125 34 1.9958 1.5031 1.6561 1.27 2.1 12.0 2.53 156.59541.9940 1.3563 1.7425 1.25 1.6 12.0 2.50 154.2314

30.0 0.345 0.275 60 1.9829 0.6500 1.1091 1.20 0.4 7.5 2.41 63.68891.9584 0.6054 1.4888 1.10 0.1 5.0 2.21 31.1016

30.0 0.345 0.425 35 1.8398 0.4681 2.0000 0.81 0.0 6.0 1.63 39.31111.9412 0.4852 1.1146 1.06 0.0 5.0 2.12 30.6049

30.0 0.545 0.125 58 1.9604 1.1930 1.9531 1.02 1.2 12.0 2.04 150.55781.9867 1.8750 1.9957 1.01 1.2 12.0 2.02 150.4346

30.0 0.545 0.275 33 1.6531 0.5523 1.5977 0.93 0.1 5.5 1.86 34.60261.8301 0.5967 1.4432 0.96 0.1 5.0 1.93 29.6584

30.0 0.545 0.425 59 1.8759 0.5418 1.2186 0.89 0.1 5.0 1.78 28.96881.9222 0.5588 1.1015 0.93 0.1 4.0 2.58 23.5466

Total genetic cycles=100. Results shown in bold form indicate more prominent effect of GA-SA method over GA method regarding transient performancecharacteristics.

122 S.P. Ghoshal / Electric Power Systems Research 70 (2004) 115–127

There exists a definite numerical value(s) of optimal gainK (integral gain value for integral control or proportional,integral, derivative gain values in PID control, as determinedby GA/GA-SA methods), corresponding to each definitegroup SSS, MLM, LSM etc. of input parameters.

Table 3contains off-nominal input parameters and corre-sponding off-nominal integral and PID gains for both GAand GA-SA based methods using fast computing, on-lineSugeno fuzzy inference (Section 5.2)and Sugeno defuzzifi-cation technique (Section 5.3). All these tables also containcorresponding undershoot, overshoot, settling time and df/dtvalues.

5.2. Sugeno fuzzy inference[11]

For some imprecise numerical values oftp, t12 and b,firstly their fuzzy subsets in which the values lie are deter-mined with the help of “IF,” “THEN” logic and correspond-ing membership values are determined from the membershipfunctions of the subsets. From the Sugeno fuzzy rule-basetable corresponding group numbers and integral gain (K)values are determined. Now, for each group number say (i)being satisfied, three number of membership values ofµ

(1)tp

,

µ(i)t12

andµ(i)b are computed from the membership functions

and the ultimate membership value for each group is cal-

Table 3Off-nominal input parameters vs. various performance parameters (different rows), for columns 5–12 of each row, first sub-row representing GA basedfuzzy integral control, second sub-row for GA-SA based fuzzy integral control, third sub-row for GA based fuzzy PID control, fourth sub-row for GA-SAbased fuzzy PID control

tp t12 b Figureno.

Kp

(−ve)Ki

(−ve)Kd

(−ve)USH (−ve)× 102

OSH× 103

Settlingtime (s)

df/dt× 102

Figure ofdemerit

11 0.33 0.41 3 – 0.1839 – 3.45 10.4 30.2 10.71 1147.7– 0.2077 – 3.44 11.0 30.0 6.89 1081.41.609 1.4804 1.244 1.45 1.70 7.2 7.23 109.01.866 0.5528 1.6295 0.72 0.10 5.0 1.44 27.6

21 0.16 0.29 4 – 0.2729 – 3.14 8.9 26.6 5.68 824.1554– 0.2710 – 3.14 8.9 26.5 5.68 823.06221.9108 1.3206 1.1238 1.59 2.1 8.0 5.00 95.98761.8151 0.5130 1.0937 1.64 0.0 4.4 5.02 47.2170

31 0.2 0.2 5 – 0.3165 – 2.45 7.6 30.5 3.46 1006.5– 0.3155 – 2.45 7.6 30.5 3.46 1005.31.9719 1.0755 1.2493 1.43 2.2 11.6 3.58 154.41.9655 1.0013 1.3791 1.40 1.6 8.8 3.56 94.8

21 0.21 0.3 – – 0.2518 – 2.90 8.6 26.7 5.67 827.4350– 0.2489 – 2.90 8.5 26.7 5.67 825.72141.8424 1.1158 1.1329 1.46 1.6 8.8 4.95 106.77641.8124 0.5250 1.0773 1.51 0.0 5.2 4.98 54.1014

11 0.20 0.3 – 1.8099 1.0022 1.3886 1.64 1.0 9.6 7.91 153.39931.8480 0.5647 1.2783 1.73 0.0 5.0 8.07 93.0506

15 0.35 0.35 – 1.6673 1.1511 1.3669 1.28 1.3 8.6 5.99 113.17091.7925 0.6027 1.3624 1.29 0.1 7.4 6.06 93.5180

25 0.15 0.4 6 1.8447 1.0601 1.1982 1.31 1.5 9.0 4.13 102.11511.8246 0.4458 0.9920 1.43 0.1 6.0 4.22 55.8250

35 0.4 0.4 – 1.8418 0.5166 1.6774 0.83 0.1 8.4 2.91 79.81.9255 0.5293 1.1997 0.94 0.1 6.2 3.02 48.5

The last four results (four rows) in cursive form indicate comparison between GA (first sub-row) and GA-SA methods (second sub-row).

culated as the minimum (µ(i)min) of µ

(i)tp

, µ(i)t12

, andµ(i)b be-

cause the fuzzy decision is obtained by the intersection ofthe fuzzy subsets in which the numerical values of the in-putstp, t12, andb lie. For the group numbers which are notsatisfied because values oftp, t12, andb do not lie in thecorresponding fuzzy subsets (vide Sugeno fuzzy rule basetables[11]), µ

(i)min will be zero. The execution time for this

process is negligible.

5.3. Sugeno defuzzification technique[11]

The defuzzified, crisp value of gain,Kcrisp is computed asfollows:

Final crisp gain(Kcrisp) =∑

i µ(i)minKi∑

i µ(i)min

,

wherei = 1–27 sets (1)

In this equation, only a few values ofµ(i)min are not

equal to zero, majority values ofµ(i)min will be zero because

off-nominal input parameters do not satisfy the nominalgroups as shown in the tables. The defuzzified crisp gain(s)are computed in negligible time, which are then utilizedto determine the transient response. These gains can beutilized on-line because of fast computation.

S.P. Ghoshal / Electric Power Systems Research 70 (2004) 115–127 123

Fig. 4. The overlapping curves with high frequency oscillations of�f1(t)correspond to GA/GA-SA based fuzzy integral control, highly dampedcurve having overshoot corresponds to GA based fuzzy PID control,the one critically damped with no overshoot corresponds to GA-SAbased fuzzy PID control. Off-nominal input parameters are: 11, 0.33,0.41.

6. Input data and computational results

Total binary string size of each PID gain string is 48,each 16 binary bits for each of proportional, integral andderivative gains. Population size chosen for GA or hybridGA-SA based off-line optimization of integral or PID gains= 50, crossover rate=100%, mutation probability= 0.004,maximum number of genetic iteration cycles chosen= 100.For simulated annealing, initial temperature,C0 = 75◦Candβ = 0.995.

Three equal area are having the following data: governorregulation,R = 2.4 Hz/p.u., governor time constant,Tg =0.08 s, non-reheat time constant,Tt = 0.3 s. Reheat timeconstant,Tr = 4.2 s, reheat parameterc = 0.35, area gainconstant,Kp = 120 Hz/p.u., other nominal/off-nominal in-put parameters,tp, t12 andb are shown in the tables.

Load change in area 1,∆PD = 0.01 p.u. Computationalresults for nominal input parameters versus various perfor-mance parameters are shown inTables 1 and 2. Table 3shows results for off-nominal input parameters versus var-ious performance parameters. Some results are also shownin theFigs. 4–8. USH, OSH are undershoot and overshoot,respectively.

Fig. 5. The overlapping curves with high frequency oscillations of�f1(t)correspond to GA/GA-SA based fuzzy integral control, highly dampedcurve having overshoots corresponds to GA based fuzzy PID control,the one critically damped with no overshoot corresponds to GA-SAbased fuzzy PID control. Off-nominal input parameters are: 21, 0.16,0.29.

7. Discussion on the computational results

1. Lower values oftp (around 10 s) but higher values oft12(around 0.52) andb (around 0.4) yield very high settlingtimes, high frequency transient oscillations specially foroptimal integral control, which are drastically damped byoptimal PID control. Maximum undershoot, maximumovershoot and settling time are also drastically reducedin case of optimal PID control, as shown in theFigs. 4–8.Optimal PID control (shown in theFig. 3(b), zero over-shoot and 5 s settling time of�f1(t)) is superior to ar-bitrary, suboptimal PID control (shown in theFig. 3(c),more undershoot (0.0185 p.u.), larger overshoot (3×10−3) and larger settling time� 10 s). These figures areobtained by START-ing the “MATLAB–SIMULINK”software based three-area automatic generation controlblock diagram, as shown in theFig. 3(a). Thus, though aPID controller is always superior to an integral controller,there is every necessity of optimizing the PID gains ifone desires to get optimal response with a PID controller,which would be further superior to an optimal integralcontroller.

2. The function ‘figure of demerit’ chosen by the authorcorrectly determines GA/GA-SA based optimum valuesof integral gains and PID gains. The lesser is the value of

124 S.P. Ghoshal / Electric Power Systems Research 70 (2004) 115–127

Fig. 6. The overlapping curves with high frequency oscillations of�f1(t) correspond to GA/GA-SA based fuzzy integral control, highly damped curvehaving a little more overshoot corresponds to GA based fuzzy PID control, the other damped one with a little less overshoot corresponds to GA-SAbased fuzzy PID control. Off-nominal input parameters are: 31, 0.20, 0.20.

figure of demerit, the better is the transient performance,therefore, more optimal are the gains[11].

3. Transient response in case of optimal PID control showsmuch improvement in undershoots, overshoots and set-tling times (i.e. very much less overall figure of demerit,every third and fourth sub-row of 12th column ofTable 3)as compared with that in case of optimal integral control.This is true for both GA-fuzzy or GA-SA-fuzzy basedgains (Table 3) and the optimizing effect of GA-SA-fuzzybased PID gains is more pronounced with smaller over-shoots as shown in theFigs. 4–8. GA-SA-fuzzy basedPID control yields better performance than GA-fuzzybased PID control as shown in theFig. 7. Area 1 gen-eration settles down at�PD = 0.01 p.u (as desired) at13.5 s for GA-SA-fuzzy PID control, 14 s for GA-fuzzyPID control and more than 25 s for GA-/GA-SA-fuzzyIntegral control as shown in theFig. 8.

4. Because GA based optimization is a random globalsearch method it may converge at local optimal (sub-optimal) gain values, hence rigorous experiments withpopulation size, crossover rate and mutation probabilityare performed for determining optimal gains for opti-mal transient response (lower overall figure of demerit).Finally, the values settle down at some stable values asgiven inSection 6. To strengthen the searching ability andto get rid of local optimum, SA is introduced by choos-ing trial solutions in the neighborhood of GA computed

solutions and subjecting them to the exponential accep-tance probability for rejecting or accepting, if figures ofdemerit corresponding to neighboring solutions becomemore. In all fourth sub-rows of 10th and 12th columns oftheTable 3, the effect of SA is clearly reflected in the re-duction of settling times and figures of demerit. For PIDgains, this effect is very pronounced, clearly indicatingPID gains computed in GA are sub-optimal. For integralgains, GA is in most cases sufficient to obtain optimalvalues; the effect of SA being shown a little, even witha lot of experimentation with initial temperature,C0andβ.

5. GA and hybrid GA/SA methods are much simplerin computation as compared to earlier methods ofMatrix–Riccati solutions[7,9]. Though GA and GA-SAoptimization techniques take on the average 3–3.7 min(max.) and 6.0–6.6 min (max.) (for 1.8 GHz Pentium-IVcomputer) respectively, GA and hybrid GA-SA op-timized gains are off-line values for nominal systemoperating parameters. The time variation in both GAand GA-SA methods is due to initial random selectionof gain strings, random crossover and mutation. GA-SAmethod takes more time than GA method because theformer involves cascade optimization by first GA andthen SA. The off-line optimal gains are stored in Sugenofuzzy rule base look-up tables as shown inTables 1and 2of the paper. Sugeno fuzzy logic computation (i.e.

S.P. Ghoshal / Electric Power Systems Research 70 (2004) 115–127 125

Fig. 7. The curve having overshoot corresponds to GA based fuzzyPID control, the one critically damped with no overshoot corresponds toGA-SA based fuzzy PID control. Off-nominal input parameters are: 25,0.15, 0.40.

Sugeno inference based on look up tables and subse-quent Sugeno defuzzification) take only 0.265 or 0.281 sexecution time, depending on how many groups SSS,MLM, LSM etc. of input parameters are satisfied inSugeno fuzzy inference. Hence, Sugeno fuzzy logic isvery fast acting, very much suitable for on-line, adaptivecomputation of integral or PID gains for any type of de-viation of system parameters from nominal values. Thisis the merit of Sugeno fuzzy based computation overother complicated state adaptive optimal control tech-niques[7]. Moreover, overall satisfactory performanceis achieved over a wide range of variation of operatingparameters.

6. Solution of Matrix–Riccati equation in state adaptiveoptimal controllers[7,9] to find the state feedback coef-ficients involves heavy computational complexity, mem-ory burden and execution time for determining a largenumber of Eigen values, Eigen vector matrix, matrixinversions, a lot of matrix multiplications and additionsetc. Hence, these techniques are impractical for on-lineautomatic generation control. But, Sugeno fuzzy gaincontrol based on inference and defuzzification involveimplementation of a few “IF” logic statements and thenonly a very few simple mutiplications and additions re-

Fig. 8. These are the curves of�PG1(t), all settling at�PD = 0.01 p.u.in area 1. The overlapping oscillating curves with large oscillating under-shoots and large overshoot correspond to GA and GA-SA based fuzzy In-tegral Control, the damped curve with less undershoot and less overshootcorresponds to GA based fuzzy PID control and the other damped curvehaving more undershoot but no overshoot and less settling time i.e. thecritically damped curve corresponds to GA-SA based fuzzy PID control.

spectively. So, both computational complexity, memoryburden and execution times are lower.

7. It is well known that fuzzy logic techniques provide amodel-free description of control systems and do not re-quire model identification. Moreover, Sugeno type fuzzyinference and defuzzification are extremely well suitedto the task of smoothly interpolating linear gains acrossthe input space when a very non-linear system movesaround in its operating space. The multi-area block dia-grams (Figs. 1 and 3(a)) in the paper are very non-linear.Sugeno fuzzy logic system well interpolates nominal,optimal gains of the Sugeno fuzzy rule base look upTables 1 and 2to off-nominal gains smoothly and lin-early irrespective of the complicated, non-linear powersystem model. So, with all these above viewpoints andwith the exception of off-line predetermination of rulebase tables (input sets versus output values), one mayconclude that Sugeno fuzzy control is very suitable forvery fast, adaptive, on-line automatic generation controlfor any highly non-linear interconnected power systems.

126 S.P. Ghoshal / Electric Power Systems Research 70 (2004) 115–127

8. Conclusion

1. The author has designed a new method of evaluatingthe fitness of GA/GA-SA optimization by choosing afunction like ‘figure of demerit’ which directly dependson transient performance characteristics like undershoot,overshoot, df/dt and settling times. More weight has beengiven to settling time and overshoot. That is why, bothGA-fuzzy [11] and GA-SA-fuzzy based results showmore improvement over Matrix–Riccati based results[9]in overshoots and settling times.

2. Optimal PID control yields much improvement in tran-sient performance characteristics in comparison witharbitrary, suboptimal PID control and optimal integralcontrol.

3. GA and GA-SA optimization methods are much simpler,involve less computational complexity, memory burdenand yield more optimal gains than other state adaptivetechniques and solution of Matrix–Riccati equations[11].

4. The hybrid GA-SA technique yields more optimal gainvalues than GA method, specially for PID control. Allsuch off-line, nominal gains and corresponding nominalsystem parameters can be stored as tables for the use ofon-line Sugeno fuzzy logic control for varying systemparameters.

5. Sugeno fuzzy logic technique is very simple to imple-ment and very fast acting, which makes it suitable foron-line adaptive gain scheduling for varying off-nominalinput parameters. Computational complexity and mem-ory burden are not heavy, which is an advantage overANN and other state adaptive techniques.

Appendix A

Block diagram model parameters and equations for ringconnected three power system areas are as follows:

Block diagram model parameters (refer toFig. 1):

c reheat parameterb frequency bias constants of the areas (p.u./Hz)f system frequency (Hz)Gi −Ki/s for Integral control or

(−Kp − Ki/s − Kds) for PID control,respectively,i is Ki integral gain constant,Kp, proportional gain constant;Kd, derivativegain constant;s, Laplace variable

R speed governor regulation (Hz/p.u.)Tg time constant of speed governors of the

areas (s)Tt normal time constant of non-reheat turbines

of the areas (s)Tr reheat time constants of reheat turbines (s)tp time constants of the power system areas (s)t12 tie-line synchronising power co-efficient

(p.u.) between area 1 and area 2�fm Laplace transform of small change in

frequency in aream area identification number�PD(s) Laplace transform of small load demand step

input �PD p.u. in area 1= �PD/s

�Ptie,m,n Laplace transform of small change in tie-linepower interchange between areasm andn (p.u.)

Equations for three-area block diagram using Integral orPID gains:

�f1 =([{

b1�f1 +( t12

s

)(�f1 − �f2) +

( t13

s

)(�f1 − �f3)

}(G1) −

(�f1

R1

)] ((1 + sc1Tr1)

(1 + sTg1)(1 + sTt1)(1 + sTr1)

)

−�PD(s) −( t12

s

)(�f1 − �f2) −

( t13

s

)(�f1 − �f3)

)(Kp1

1 + stp1

)(A.1)

�f2 =([{

b2�f2 −( t12

s

)(�f1 − �f2) +

( t23

s

)(�f2 − �f3)

}(G2) −

(�f2

R2

)]

×(

1 + sc2Tr2

(1 + sTg2)(1 + sTt2)(1 + sTr2)

)+

( t12

s

)(�f1 − �f2) −

( t23

s

)(�f2 − �f3)

)×

(Kp2

1 + stp2

)(A.2)

�f3 =([{

b3�f3 −( t12

s

)(�f1 − �f3) −

( t23

s

)(�f2 − �f3)

}(G3) −

(�f3

R3

)]

×(

1 + sc3Tr3

(1 + sTg3)(1 + sTt3)(1 + sTr3)

)+ t13

s(�f1 − �f3) + t23

s(�f2 − �f3)

)(Kp3

1 + stp3

)(A.3)

�PG1, �PG2, �PG3 are Laplace transforms of area gener-ation deviations, functions of area frequency deviations.

�Ptie,1,2 =( t12

s

)(�f1 − �f2),

�Ptie,1,3 =( t13

s

)(�f1 − �f3),

�Ptie,2,3 =( t23

s

)(�f2 − �f3) (A.4)

G1, G2, andG3 can be replaced by PID1, PID2, and PID3,respectively, for PID control.�f1(s), �f2(s), �Ptie12(s)etc. are algebraically solved and explicitly computed bysolving Eqs. (A.1)–(A.4)etc. Then, step responses by in-verse Laplace transforms with step input of (�PD/s) of

S.P. Ghoshal / Electric Power Systems Research 70 (2004) 115–127 127

these transform quantities yield�f1(t), �f2(t), �Ptie12(t)etc. respectively. With the help of “MATLAB 6.1 version”software run on 1.8 GHz. Pentium-IV computer, all thesetime functions are plotted to show transient responses.The same transient responses are also easily verified by“MATLAB–SIMULINK” software by redrawing theFig. 1block diagram asFig. 3(a), substituting all the necessaryparameters and nominal or off-nominal gains, starting thediagram (Fig. 3(a)) and displaying the responses in scopes,two such responses are shown inFig. 3(b) and (c).

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