# Solution Approach to Automatic Generation Control Problem Using Hybridized Gravitational Search Algorithm Optimized PID and FOPID Controllers.

I. INTRODUCTIONIn an interconnected power system, the purpose of automatic generation control (AGC) is to achieve better frequency regulation and maintain the tie-line power flow at scheduled level irrespective of load changes in an area. In order to implement AGC, the system parameters such as frequency, tie line power flow and individual generator outputs are monitored continuously. Based upon the monitored information and reference values of frequency and tie line power, area control errors (ACE) which are a linear combination of tie line power mismatch and frequency deviations are computed. The area control errors are sensed by the designed controllers, which generate control signals for establishing new generator set points whenever load occurs in the system. The minimization of ACE results in reduction in both frequency and tie-line power errors [1]. The researchers in this field are trying to understand and implement several strategies for AGC of power systems. The authors in [2-4] critically reviewed various control schemes such as classical, optimal [5], sub-optimal [6], adaptive, soft computing [7-9] etc. for conventional and distributed generation power systems. These approaches are not simple and need thorough knowledge and familiarity among users to implement these methods. The proportional integral derivative (PID) controller is structurally simple and reliable for implementation. The fractional order proportional integral derivative (FOPID) controller has two additional parameters of non-integer order of integration and differentiation and this feature of FOPID provides more flexibility and accuracy to the controller design [10]. The major concern of these controllers is the optimal tuning of their several parameters. Literature survey shows that in recent past, meta-heuristic optimization techniques have been used to optimize the controller parameters for AGC system such as genetic algorithm (GA), particle swarm optimization (PSO) [11-13], bacterial foraging optimization (BFO) [14], differential evolution (DE) [15], imperialist competitive algorithm (ICA) [16], firefly algorithm (FA) [17] etc. These types of heuristic optimization approaches are frequently being used because they easily handle the inherent non-linearities such as generation rate constraint (GRC), governor deadband etc. present in the AGC system. In the literature, various authors have tried to prove the superiority of one technique over the other in terms of performance parameters like premature convergence, reduction in search capability, robustness and precision etc. Therefore, the researchers are trying to develop and implement more and more efficient algorithms to deal with the complex AGC problems. The present work is an attempt towards tuning of PID and FOPID controller parameters for optimum performance using gravitational search algorithm and its variants. The gravitational search algorithm (GSA) introduced in [18] is based on Newtonian gravity and it has been reported in the literature that GSA is efficient in terms of computation time and gives more accuracy while solving optimal problems [19-20]. The convergence speed and solution quality are affected by population initialization in all the heuristic algorithms. The quality of population initialization is enhanced by using the opposition based learning concept [21-22]. A balance between exploration and exploitation ability for optimal search is maintained by employing a disruption operator introduced in [23]. The investigation of the search space for new and better solutions refers to exploration and exploitation is the capability of algorithm to search a better solution near a good one. Having known all this, an attempt has been made in this paper to tune the PID and FOPID controller parameters using a novel disruption based opposition learned gravitational search algorithm. The objectives of the present work are as follows:

i. to tune the PID and FOPID controller parameters of four area interconnected power system using hybridized disruption based opposition learned gravitational search algorithm.

ii. to compare the results of DOGSA tuned controllers with GSA, OGSA and DGSA to investigate the superiority of the optimization algorithm.

iii. to demonstrate the robustness of DOGSA tuned PID and FOPID controllers to wide variations in the system parameters, sensitivity analysis is performed.

iv. to study the effect of the physical constraints such as thermal turbine generation rate constraint, speed governor dead band and time delay caused during signal processing on the system performance.

The description of modeling of four area hydro-thermalgas interconnected power system is presented in Section II. The mathematical model of PID and FOPID controllers are discussed in Section III. Section IV presents the optimization techniques used for tuning the gain parameters of the controllers. The simulation results of considered power system with PID and FOPID controllers optimized by GSA and its variants are presented in Section V. The next subsection gives the sensitivity analysis followed by the effect of generation rate constraint, time delay and governor deadband. The conclusion based on the simulation results is discussed in Section VI.

II. SYSTEM MODELING

The four area power system hydro-thermal-gas plant connected by tie-line is shown in Fig. 1. The fourth area is considered to have infinite kinetic energy i.e. free energy source which implies that the frequency deviations did not matter in this area [5, 24]. The inputs to each control area are the controller outputs ([u.sub.k]), load disturbances ([d.sub.k]), and tie-line power error [DELTA][P.sub.tiek] for an area k. The outputs are the generator frequency deviations ([[DELTA].sub.fk]) and area control error ([ACE.sub.k]) given as:

[mathematical expression not reproducible] (1)

[DELTA][P.sub.kj] is the tie line power flow from area k to area j and [B.sub.k] is the bias factor. Each control area comprises of speed governor, turbine and load whose transfer functions are given in Fig. 1. The symbols and parameters of the considered interconnected power system are given in Appendix A and Appendix B.

III. MATHEMATICAL MODEL OF PID AND FOPID CONTROLLER

The fractional order (FO) theory deals with the differential equations of non-integer order. The fractional calculus is the generalization of integer order (IO) calculus. The FOPID controller has the notion given by [PI.sup.[lambda]] [D.sup.u] where [lambda] and [mu]. are non-integer orders of integrator and differentiator respectively. The controlled action of PI D controller for considered interconnected power system can be expressed as:

[mathematical expression not reproducible] (2)

where e(t) is the input area control error signal, u(t) is controller output, [K.sub.p], [K.sub.i] and [K.sub.d] are proportional, integral and derivative gains respectively. The additional tuning knobs [lambda] and [mu] in FOPID controller offers better flexibility and system dynamics than the integral order proportional integral derivative (IOPID) controller [10]. The laplace transfer function of FOPID controller of generator n is given as:

[mathematical expression not reproducible] (3)

When the value [lambda] and [mu] are equal to unity, it gives the transfer function of IOPID controller. The fractional derivative or integral [s.sup.[rho]], where [rho] is a non-integer number can be approximated by Oustaloup recursive filter in pre-specified frequency range [[[omega].sub.l], [[omega].sub.u] ], where [[omega].sub.l] and [[omega].sub.u] is the lower and upper frequency limits of approximate transfer function [10] expressed as:

[mathematical expression not reproducible] (4)

where, constant gain, frequency of zeros and poles are determined by:

[mathematical expression not reproducible] (5)

where [N.sub.f] is number of zeros and poles, p and q are small constant gains.

IV. CONTROLLER TUNING USING HEURISTIC OPTIMIZATION TECHNIQUES

This section gives a brief overview of basic GSA, opposition learning and disruption operator. Then, the pseudo code for hybridized DOGSA approach to obtain optimal parameter values is given which gives the advantages of fast convergence and better exploration & exploitation of the search space. The optimization of the controller gains corresponds to the minimum deviations in frequency and tie-line power. The overall fitness function, integral time absolute error (ITAE) for the optimization problem in present work is:

[mathematical expression not reproducible] (6)

where [DELTA][f.sub.k] and [DELTA][P.sub.tiek] are deviations in frequency and tie-line power in area k for k =1, 2, 3 and T is the time interval over which fitness function is evaluated. The fitness function penalizes the deviations in frequency and tie-line power that persist for long periods of time. The figure of demerit (FOD) is also calculated whose minimum value indicates the minimization of the overshoot (OS) and steady state value (SV) of deviations in frequency and tie-line power.

[mathematical expression not reproducible] (7)

The state space model of the system is developed using the state variables vector:

X = [[DELTA][f.sub.1] [DELTA][f.sub.2] [DELTA][f.sub.3] [[DELTA][P.sub.tie1] [[DELTA][P.sub.tie2] [[DELTA][P.sub.tie3] [[DELTA][P.sub.t1] [[DELTA][P.sub.r1] [[DELTA][P.sub.g1] [[DELTA][P.sub.t2] [[DELTA][P.sub.r2] [[DELTA][P.sub.g2] [[DELTA][P.sub.tw] [[DELTA][P.sub.G2] [[DELTA][P.sub.G1] [[DELTA][P.sub.TD] [[DELTA][P.sub.FC] [[DELTA][P.sub.vp] [[DELTA][P.sub.sg] [[DELTA][P.sub.t3] [[DELTA][P.sub.g3] [ACE.sub.1][G.sub.1] (s) [ACE.sub.1] [G.sub.2](s) [ACE.sub.2][G.sub.3](s) [ACE.sub.2][G.sub.4](s) [ACE.sub.3][G.sub.5](s)]

A. Gravitational Search Algorithm

GSA is a heuristic optimization algorithm based on Newton's law of gravity and motion [18]. For tuning of PID controller, the masses correspond to the controller gains and the performance is the fitness function given by equation (6). To describe the GSA, consider a system with [N.sub.p] controller gains in which the position of the [i.sub.th] gain is defined by:

[U.sub.i]=([u.sup.1.sub.i],...., [u.sup.d.sub.i],....,[u.sup.n.sub.i]) for i = 1,2,...., [N.sub.p] (8)

where [u.sup.d.sub.i] denotes the position of the [i.sup.th] gain in the [d.sup.th] dimension and n is the dimension of the search space. The positions of gains in the search space correspond to the optimal solution. The acceleration of the gain i at time t, and in direction [d.sup.th] according to law of motion ([a.sup.d.sub.i](t)), is:

[mathematical expression not reproducible] (9)

where [f.sup.d.sub.i] is force acting on gain i, [W.sub.i] is the inertial mass of [i.sup.th] gain, W is the gravitational active mass, G(t) is

gravitational constant at time t, [G.sub.0] is initial gravitational constant, [epsilon] is small constant, [alpha] is small decaying constant, [rand.sub.j] is a random number in the interval [0, 1] and [R.sub.ij](t) is the euclidian distance between two gains i and j. The fitness evaluation gives the inertial mass. An agent (controller gain) is more efficient if it has more mass. This means that better gains have higher attractions and move more slowly. The inertial masses are updated by the following equation:

[mathematical expression not reproducible] (10)

where [F.sub.i](t) denotes the fitness value of the gain i at time t, and, w(t) and b(t) are defined as follows for a minimization problem:

[mathematical expression not reproducible] (11)

[mathematical expression not reproducible] (12)

The updation of position and velocity of controller gains in search space is done as:

[mathematical expression not reproducible] (13)

[mathematical expression not reproducible] (14)

where randi is a uniform random variable in the interval [0, 1].

B. Opposition based Learning

The evolutionary optimization algorithms are population based techniques which are aimed at finding the optimal solutions to the problem. The random presumptions are made at the beginning about the solution. The computational time for searching optimal solution can be reduced by concurrently checking the opposite solution [21]. This helps in choosing the better solution as an initial solution and hence, increases the potential to accelerate convergence of the algorithm. Let U = ([u.sub.1],[u.sub.2],....,[u.sub.d]) be a point in d-dimensional space, where [u.sub.1], [u.sub.2],....,[u.sub.d] [member of] R and [u.sub.i] [member of] [[a.sub.i],[b.sub.i]] [member of] {1,2,...., d}. The opposite point is defined by [18]:

[u.sub.i]=[a.sub.i]+[b.sub.i]-[u.sub.i] (15)

The fitness function given by equation (6) is used to calculate the individual fitness at both [u.sub.i] and [u.sub.i]. If f(u) [greater than or equal to] f(u), then point ucan be replaced with uelse continue with u. The GSA based on this opposition based population initialization is called as OGSA.

C. Disruption Operator

The exploration and exploitation abilities of GSA are enhanced using novel operator called disruption which is originated from astrophysics [23]. For the simulation of the disruption process, it is assumed that the best solution (agent with heaviest mass) is the star of the system, and the other solutions can potentially disrupt and scatter in space under the gravity force of the star. The solutions that satisfy the condition given by equation (16) below become disrupted so as to prevent divergence in solutions [23].

[mathematical expression not reproducible] 16

where [R.sub.i,j] is euclidian distance between masses i and j, [R.sub.i,best] is euclidian distance between mass i and the best solution, K is threshold value, [tau] is small constant, t is current iteration and [iter.sub.max] is total number of iterations. The position of every mass (solution) that satisfies equation (16) will change, according to the following equation:

[U.sub.i](new) = [U.sub.i] (old).D (17)

where [mathematical expression not reproducible] (18)

In equation (18), c(-0.5, 0.5) returns a uniformly distributed pseudo random number in the interval [-0.5, 0.5]. [U.sub.i] is the position of mass i that should be disrupted and [beta] is a small number. The GSA incorporating the disruption operator is called DGSA.

D. Hybridized Disruption based Opposition Learned GSA (DOGSA)

A more recent hybridized DOGSA technique is introduced to incorporate the advantages of both opposition based learning to increase the convergence rate of optimization algorithm and disruption operator to have a good balance between exploration and exploitation to achieve both efficient global and local searches. The pseudo code of DOGSA optimization technique is given below: Step 1: Initialize population [P.sub.0] with controller gains Step 2: Initialize population [OP.sub.0] with opposite presumptions for (i=0;i<[N.sub.p];i++) % [N.sub.p] is the size of population for (j=0;j<d;j++) % d is the dimension of problem

[OP.sub.i,j]=[a.sub.j]+[b.sub.j]-[P0.sub.i,j]

end

Step 3: Evaluate the fitness function using equation (6)

Step 4: Rank the fitness function based on their evaluated value. Select [N.sub.p] individuals from the set of {[P.sub.0] [OP.sub.0]} based on the minimum fitness function as initial population P

Step 5: Calculate [W.sub.i](t), b(t) and w(t) using equation (10), equation (11) and equation (12)

Step 6: Calculate forces in all directions according to equation (9)

Step 7: Update positions and velocities of all gains using equation (13) and equation (14)

Step 8: Disrupt the positions [u.sub.i] using equation (17) and equation (18)

Step 9: Repeat steps 3-7 until stopping criterion is met

V. RESULTS AND DISCUSSIONS

This section presents simulation results of four area AGC interconnected power system using stochastic heuristic approach GSA and its variants such as OGSA, DGSA and DOGSA tuned PID and FOPID controllers. For each of the algorithms, integral time absolute error fitness function has been used to find the optimal gains of controllers. In this paper, the minimum limit of each of [K.sub.p], [K.sub.i], [K.sub.d] is -10 and the maximum limit is 10 and for [lambda] and [mu], minimum and maximum values are chosen as 0 and 2. Each generating area has its individual controller which increases the flexibility of the system. The controller parameters chosen for the application of optimization algorithms are: population size [N.sub.p]=30; maximum iteration [iter.sub.max]=1000, [G.sub.0]=100, [alpha]=20, [tau]=100 and [beta]=[10.sup.-16]. The frequency range for Oustalop approximation is taken to be [0.01 100] and [N.sub.f] =5. The settling time and steady state values are calculated considering an error band of 1%.

A. Comparison of GSA, OGSA, DGSA and DOGSA Tuned PID Controller

The optimization techniques GSA, OGSA, DGSA and DOGSA are implemented to tune the PID controller gains for four area AGC system when a step load perturbation of 0.1 p.u MW is considered in all the areas. The convergence profiles of all the optimization algorithms are shown in Fig. 2. The general trend observed from these characteristics is that all the presented algorithms quickly bring down the fitness function during initial iterative process. As expected GSA, OGSA, DGSA and DOGSA saturate to fitness values which are 0.1046, 0.0945, 0.0874 and 0.0823 respectively. The best performance of DOGSA giving least fitness value is attributed to its better exploration and exploitation capabilities. The eigen values depicting the relative stability of the system, damping ratio, settling time and peak overshoot of frequency and tie-line power deviations of the system, ITAE fitness function and figure of demerit (FOD) values are given in Table I for PID controller. The dynamic responses are shown for area 1 only.

Although, all the eigen values of system without controller lies on left side of s-plane, yet the tie-line power deviations does not reach their steady state. The optimized PID diminishes the perturbation in frequency and scheduled tie line power to their steady state values. Examining the performance criterion values from Table I and dynamic responses from Fig. 3 and Fig. 4, it is observed that the DOGSA tuned PID controlled system has minimum integral time absolute error, damping ratio, settling time and peak overshoot as compared to its other variants.

The figure of demerit is also minimum for DOGSA optimized PID controlled system which signifies the minimum settling time and peak overshoot of deviations in frequency and tie-line power. The optimized gains of these controllers are given in Table II.

B. Comparison of GSA, OGSA, DGSA and DOGSA Tuned FOPID Controller

The convergence profiles of fitness values of GSA, OGSA, DGSA and DOGSA tuned FOPID controlled interconnected system is shown in Fig. 5. The fitness values are saturated to 1.6736, 1.4742, 1.2238 and 1.1522 for GSA, OGSA, DGSA and DOGSA respectively. The exploration and exploitation capabilities of DOGSA enable it to achieve the minimum fitness value in 55 iterations only. The eigen values, damping ratio, settling time, peak overshoot, fitness function and figure of demerit values for optimal FOPID controlled system are given in Table III.

The eigen values of DOGSA optimized FOPID controlled system lies on left side of s-plane far away from origin as compared to its other variants which signifies more stability of the system. The least figure of demerit 0.00004339 is given by DOGSA which shows minimum steady state value and peak overshoot of deviations in frequency and tie-line power. The dynamic responses of area 1 are shown in Fig. 6 and Fig. 7.

C. Sensitivity Analysis

The robustness of the DOGSA tuned PID and FOPID controllers are also investigated against wide variations of system parameters. The operating load conditions, tie-line synchronizing coefficients, time constants of speed governor and turbine are varied in the range of +50% to -50% in steps of 25% taking at a time from their nominal values.

a. Sensitivity Analysis of DOGSA Optimized PID Controller

The settling time and peak overshoot of the frequency and tie-line power deviations and damping ratio of the optimal PID controlled system are given in Table IV. It is observed that the overall system performance is hardly changed when the operating load condition and system parameters are varied. The frequency deviation responses of area 1 with different load conditions are shown in Fig. 8. which show that the DOGSA optimized PID controller gives robust and stable control to the system in case of wide variations in the nominal loading and other system parameters.

b. Sensitivity Analysis of DOGSA Optimized FOPID Controller

The damping ratio, settling time and peak overshoot of deviations in frequency and tie-line power for DOGSA optimized FOPID controller are given in Table V. From the simulation responses shown in Fig. 9 and system performance given in Table V, it has been concluded that the proposed control strategy provides a robust and stable control of the system and the controller parameters need not to be reset for wide changes in the system loading or parameters.

D. Handling of Physical Constraints

It is necessary to include the basic constraints imposed by the physical system dynamics and model them for the sake of performance evaluation to get an accurate discernment into the AGC problem. The generation rate constraint (GRC), time delay during signal processing in interconnected system and governor dead band are the important constraints which affects the power system. The values considered for these constraints are given in Table VI.

a. DOGSA Tuned PID Controller

The system dynamic responses of deviations in frequency and tie line power of area 1 including the physical constraints with optimized PID controller are shown in Fig. 10 and Fig. 11. The settling time and peak overshoot of deviations in frequency and tie-line power are given in Table VII. It is observed that the DOGSA optimized PID controller stabilizes the system even if the system becomes highly non-linear. The time delay and governor deadband cause a great overshoot/undershoot after a disturbance. But the controller successfully minimizes the frequency and tie-line power deviations to steady state in a considerable time.

b. DOGSA Tuned FOPID Controller

The simulation responses of DOGSA optimized FOPID controller including the physical constraints are shown in Fig. 12 and Fig. 13 for area 1. The settling time and

overshoot of deviations in frequency and tie-line power given in Table VIII reveals that the proposed approach stabilizes the system rapidly as compared to PID controller.

VI. CONCLUSION

The PID and FOPID controllers for AGC four area interconnected power system are tuned using GSA, OGSA, DGSA and DOGSA stochastic heuristic optimization techniques. The superiority of the hybridized DOGSA technique is investigated by comparing the convergence profile with other optimization techniques. The comparison of eigen values, damping ratio, settling time, peak overshoot and integral time absolute error shows the promising nature of proposed DOGSA technique. The optimized FOPID controller minimizes the frequency and tie line deviations faster as compared to tuned PID controller. The results obtained from the sensitivity analysis of the optimal PID and FOPID controlled system to wide variations in the system loading and parameters shows the robustness of the proposed controller. The proposed controller technique is also able to stabilize the power system effectively in the presence of non-linearities in the system.

APPENDIX A

Nominal values of parameters used in four area system models [25-27]

Parameter Description f Nominal frequency [P.sub.r1], [P.sub.r2], [P.sub.r3] Rated generation of thermal, hydro and gas plant respectively [P.sub.d Nominal load [R.sub.1],[R.sub.2],[R.sub.3], Regulations of [R.sub.4],[R.sub.5] governors in areas 1,2 and 3 [B.sub.1],[B.sub.2],[B.sub.3] Tie line frequency bias in areas 1, 2 and 3 Governor time [T.sub.g1],[T.sub.g2],[T.sub.g3] constants for thermal areas 1 and 3. Turbine time constants [T.sub.t1],[T.sub.t2],[T.sub.t3] for thermal areas 1 and 3 Reheater constraints [K.sub.r1],[K.sub.r2] (gains) for thermal reheat area 1 Reheater time [T.sub.t1],[T.sub.t2] constants for thermal (reheat) area 1 [K.sub.p1], [K.sub.p2], [K.sub.p3] Power system gains of area 1, 2 and 3 Power system load [T.sub.p1], [T.sub.p2], [T.sub.p3] time constants of area 1, 2 and 3 Hydro governor (Stage [T.sub.1] 1) time constant for area 2 Hydro governor (Stage [T.sub.2] 2) time constant for area 2 Hydro governor (Stage [T.sub.3] 2) time constant for area 2 [T.sub.w] Water starting time for water turbine in area 2 Synchronizing coefficients for tie lines [T.sub.ij] between pair of areas for the four area system X Gas governor lead time constant Y Gas governor lag time constant a,c Valve positioner constants b Valve positioner constant [T.sub.CR] Combustion reaction time delay [T.sub.F] Fuel time constant [T.sub.CD] Compressor discharge volume time constant Parameter Value f 60 [P.sub.r1], [P.sub.r2], [P.sub.r3] 1000, 600, [P.sub.d 1000 [R.sub.1],[R.sub.2],[R.sub.3],[R.sub.4],[R.sub.5] 2.4 [B.sub.1],[B.sub.2],[B.sub.3] 0.425 [T.sub.g1],[T.sub.g2],[T.sub.g3] 0.08 [T.sub.t1],[T.sub.t2],[T.sub.t3] 0.4 [K.sub.r1],[K.sub.r2] 0.333 [T.sub.t1],[T.sub.t2] 10 [K.sub.p1], [K.sub.p2], [K.sub.p3] 120 [T.sub.p1], [T.sub.p2], [T.sub.p3] 20 [T.sub.1] 48.7 [T.sub.2] 0.513 [T.sub.3] 10 [T.sub.w] 1 [T.sub.ij] 0.0707 X 0.6 Y 1.0 a,c b 0.5 [T.sub.CR] 0.3 [T.sub.F] 0.23 [T.sub.CD] 0.2 Parameter Unit f Hz [P.sub.r1], [P.sub.r2], [P.sub.r3] MW [P.sub.d MW [R.sub.1],[R.sub.2],[R.sub.3],[R.sub.4],[R.sub.5] Hz/p.u MW [B.sub.1],[B.sub.2],[B.sub.3] p.u MW/Hz [T.sub.g1],[T.sub.g2],[T.sub.g3] second [T.sub.t1],[T.sub.t2],[T.sub.t3] second [K.sub.r1],[K.sub.r2] Thermal Unit [T.sub.t1],[T.sub.t2] second [K.sub.p1], [K.sub.p2], [K.sub.p3] Hz/p.u [T.sub.p1], [T.sub.p2], [T.sub.p3] second [T.sub.1] second [T.sub.2] second [T.sub.3] second [T.sub.w] second [T.sub.ij] MW/radian X second Y second a,c b - [T.sub.CR] second [T.sub.F] second [T.sub.CD] second

APPENDIX B

[[DELTA].sub.fk] incremental frequency deviation in area k [DELTA][P.subtiek] net incremental real power exported from area k [[DELTA].sub.int[infinity]] net incremental real power exported from area considered to have infinite kinetic energy [DELTA][P.sub.tk] incremental real power generated from thermal reheat and non-reheat turbines [DELTA][P.sub.rk] incremental real power generated from reheat turbine [DELTA][P.sub.gk] change in steam valve position [DELTA][P.sub.tw] incremental real power generated from hydro turbine [DELTA][P.sub.Gk] change in hydro speed governor [DELTA][P.sub.TD] incremental real power generated from gas turbine [DELTA][P.sub.FC] incremental change in fuel combustor output [DELTA][P.sub.vp] change in gas valve positioner [DELTA][P.sub.sg] change in gas speed governor output

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Preeti DAHIYA (1), Veena SHARMA (2), Ram NARESH (3)

(1,2,3) National Institute of Technology, Hamirpur, Himachal Pradesh, 177005, India

(1) preetieed@nith.ac.in, (2) veena@nith.ac.in, (3) rnaresh@nith.ac.in

Digital Object Identifier 10.4316/AECE.2015.02004

TABLE I. COMPARISON OF EIGEN VALUES, DAMPING RATIO, SETTLING TIME, PEAK OVERSHOOT, FITNESS FUNCTION AND FIGURE OF DEMERIT VALUES FOR OPIMIZED PID CONTROLLED SYSTEM Parameters Without Controller GSA -12.5034 -12.5065 -12.5104 -12.5015 -4.7057 [+ or -] 0.2425i -4.9778 Eigen Values -0.0446 [+ or -] 1.0687i -4.3221 -0.0539 [+ or -] 1.0646i 0.0765 [+ or -] 1.2800i -0.0561 [+ or -] 0.5354i 0.0450 [+ or -] 1.1070i -0.9574 -2.6734 -2.6595 -2.5357 [+ or -] 0.0779i -2.3177 -0.0087 [+ or -] 0.6130i -2.4522 -1.1277 -1.9675 -2.0461 -0.0205 -2.0000 -0.0990 -2.5000 -0.1002 -0.0488 -2.5000 -0.0198 [+ or -]0.0208i -2.0000 -0.0158 -0.1000 -0.1044 -0.0989 -0.1000 -0.0000 -0.0000 Damping 0.0417 0.0141 Ratio Settling [DELTA][f.sub.1] 73.10 34.37 [DELTA][f.sub.2] 78.80 37.67 [DELTA][f.sub.3] 68.20 38.09 [DELTA][P.sub.tie1] 71.50 101.80 [DELTA][P.sub.tie2] 88.40 211.10 [DELTA][P.sub.tie3] 66.30 294.70 Peak overshoot [DELTA][f.sub.1] 0.001175 0.0008501 [DELTA][f.sub.2] 0.001148 0.0008772 [DELTA][f.sub.3] 0.001190 0.0008293 [DELTA][P.sub.tie1] -0.01704 -0.009886 [DELTA][P.sub.tie2] -0.01680 -0.013410 [DELTA][P.sub.tie3] -0.01678 -0.008562 Steady state values [DELTA][f.sub.1] -0.0000311 0.0000561 [DELTA][f.sub.2] -0.0000212 0.0000017 [DELTA][f.sub.3] 0.0000408 0.0000135 [DELTA][P.sub.tie1] -0.0098410 -0.0000185 [DELTA][P.sub.tie2] -0.0100060 0.0000499 [DELTA][P.sub.tie3] -0.0101300 -0.0000514 ITAE - 0.1046 fitness value FOD - 0.000353 Parameters OGSA DGSA -12.5064 -12.5029 -12.5003 -12.5098 -4.7027 [+ or -] 0.2196i -4.7070 [+ or -] 0.2482i Eigen Values -2.7270 -2.5528 [+ or -] 0.3420i -2.6692 0.0435 [+ or -] 1.1325i -2.3718 -0.0301 [+ or -] 1.0814i 0.0836 [+ or -] 1.3133i -2.4800 -0.0014 [+ or -] 1.1092i -1.9630 -1.9685 -2.0000 -2.0000 -0.9282 -0.9423 -0.0285 [+ or -]0.5611i -0.0283[+ or -]0.5977i -0.0285 - 0.5611i -2.5000 -2.5000 -0.1388 -0.0943 [+ or -]0.0408i -0.1084 -0.0943 - 0.0408i -0.0308 [+ or -] 0.0254i -0.1115 -0.0063 -0.0254 [+ or -] 0.0313i -0.0104 -0.0234 -0.1000 -0.1000 -0.0000 -0.0000 -0.0000 -0.0000 Damping 0.0012 0.0279 Ratio Time (sec) [DELTA][f.sub.1] 25.03 25.01 [DELTA][f.sub.2] 26.42 29.48 [DELTA][f.sub.3] 28.04 24.86 [DELTA][P.sub.tie1] 144.37 96.57 [DELTA][P.sub.tie2] 145.10 144.50 [DELTA][P.sub.tie3] 243.10 248.80 [DELTA][f.sub.1] 0.0006584 0.0002684 [DELTA][f.sub.2] 0.0006091 0.0003999 [DELTA][f.sub.3] 0.0006185 0.0003469 [DELTA][P.sub.tie1] -0.005301 -0.004194 [DELTA][P.sub.tie2] -0.012800 -0.012600 [DELTA][P.sub.tie3] -0.007700 -0.001470 [DELTA][f.sub.1] 0.0000470 0.0000555 [DELTA][f.sub.2] 0.0000120 0.0000117 [DELTA][f.sub.3] 0.0000118 0.0000115 [DELTA][P.sub.tie1] -0.0000336 -0.00000075 [DELTA][P.sub.tie2] 0.0000729 -0.0000268 [DELTA][P.sub.tie3] -0.0000480 -0.0000725 ITAE 0.0945 0.0874 fitness value FOD 0.000252 0.000179 Parameters DOGSA -12.5066 -12.4997 -4.6900 [+ or -] 0.1068i Eigen Values -2.6860 -2.5723 [+ or -] 0.1984i -0.0944 [+ or -] 1.3170i -0.0373 [+ or -] 1.1395i -1.9863 -2.0000 -1.0052 -0.0173 [+ or -] 0.6194i -2.5000 -0.1488 -0.0651 [+ or -] 0.0306i -0.0328 -0.0089 -0.0080 -0.1000 -0.0000 [+ or -] 0.0000i Damping 0.0236 Ratio [DELTA][f.sub.1] 21.33 [DELTA][f.sub.2] 25.24 [DELTA][f.sub.3] 21.94 [DELTA][P.sub.tie1] 63.09 [DELTA][P.sub.tie2] 160.90 [DELTA][P.sub.tie3] 174.90 [DELTA][f.sub.1] 0.0002238 [DELTA][f.sub.2] 0.0003496 [DELTA][f.sub.3] 0.0002345 [DELTA][P.sub.tie1] -0.006406 [DELTA][P.sub.tie2] -0.009537 [DELTA][P.sub.tie3] -0.005298 [DELTA][f.sub.1] 0.0000272 [DELTA][f.sub.2] 0.0000115 [DELTA][f.sub.3] 0.0000113 [DELTA][P.sub.tie1] 0.0000167 [DELTA][P.sub.tie2] -0.0000916 [DELTA][P.sub.tie3] -0.0000725 ITAE 0.0823 fitness value FOD 0.000160 TABLE II. OPTIMIZED PARAMETERS OF PID AND FOPID CONTROLLERS PID Initial values Parameter (Before GSA OGSA DGSA DOGSA optimization) [K.sub.p1] -3.5010 4.1205 3.0257 1.8027 -4.8601 [K.sub.i1] 2.0308 1.5353 -2.8987 3.3507 4.7489 [K.sub.d1] -4.5438 2.2709 3.6301 4.4628 4.7406 [[lambda].sub.1] - - - - - [[mu].sub.1] - - - - - [K.sub.p2] -4.3785 -1.4271 -3.7641 -4.9564 -3.2687 [K.sub.i2] -3.5042 4.2183 2.2584 3.0313 2.8394 [K.sub.d2] -1.2064 -0.4860 1.3285 -0.2105 -4.8444 [[lambda].sub.2] - - - - - [[mu].sub.2] - - - - - [K.sub.p3] -2.8794 0.2738 -4.0111 2.7211 -2.5686 [K.sub.i3] -3.9710 -3.4289 -4.8503 2.9749 -1.7065 [K.sub.d3] 4.8681 0.6248 4.7562 -4.5950 0.0836 [[lambda].sub.3] - - - - - [[mu].sub.3] - - - - - [K.sub.p4] 4.1579 -2.5089 0.0513 -1.4624 0.5753 [K.sub.i4] 3.0077 0.8999 2.2554 -2.3394 -1.2381 [K.sub.d4] -4.5800 -3.4513 -0.4474 4.3253 3.6706 [[lambda].sub.4] - - - - - [[mu].sub.4] - - - - - [K.sub.p5] 1.5249 -2.0176 -0.1260 -2.7869 -1.2885 [K.sub.i5] 3.0571 -1.2252 0.4366 -1.4046 -1.4560 [K.sub.d5] 1.8484 -1.0604 -3.7738 -0.3234 2.0932 [[lambda].sub.5] - - - - - [[mu].sub.5] - - - - - FOPID Initial values Parameter (Before GSA OGSA DGSA DOGSA optimization) [K.sub.p1] 0.8042 0.5254 0.5729 0.3524 1.6662 [K.sub.i1] 0.0314 0.0447 0.0312 0.0511 0.0580 [K.sub.d1] 4.6400 1.0844 6.0844 5.8209 9.3220 [[lambda].sub.1] 1.9627 1.5316 1.3120 1.6370 1.6721 [[mu].sub.1] 0.7233 0.7623 0.7115 0.7716 0.9127 [K.sub.p2] 0.6804 0.5680 0.7498 4.7833 2.8702 [K.sub.i2] 0.0079 0.0874 0.0128 0.0662 0.0960 [K.sub.d2] 11.6187 2.6566 1.4086 6.3415 1.8104 [[lambda].sub.2] 1.2549 1.2222 1.0619 1.6074 1.4603 [[mu].sub.2] 0.7654 0.7192 0.7159 0.7259 0.7015 [K.sub.p3] 0.9972 0.0965 0.0166 0.0244 0.0236 [K.sub.i3] 0.0295 0.0299 0.0234 0.0255 0.0229 [K.sub.d3] 1.5793 0.7243 4.0751 0.9766 5.7880 [[lambda].sub.3] 0.1356 1.4201 1.4441 1.5568 1.6530 [[mu].sub.3] 0.7337 0.7581 0.7706 0.7363 0.7728 [K.sub.p4] 0.3602 0.0673 0.0116 0.0924 0.0001 [K.sub.i4] 0.0040 0.0247 0.0345 0.0291 0.0330 [K.sub.d4] 4.5944 0.0136 0.0256 0.0216 0.0195 [[lambda].sub.4] 1.7005 1.5766 1.6032 1.3037 1.4672 [[mu].sub.4] 0.7896 0.7499 0.7805 0.7999 0.7260 [K.sub.p5] 0.2486 1.6304 0.8311 0.2246 1.0928 [K.sub.i5] 0.0108 0.0633 0.0215 0.0918 0.0993 [K.sub.d5] 4.9881 0.6657 0.9569 1.0191 8.2022 [[lambda].sub.5] 1.7004 1.7751 1.5437 1.4924 1.3127 [[mu].sub.5] 0.7086 0.7371 0.7869 0.7109 0.7545 TABLE III. COMPARISON OF EIGEN VALUES, DAMPING RATIO, SETTLING TIME, PEAK OVERSHOOT, FITNESS FUNCTION AND FIGURE OF DEMERIT VALUES FOR OPTIMAL FOPID CONTROLLED SYSTEM Parameters GSA OGSA -71.0995 -90.9355 -64.9894 -72.1076 -66.7984 -70.3182 -69.4306 -71.3948 -68.9816 -61.9441 -68.1187 -64.2337 -52.4727 -64.6581 -44.2181 -50.5747 -41.2667 -43.4044 Eigen Values -30.4275 -39.6156 -12.5396 -12.5790 -12.5106 -12.5120 -12.5000 -12.5000 -4.7443 -0.1726 [+ or -] 2.1280i -4.5920 -4.6940 -0.0353 [+ or -] 1.7383i -4.5966 -3.9408 -4.2244 -0.0920 [+ or -]1.4084i -4.2115 -3.6261 -3.8664 -3.2998 -0.1542 [+ or -] 1.4184i -3.1534 -3.3596 -3.2015 -3.2636 -3.1908 -2.9972 -0.0629 [+ or -] 0.6624i -2.8752 -2.4356 -0.0848 [+ or -] 0.6590i -1.4050 -2.3475 -1.3196 -1.6999 -0.9820 -1.8414 -1.8805 -2.0252 -2.0396 [+ or -] 0.0325i -1.9723 -2.0553 -2.0000 -2.0000 -0.9085 -2.5000 -0.9212 -0.2101 -2.5000 -0.0162 [+ or -] 0.1247i -0.0311 [+ or -] 0.1277i -0.0257 [+ or -] 0.1055i -0.2162 -0.0234 [+ or -] 0.0949i -0.1949 -0.0210 -0.0294 [+ or -] 0.0524i -0.0197 -0.0217 [+ or -] 0.0303i -0.0351 -0.0200 -0.0577 -0.0290 -0.1005 [+ or -]0.0200i -0.0362 -0.1163 -0.0433 -0.1506 -0.1017 [+ or -] 0.0189i -0.1449 -0.1538 -0.1471 -0.1345 -0.1460 -0.1478 -0.1410 -0.1391 -0.1000 -0.1399 -0.1000 Damping Ratio 0.0203 2.10*[10.sub.-5] Settling time (sec) [DELTA][f.sub.1] 53.99 49.27 [DELTA][f.sub.2] 70.05 69.11 [DELTA][f.sub.3] 63.45 55.46 [DELTA][P.sub.tie1] 145.60 119.60 [DELTA][P.sub.tie2] 156.30 132.10 [DELTA][P.sub.tie3] 161.80 106.50 Peak overshoot [DELTA][f.sub.1] 0.001045 0.0009196 [DELTA][f.sub.2] 0.001267 0.0012400 [DELTA][f.sub.3] 0.001082 0.0009862 [DELTA][P.sub.tie1] 0.003367 0.0014020 [DELTA][P.sub.tie2] 0.008158 0.0067840 [DELTA][P.sub.tie3] 0.003063 0.0022180 Steady state values [DELTA][f.sub.1] 0.0000185 -0.0000101 [DELTA][f.sub.2] 0.00000973 0.00000506 [DELTA][f.sub.3] 0.0000353 0.0000429 [DELTA][P.sub.tie1] 0.0000667 -0.0000616 [DELTA][P.sub.tie2] 0.000175 0.000123 [DELTA][P.sub.tie3] -0.000121 -0.0000221 Fitness value 1.6736 1.4742 FOD 0.00009118 0.00005631 Parameters DGSA DOGSA -73.5529 -87.4911 -70.4417 -70.5561 -66.7114 -68.6652 -65.6702 -65.6649 -64.1718 -63.2453 -62.7384 -61.8776 -46.9606 -49.3325 -42.5402 -48.8128 -39.3610 -35.6396 Eigen Values -37.6126 -36.7001 -12.6267 -12.8588 -12.5135 -12.5241 -12.5000 -12.5000 -0.0499 [+ or -] 2.6459i -0.0291 [+ or -] 3.5820i -4.8777 -4.9513 -4.4495 [+ or -] 0.1334i -4.6478 -3.9327 -4.5916 [+ or -] 0.1926i -0.3113 [+ or -] 1.5447i -0.4119 [+ or -] 2.2978i -3.4397 -3.2738 -3.1604 -3.1215 -3.0963 -3.0691 -2.9187 -2.8725 -0.0813 [+ or -] 0.6953i -0.0788 [+ or -] 0.7119i -0.8256 -2.2917 -1.0056 [+ or -] 0.0744i -2.2812 -2.1877 -1.6570 -1.7472 -1.7035 -1.8287 -1.9442 -1.9646 -2.0000 -1.9745 -2.5000 -2.0000 -0.9031 -2.5000 -0.9262 -0.0340 [+ or -] 0.1607i -0.4733 -0.2473 -0.2382 -0.0126 [+ or -] 0.0820i -0.0181 [+ or -] 0.0964i -0.0098 [+ or -] 0.0513i -0.0719 [+ or -] 0.0925i -0.1024 [+ or -] 0.0642i -0.0319 [+ or -] 0.0822i -0.0301 -0.1737 -0.0225 -0.0334 [+ or -] 0.0539i -0.0235 -0.0186 -0.0827 -0.0281 -0.1583 -0.0273 -0.1479 -0.0911 -0.1425 -0.1325 -0.1329 -0.1418 -0.1247 -0.1398 -0.1181 -0.1359 -0.1000 -0.1016 -0.1000 Damping Ratio 0.0189 6.57*[10.sup.-7] [DELTA][f.sub.1] 47.29 46.61 [DELTA][f.sub.2] 68.43 63.41 [DELTA][f.sub.3] 52.70 50.85 [DELTA][P.sub.tie1] 104.60 102.40 [DELTA][P.sub.tie2] 112.80 112.70 [DELTA][P.sub.tie3] 112.60 77.50 [DELTA][f.sub.1] 0.008673 0.000690 [DELTA][f.sub.2] 0.001195 0.001165 [DELTA][f.sub.3] 0.000978 0.000688 [DELTA][P.sub.tie1] 0.001324 0.001571 [DELTA][P.sub.tie2] 0.005074 0.006044 [DELTA][P.sub.tie3] 0.004917 0.001443 [DELTA][f.sub.1] -0.0000129 0.000000199 [DELTA][f.sub.2] -0.0000461 -0.0000269 [DELTA][f.sub.3] -0.0000177 -0.0000140 [DELTA][P.sub.tie1] -0.0000741 0.0000164 [DELTA][P.sub.tie2] 0.0000429 -0.0000246 [DELTA][P.sub.tie3] -0.000152 -0.0000831 Fitness value 1.2238 1.1522 FOD 0.00005482 0.00004339 TABLE IV. SENSITIVITY ANALYSIS OF DOGSA TUNED PID CONTROLLER Parameter Settling Time (sec) Variation % change [DELTA][f.sub.1] [DELTA][f.sub.2] +25 55.84 60.1 Loading -25 57.35 58.35 condition +50 65.4 60.11 -50 63.76 61.98 [T.sub.g] +25 65.64 62.12 -25 63.53 61.64 +50 65.84 60.72 -50 69.92 60.21 +25 67.35 65.58 -25 75.7 65.51 [T.sub.t] +50 78.89 61.88 -50 78.01 63.34 +25 42.83 52.9 -25 185.5 191.5 [T.sub.ij] +50 70.45 68.17 -50 66.17 54.7 Peak Overshoot +25 0.0004246 0.0006392 Loading -25 0.0005906 0.0009132 condition +50 0.001027 0.001431 -50 0.000334 0.0003779 +25 0.000588 0.001040 -25 0.0002029 0.000300 [T.sub.g] +50 0.0006974 0.0008207 -50 0.0002539 0.0004113 +25 0.0004323 0.0006425 -25 0.0007153 0.0009800 [T.sub.t] +50 0.0007612 0.001078 -50 0.0005571 0.0009738 +25 0.0004857 0.0007136 -25 0.0007615 0.0012840 [T.sub.ij] +50 0.0005427 0.0009845 -50 0.0006897 -0.0001836 Parameter Settling Time (sec) Variation [DELTA][f.sub.3] [DELTA][P.sub.tie1] [DELTA][P.sub.tie2] 58.85 107.7 180.1 Loading 60.61 111.9 188.8 condition 64.89 109.6 187.2 68.09 111.3 186.1 [T.sub.g] 68.5 107.6 182.2 62.96 108.2 185.6 69.97 111.8 184.8 61.73 109.8 187.5 67.61 117.6 271.8 69.32 115.8 293.3 [T.sub.t] 68.95 119.3 275.8 71.29 113.5 294.6 57.38 187.4 130.2 179.8 178.6 169.5 [T.sub.ij] 58.7 169.7 87.28 68.67 179.4 70.4 0.0004414 -0.00512 -0.010070 Loading 0.0007680 -0.003345 -0.009928 condition 0.0012320 -0.01107 -0.019020 0.0003949 -0.004437 -0.005142 0.0006122 -0.004334 -0.01421 0.0002638 -0.003642 -0.008545 [T.sub.g] 0.0006411 -0.0082 -0.01494 0.0002750 -0.003813 -0.009763 0.0005860 -0.004286 -0.01343 0.0010450 -0.004331 -0.01453 [T.sub.t] 0.0010760 -0.004428 -0.01402 0.0006612 -0.004384 -0.01225 0.0005422 -0.004591 -0.01307 0.0007521 -0.004043 -0.01217 [T.sub.ij] 0.0005458 -0.003534 -0.01669 0.0008677 -0.005182 -0.01333 Parameter Settling Time (sec) Damping Variation [DELTA][P.sub.tie3] ratio 180.9 0.0280 Loading 184.2 0.0280 condition 185.6 0.0280 192.4 0.0280 [T.sub.g] 180.9 0.0296 187.9 0.0263 181.3 0.0283 173.5 0.0244 378.3 0.0252 394.1 0.0219 [T.sub.t] 403.7 0.0199 401.9 0.0138 142.1 0.0181 139.1 0.0086 [T.sub.ij] 142.3 0.0046 159.1 0.0054 -0.006857 0.0280 Loading -0.006478 0.0280 condition -0.013220 0.0280 -0.004828 0.0280 -0.008825 0.0296 -0.007196 0.0263 [T.sub.g] -0.006635 0.0283 -0.005753 0.0244 -0.012510 0.0252 -0.013690 0.0219 [T.sub.t] -0.012070 0.0199 -0.008827 0.0138 -0.008720 0.0181 -0.008085 0.0086 [T.sub.ij] -0.007275 0.0046 -0.012760 0.0054 TABLE V. SENSITIVITY ANALYSIS OF DOGSA TUNED FOPID CONTROLLER Settling Time (sec) Parameter % change [DELTA][f.sub.1] [DELTA][f.sub.2] Variation +25 57.73 59.20 Loading -25 51.76 54.35 condition +50 57.77 59.51 -50 50.64 54.38 +25 78.95 57.56 -25 67.80 60.13 [T.sub.g] +50 51.01 58.94 -50 69.13 62.39 +25 59.70 59.22 -25 58.95 60.31 [T.sub.t] +50 60.69 62.43 -50 60.09 60.40 +25 73.31 88.78 -25 71.94 99.59 [T.sub.ij] +50 82.14 108.80 -50 150.60 167.60 Peak Overshoot Loading +25 0.0008831 0.0014560 condition -25 0.0005296 0.0008714 +50 0.0010600 0.0017480 -50 0.0003522 0.0005824 +25 0.0010040 0.0011560 -25 0.0011870 0.0007375 [T.sub.g] +50 0.0003499 0.0006693 -50 0.0012160 0.0007384 +25 0.0011720 0.0011720 -25 0.0006865 0.0007068 [T.sub.t] +50 0.0006851 0.0006865 -50 0.0007059 0.0006709 +25 0.0017840 0.0027850 -25 0.0024700 0.0024400 [T.sub.ij] +50 0.0015960 0.0015960 -50 0.0032060 0.0032060 Settling Time (sec) Parameter [DELTA][f.sub.3] [DELTA][P.sub.tie1] [DELTA][P.sub.tie2] Variation 57.27 79.88 78.41 52.03 78.40 77.54 Loading 54.24 79.43 78.52 condition 50.18 79.16 78.14 56.42 77.08 78.45 55.30 78.87 78.23 59.34 76.20 77.43 [T.sub.g] 60.61 78.56 76.85 64.76 78.65 76.81 65.73 77.50 78.98 68.15 78.67 77.30 [T.sub.t] 60.13 78.93 74.19 81.47 185.40 110.60 96.70 188.50 99.80 84.03 178.90 102.40 [T.sub.ij] 169.30 175.40 177.20 Peak Overshoot 0.0008597 0.001958 0.007523 Loading 0.0005150 0.001178 0.004544 condition 0.0010330 0.002353 0.009085 0.0003443 0.000785 0.003026 0.0007496 -0.00010 0.004834 0.0007564 -0.00024 0.004222 0.0006867 -0.00027 0.004843 [T.sub.g] 0.0007447 -0.00061 0.005129 0.0007126 0.001722 0.006058 0.0006768 0.001570 0.006050 0.0006342 0.001509 0.006044 [T.sub.t] 0.0006813 0.001516 0.006025 0.0017220 0.001854 0.009176 0.0024100 0.002567 0.012220 0.0015300 0.001547 0.007925 [T.sub.ij] 0.003223 0.002631 0.014090 Settling Time (sec) Parameter [DELTA][P.sub.tie3] Damping Variation ratio 78.42 6.57*[10.sup.-7] 79.41 6.57*[10.sup.-7] Loading 75.79 6.57*[10.sup.-7] condition 79.04 6.57*[10.sup.-7] 78.80 1.77*[10.sup.-7] 77.64 2.12*[10.sup.-8] 77.16 0.0363 [T.sub.g] 79.26 9.07*[10.sup.-7] 76.09 4.16*[10.sup.-8] 77.53 5.96*[10.sup.-7] 75.01 6.57*[10.sup.-7] [T.sub.t] 76.73 0.1094 144.50 2.17*[10.sup.-7] 142.60 2.22*[10.sup.-7] 143.60 3.06*[10.sup.-7] [T.sub.ij] 166.20 2.73*[10.sup.-8] Peak Overshoot 0.001817 6.57*[10.sup.-7] Loading 0.001090 6.57*[10.sup.-7] condition 0.002187 6.57*[10.sup.-7] 0.000728 6.57*[10.sup.-7] -0.001831 1.77*[10.sup.-7] -0.002003 2.12*[10.sup.-8] -0.001549 0.0363 [T.sub.g] -0.002011 9.07*[10.sup.-7] 0.002002 4.16*[10.sup.-8] 0.001459 5.96*[10.sup.-7] 0.001422 6.57*[10.sup.-7] [T.sub.t] 0.001536 0.1094 0.001667 2.17*[10.sup.-7] 0.002714 2.22*[10.sup.-7] 0.001741 3.06*[10.sup.-7] [T.sub.ij] 0.003991 2.73*[10.sup.-8] TABLE VI. PHYSICAL CONSTRAINTS Parameter Values GRC 3% p.u MW Governor deadband 0.036Hz Time delay 2 sec TABLE VII. SETTLING TIME AND PEAK OVERSHOOT OF PID CONTROLLED SYSTEM INCLUDING PHYSICAL CONSTRAINTS Parameter Settling time (sec) Peak overshoot [DELTA][f.sub.1] 357.50 0.01331 [DELTA][f.sub.2] 367.10 0.01335 [DELTA][f.sub.3] 369.80 0.01332 [DELTA][P.sub.tie1] 329.40 -0.1787 [DELTA][P.sub.tie2] 381.70 -0.1782 [DELTA][P.sub.tie3] 492.90 -0.1785 TABLE VIII. SETTLING TIME AND PEAK OVERSHOOT OF FOPID CONTROLLED SYSTEM INCLUDING PHYSICAL CONSTRAINTS Parameter Settling time (sec) Peak overshoot [DELTA][f.sub.1] 57.83 0.009630 [DELTA][f.sub.2] 66.90 0.009326 [DELTA][f.sub.3] 75.19 0.009329 [DELTA][P.sub.tie1] 142.50 0.012930 [DELTA][P.sub.tie2] 153.30 -0.17770 [DELTA][P.sub.tie3] 125.30 -0.17870