A] -

Alexandria Engineering Journal (2014) 53, 537-552

faculty of engineering alexandria university

Alexandria University Alexandria Engineering Journal

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ORIGINAL ARTICLE

Differential evolution algorithm based c^ark

automatic generation control for interconnected power systems with non-linearity

Banaja Mohanty a, Sidhartha Panda b'*, P.K. Hota a

a Department of Electrical Engineering, Veer Surendra Sai University of Technology (VSSUT), Burla 768018, Odisha, India b Department of Electrical and Electronics Engineering, Veer Surendra Sai University of Technology (VSSUT), Burla 768018, Odisha, India

Received 29 July 2013; revised 12 June 2014; accepted 14 June 2014 Available online 10 July 2014

KEYWORDS

Automatic Generation Control (AGC); Boiler dynamics; Generation Rate Constraint (GRC);

Governor dead band; Proportional Integral Derivative (PID) controller; Differential Evolution (DE) algorithm

Abstract This paper presents the design and performance analysis of Differential Evolution (DE) algorithm based Proportional-Integral (PI) and Proportional-Integral-Derivative (PID) controllers for Automatic Generation Control (AGC) of an interconnected power system. Initially, a two area thermal system with governor dead-band nonlinearity is considered for the design and analysis purpose. In the proposed approach, the design problem is formulated as an optimization problem control and DE is employed to search for optimal controller parameters. Three different objective functions are used for the design purpose. The superiority of the proposed approach has been shown by comparing the results with a recently published Craziness based Particle Swarm Optimization (CPSO) technique for the same interconnected power system. It is noticed that, the dynamic performance of DE optimized PI controller is better than CPSO optimized PI controllers. Additionally, controller parameters are tuned at different loading conditions so that an adaptive gain scheduling control strategy can be employed. The study is further extended to a more realistic network of two-area six unit system with different power generating units such as thermal, hydro, wind and diesel generating units considering boiler dynamics for thermal plants, Generation Rate Constraint (GRC) and Governor Dead Band (GDB) non-linearity.

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* Corresponding author. Tel.: +91 9438251162. E-mail addresses: banaja_m@yahoo.com (B. Mohanty), panda_sidhartha@rediffmail.com (S. Panda), p_hota@rediffmail.com (P.K. Hota).

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1. Introduction

An interconnected power system is made up of several areas and for the stable operation of power systems; both constant frequency and constant tie-line power exchange should be provided. In each area, an Automatic Generation Controller (AGC) monitors the system frequency and tie-line flows, computes the net change in the generation required (generally

1110-0168 © 2014 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. http://dx.doi.Org/10.1016/j.aej.2014.06.006

referred to as area control error - ACE) and changes the set position of the generators within the area so as to keep the time average of the ACE at a low value [1]. Therefore ACE, which is defined as a linear combination of power net-interchange and frequency deviations, is generally taken as the controlled output of AGC. As the ACE is driven to zero by the AGC, both frequency and tie-line power errors will be forced to zeros [2]. AGC function can be viewed as a supervisory control function which attempts to match the generation trend within an area to the trend of the randomly changing load of the area, so as to keep the system frequency and the tie-line power flow close to scheduled value. The growth in size and complexity of electric power systems along with an increase in power demand has necessitated the use of intelligent systems that combine knowledge, techniques and methodologies from various sources for the real-time control of power systems.

Researchers all over the world are trying to understand several strategies for AGC of power systems in order to maintain the system frequency and tie line flow at their scheduled values during normal operation and also during small perturbations. A critical literature review on the AGC of power systems has been presented in [3] where various control aspects concerning AGC problem have been studied. Moreover the authors have reported various AGC schemes, AGC strategies and AGC system incorporating BES/SMES, wind turbines, FACTS devices and PV systems. There has been a considerable research work attempting to propose better AGC systems based on modern control theory [4,5], neural network [6-9], fuzzy system theory [10-12], reinforcement learning [13] and ANFIS approach [14,15]. From the literature survey, it may be concluded that there is still scope of work on the optimization of controller parameters to further improve the AGC performance. For this, various novel evolutionary optimization techniques can be proposed and tested for comparative optimization performance study. New artificial intelligence-based approaches have been proposed recently to design a controller. These approaches include particle swarm optimization [16,17], differential evolution [18,19], multi-objective evolutionary algorithm [20], NSGA-II [21,22], etc. Nanda et al. [23] have demonstrated that bacterial foraging, a powerful evolutionary computational technique, based integral controller provides better performance as compared to that with integral controller based on classical and GA techniques in a three unequal area thermal system. Ali and Abd-Elazim [24] have reported recently that Bacterial Foraging Optimization Algorithm (BFOA), based Proportional Integral (PI) controller provides better performance as compared to that with GA based PI controller in two area non-reheat thermal system. Gozde and Taplamacio-glu [25] proposed a gain scheduling PI controller for an AGC system consisting of two area thermal power system with governor dead-band nonlinearity. The authors have employed a Craziness based Particle Swarm Optimization (CRAZYPSO) with different objective functions to minimize the settling times and standard error criteria.

Differential Evolution (DE) is a branch of evolutionary algorithms developed by Stron and Price in 1995 for optimization problems [26]. It is a population-based direct search algorithm for global optimization capable of handling non-dif-ferentiable, non-linear and multi-modal objective functions, with few, easily chosen, control parameters. It has demonstrated its usefulness and robustness in a variety of applications such as, Neural network learning, Filter design and the

optimization of aerodynamics shapes. DE differs from other Evolutionary Algorithms (EA) in the mutation and recombination phases. DE uses weighted differences between solution vectors to change the population whereas in other stochastic techniques such as Genetic Algorithm (GA) and Expert Systems (ES), perturbation occurs in accordance with a random quantity. DE employs a greedy selection process with inherent elitist features. Also it has a minimum number of EA control parameters, which can be tuned effectively [18,19]. In view of the above, an attempt has been made in this paper for the optimal design of DE based PI/PID controller for LFC in two area interconnected power system considering the governor dead-band nonlinearity. The design problem of the proposed controller is formulated as an optimization problem and DE is employed to search for optimal controller parameters. By minimizing the proposed objective functions, in which the deviations in the frequency and tie line power and settling times are involved; dynamic performance of the system is improved. Simulation results are presented to show the effectiveness of the proposed controller in providing good damping characteristic to system oscillations over a wide range of loading conditions, disturbance and system parameters. Further, the superiority of the proposed design approach is illustrated by comparing the proposed approach with a recently published CPSO approach [25] for the same AGC system.

2. System under study

The Automatic Generation Control (AGC) provides the control only during normal changes in load which are small and slow. So the nonlinear equations which describe the dynamic behavior of the system can be linearized around an operating point during these small load changes and a linear model can be used for the analysis thus making the analysis simpler. The system under investigation consists of a two area interconnected power system of thermal plant as shown in Fig. 1. The system is widely used in the literature for the design and analysis of automatic load frequency control of interconnected areas [25]. In Fig. 1, Bl and B2 are the frequency bias parameters; ACEi and ACE2 are area control errors; ul and u2 are the control outputs from the controller; Rl and R2 are the governor speed regulation parameters in p.u. Hz; TGl and TG2 are the speed governor time constants in seconds; APGl and APG2 are the changes in governor valve positions (p.u.); TTl and TT2 are the turbine time constants in seconds; DPTl and DPT2 are the changes in turbine output powers; APDl and APD2 are the load demand changes; APTie is the incremental change in tie line power (p.u.); KPSl and KPS2 are the power system gains; TPSl and TPS2 are the power system time constants in seconds; Tl2 is the synchronizing coefficient and Af and A/2 are the system frequency deviations in Hz. The relevant parameters are given in Appendix. The transfer function of governor with non-linearity is given by [25]:

0.8 - 02s

G, = - p

1 + sTg

3. The proposed approach

The Proportional Integral Derivative controller (PID) is the most popular feedback controller used in the process

ACE, Ml|

Controller

Controller

0.8- —j

n 1 + sTTl

1 + ^GI

Governor with deadband

Turbine

l + s7>sl

Power System

AC\ Controller %

Controller

0.8-Mf —

1 APT1^f

1 + sTT2

1 + sTGl + v

Governor with deadband

Turbine

1 + »T»2

Power System

Figure 1 Transfer function model of two-area thermal system with governor dead band.

industries. It is a robust, easily understood controller that can provide excellent control performance despite the varied dynamic characteristics of the process plant. As the name suggests, the PID algorithm consists of three basic modes, the proportional mode, the integral and the derivative modes. A proportional controller has the effect of reducing the rise time, but never eliminates the steady-state error. An integral control has the effect of eliminating the steady-state error, but it may make the transient response worse. A derivative control has the effect of increasing the stability of the system, reducing the overshoot, and improving the transient response. Proportional Integral (PI) controllers are the most often type used today in industries. A control without Derivative (D) mode is used when: fast response of the system is not required, large disturbances and noises are present during operation of the process and there are large transport delays in the system. Derivative mode improves stability of the system and enables increase in proportional gain and decrease in integral gain which in turn increases speed of the controller response. PID controller is often used when stability and fast response are required. In view of the above, both PI and PID structured controllers are considered in the present paper. Design of PID controller requires determination of the three main parameters, Proportional gain (KP), Integral time constant (Kj) and Derivative time constant (KD). For PI controller KP and Kj are to be determined. The controllers in both the areas are considered to be identical so that KP1 = KP2 = KP, Kji = Kj2 = Kj and Kdi = Kd2 = Kd.

The error inputs to the controllers are the respective area control errors (ACE) given by:

ui = KpiACEi + Kn ACEi

U2 = KP2ACE2 + K21 ACE2

The control inputs of the power system ul and u2 with PID structure are obtained as:

ui = KpiACEi + Kn J ACEi + KDi U2 = KP2ACE2 + Kn f ACE2 + KD2

dACEi dt dACE2 dt

(6) (7)

ei(t)= ACEi = BiDfi + APm e2(t)= ACE2 = B2Äf, - DPTle

(2) (3)

The control inputs of the power system ul and u2 are the outputs of the controllers. With PI structure (KDl = KD2 = 0) the control inputs are obtained as:

In the design of a PI/PID controller, the objective function is first defined based on the desired specifications and constraints. The design of objective function to tune the controller is generally based on a performance index that considers the entire closed loop response. Typical output specifications in the time domain are peak overshooting, rise time, settling time, and steady-state error. Four kinds of performance criteria usually considered in the control design are the Integral of Time multiplied Absolute Error (ITAE), Integral of Squared Error (ISE), Integral of Time multiplied Squared Error (ITSE) and Integral of Absolute Error (IAE).

Some of the realistic control specifications for Automatic Generation Control (AGC) are [2]:

(i) The frequency error should return to zero following a load change.

(ii) The integral of frequency error should be minimum.

(iii) The control loop must be characterized by a sufficient degree of stability.

(iv) Under normal operating conditions, each area should carry its own load.

To meet the above design specifications, three different objective functions are employed in the present paper as given by Eqs. (8)-(10). The first and second objective functions con-

sider only ISE and ITSE criteria given by Eqs. (8) and (9) respectively. The third objective function aims to minimize the ITSE as given by Eq. (9). The third objective function J3 tries to minimize the settling times of A/b A/2 and APTie in addition to the minimization of all the conventional integral based error criteria.

Ji = ISE = Г" (A/1 )2 + (A/2)2 + {APTle)2 ■ dt (8)

J2 = ITSE = Г" [(A/i)2 + (A/2)2 + (APTie)2]■ t ■ dt (9)

J3 = x1 ■ ISE + x2 ■ ITSE+x ■ ITAE+x4 ■ IAE + x5 ■ TS (10)

where A/1 and A/2 are the system frequency deviations; APTie is the incremental change in tie line power; tslm is the time range of simulation; TS is the sum of the settling times of frequency and tie line power deviations; ю1-ю5 are weighting factors. Inclusion of appropriate weighting factors to the right hand individual terms helps to make each term competitive during the optimization process. Wrong choice of the weighting factors leads to incompatible numerical values of each term involved in the definition of fitness function which gives misleading result. The weights are so chosen that numerical value of all the terms in the right hand side of equation 100 lie in the same range. Repetitive trial run of the optimizing algorithms reveals that numerical value of ISE lies in the range 0.00020.02, ITSE value lies in the range 0.002-0.01, ITAE value lies in the range 0.04-1, IAE value lies in the range 0.03-0.5 and total settling times of A/1, A/2 and APTle lie in the range 1550. To make each term competitive during the optimization process the weights are chosen as: ю1 = 10,000, x2 = 1000, ю3 = 50, ю4 = 70 and rn5 = 0.1.

The problem constraints are the controller parameter bounds. Therefore, the design problem can be formulated as the following optimization problem:

Minimize J (11)

Subject to

KPmin 6 Kp 6 KPmax, Klmin 6 KI 6 KImax and KDmin

6 Kd 6 Kd max (12)

where J is the objective function (J1, J2, and J3) and KPmin, KImin; KPmax, KImax and KDmax, KDmax are the minimum and maximum values of the control parameters. As reported in the literature, the minimum and maximum values of controller parameters are chosen as —1.0 and 1.0 respectively.

4. Optimization technique

4.1. Differential evolution

Differential Evolution (DE) algorithm is a population-based stochastic optimization algorithm recently introduced [26]. Advantages of DE are: simplicity, efficiency and real coding, easy use, local searching property and speediness. DE works with two populations; old generation and new generation of the same population. The size of the population is adjusted by the parameter NP. The population consists of real valued vectors with dimension D that equals the number of design parameters/control variables. The population is randomly initialized within the initial parameter bounds. The optimization

process is conducted by means of three main operations: mutation, crossover and selection. In each generation, individuals of the current population become target vectors. For each target vector, the mutation operation produces a mutant vector, by adding the weighted difference between two randomly chosen vectors to a third vector. The crossover operation generates a new vector, called trial vector, by mixing the parameters of the mutant vector with those of the target vector. If the trial vector obtains a better fitness value than the target vector, then the trial vector replaces the target vector in the next generation. The evolutionary operators are described below [18,19].

4.1.1. Initialization

For each parameter j with lower bound Xf and upper bound Xf, initial parameter values are usually randomly selected uni-

formly in the interval \xf, XfJ.

4.1.2. Mutation

For a given parameter vector XiG, three vectors (Xr1G, Xr2 G, and Xr3G) are randomly selected such that the indices i, r1, r2 and r3 are distinct. A donor vector ViiG+1 is created by adding the weighted difference between the two vectors to the third vector as:

Vifi+1 = Xrl;G + F ■(Xr2,G — Xr3,o) (13)

where F is a constant from (0, 2).

4.1.3. Crossover

Three parents are selected for crossover and the child is a perturbation of one of them. The trial vector f!>G + 1 is developed from the elements of the target vector (XiG) and the elements of the donor vector (XiG). Elements of the donor vector enter the trial vector with probability CR as:

j,l,G+1

Uji i, g+1 l/ randji i 6 CR or j = Irand / randjii > CR or j - Irand

xj l G

With randj i ~ f(0, 1), Irand is a random integer from (1,2,...,D) where D is the solution's dimension i.e. number of control variables. Irand ensures that fi;g+1— XiG.

4.1.4. Selection

The target vector XiG is compared with the trial vector fiG +1 and the one with the better fitness value is admitted to the next generation. The selection operation in DE can be represented by the following equation:

f,,G+1 iff(f;G+1) <f(XhG) Xi G otherwise

xi G 1

where i 2 [1,NP].

4.2. Craziness based Particle Swarm Optimization (CPSO)

Particle Swarm Optimization (PSO) is a population based search algorithm for solving the optimization problems. In PSO each individual is referred to as particle and represents a candidate solution. The particles fly through the search space with an adaptable velocity that is dynamically modified according to its own flying experience and also to the flying experience of the other particles. In the original PSO algorithm, the modified velocity and position of each particle are calculated as [16]:

v,^1 = vt + cxrx{p\ — X) + c2r2(gt — X) (16)

X+1 = A + vt+1 (17)

where xt is the position of ith particle of the swarm, vt is the velocity of ith particle, n is number of particles in the swarm, t is the number of iterations, c1 and c2 are cognitive and social acceleration factors respectively, r1 and r2 are random numbers uniformly distributed in the range (0, 1), pt represents the best previous position of the ith particle, and g represents the best particle among all the particles in the swarm.

The standard PSO algorithm may be trapped in local optima especially for complex problems with many local optima and variables. The Craziness based PSO (CPSO) algorithm can prevent the swarm from being trapped in local minimum, which would cause a premature convergence and lead to fail in finding the global optimum. In the CPSO algorithm, the velocity and position update formula is given by [25]:

v;+1 = rf^v, + (1 — r3)c1r1 (p, — xt) + (1 — ^(1 — n)(g' — <) (18)

= xt + v,+1 + P(r4)f(r4) Vcr (19)

where ri-r4 are random numbers uniformly distributed in the range (0, 1), f is a sign function which assigns negative values to r3 and r4 if they are less than 0.05 and 0.5 respectively, Vcr is a craziness vector linearly decreasing from 10 to 1, P(r4) is taken as r4 if r4 is less than Pcr, a predefined probability of cra-ziness, otherwise P(r4) is taken as zero.

5. Results and discussions

5.1. Application of DE

The model of the system under study has been developed in MATLAB/SIMULINK environment and DE program has been written (in .mfile). The developed model is simulated in a separate program (by .m file using initial population/controller parameters) considering a 1 % step load change in area 1. The objective function is calculated in the .m file and used in the optimization algorithm. The process is repeated for each individual in the population. Using the objective function values, the population is modified by DE for the next generation.

Implementation of DE requires the determination of six fundamental issues: DE step size function also called scaling factor (F), crossover probability (CR), the number of population (NP), initialization, termination and evaluation function. The scaling factor is a value in the range (0, 2) that controls the amount of perturbation in the mutation process. Crossover probability (CR) constants are generally chosen from the interval (0.5, 1). If the parameter is co-related, then high value of CR work better, the reverse is true for no correlation [18,19]. DE offers several variants or strategies for optimization denoted by DE/x/y/z, where x = vector used to generate mutant vectors, y = number of difference vectors used in the mutation process and z = crossover scheme used in the crossover operation. In the present study, a population size of NP = 50, generation number G = 100, step size F = 0.8 and crossover probability of CR = 0.8 have been used. The strategy employed is: DE/best/1/exp. Optimization is terminated by the prespecified number of generations for DE. One more important factor that affects the optimal solution more or less is the range for unknowns. For the very first execution of the

program, a wider solution space can be given and after getting the solution one can shorten the solution space nearer to the values obtained in the previous iteration. Here the upper and lower bounds of the gains are chosen as (1, —1). The flow chart of the DE algorithm employed in the present study is given in Fig. 2. Simulations were conducted on an Intel, core 2 Duo CPU of 2.4 GHz and 2 GB MB RAM computer in the MAT-LAB 7.10.0.499 (R2010a) environment. The optimization was repeated 20 times and the best final solution among the 20 runs is chosen as proposed controller parameters. The best final solutions obtained in the 20 runs are shown in Table 1.

5.2. Simulation results

Table 2 shows the ISE value and settling times (2% of final value) when the controller parameters are optimized using ISE error criteria. To show the effectiveness of the proposed DE method results are compared with a recently published CPSO technique for the same interconnected power system and for the same ISE objective function [25]. It can be seen from Table 2 that with the same PI controller structure, the value of ISE obtained using the proposed DE technique is less than that with CPSO technique and minimum ISE is obtained with the DE optimized PID controller. The objective function value is reduced by 5.32% and 88.59% with the proposed DE

Figure 2 Flow chart of the proposed DE optimization approach.

Table 1 Tuned controller parameters for different objective functions.

Objective function/controller parameters Ji (ISE) J2 (ITSE) J3 (proposed)

PI controller Proportional gain (KP) -0.300i -0.3586 -0.5382

Integral gain (Kj) 0.35i8 0.3i59 0.2205

PID controller Proportional gain (KP) 0.904i 0.7i46 0.2383

Integral gain (Kj) 0.9322 0.99i8 0.97i8

Derivative gain (KD) 0.958i 0.7595 0.4922

Table 2 ISE value and settling times with ISE objective function.

Parameters DE optimized PI controller DE optimized PID controller CPSO optimized PI controller [25]

ISE 2i.2i58 x i0-4 2.5559 x i0-4 22.4086 x i0-4

Ts (s)

Dfi 23.66 7.37 23.84

4/2 i6.82 4.86 i7.57

DPTle 23.66 6.73 23.83

Table 3 ITSE value and settling times with ITSE objective function.

Parameters DE optimized PI controller DE optimized PID controller CPSO optimized PI controller [25]

ITSE 35.6968 x i0-4 2.4472 x i0-4 36.2505 x i0-4

Ts (s)

Afi i8.83 7.77 26.27

Af2 i3.42 6.23 i8.0i

APTle i8.83 7.66 26.27

optimized PI and PID controllers respectively. Also, the settling time for A/1 is improved by 0.75% and 69.08% for the proposed PI and PID controllers respectively compared to the results given in [25]. The improvements in settling time for A/2 are 4.24% and 72.33% respectively with the proposed PI and PID controllers. For the tie line power deviations APtie the improvements with the proposed PI and PID controllers are 0.71% and 71.75% respectively compared to the CPSO optimized PI controller for the same system.

The ITSE value and settling times when the controller parameters are optimized using ITSE error criteria are shown in Table 3 along with the CPSO results for the same objective function. It is evident from Table 3 that the better results are obtained with DE compared to CPSO. The improvements are 1.52% and 93.24% in the objective function values with

DE optimized PI and PID controllers respectively. For the settling times the improvements are: 28.32% and 70.42% for A/1; 25.48% and 64.9%; 28.32% and 70.84% respectively with DE optimized PI and PID controllers.

To further improving the settling times the proposed objective function J3 is used and the results are summarized in Table 4. All the four error values and the settling times are compared with the best claimed objective function optimized using CPSO [25]. The respective improvements are also given in Table 4 from which it is clear that the proposed DE optimized PI controller outperforms the CPSO optimized PI controller and best performance is obtained with DE optimized PID controller.

The above analysis shows that the system performance is greatly improved by applying the proposed controllers. Time

Table 4 Error criteria and settling times with the proposed objective function J3.

Parameters DE optimized PI controller DE optimized PID controller CPSO optimized PI controller [25]

Value Improvement (%) Value Improvement (%)

ISE 37.4623 x i0-4 i4.47 4.0257 x i0-4 90.8i 43.80i6 x i0-4

ITSE 66.445 x i0-4 20.i2 3.7025 x i0-4 95.55 83.i849 x i0-4

ITAE 48.2i45 x i0-2 i7.29 7.29 x i0-2 87.49 58.2969 x i0-2

IAE i9.4063 x i0-2 ii.28 4.7644 x i0-2 78.22 2i.875 x i0-2

Ts (s)

Afi i0.67 4.65 6.87 38.6i ii.i9

Af2 9.64 4.98 4.23 62.33 ii.23

APTle i0.36 20.39 5.9i 5i.i9 i2.ii

Time (sec)

Figure 3 Change in frequency of area-1 for 1% step load increase in area-1 with ISE objective function.

Time (sec)

Figure 4 Change in frequency of area-2 for 1% step load increase in area-1 with ISE objective function.

•*v, / W

Time (sec)

Figure 5 Change in tie line power for 1% step load increase in area-1 with ISE objective function.

domain simulations are performed for step load change at different locations. A step increase in demand of 1% is applied at t = 0 s in area-1. The system dynamic responses with three objective functions (J1: ISE; J2: ITSE and J3: Proposed) are shown in Figs. 3-11. In all the figures the response with DE optimized PI and PID controllers is shown with dashed lines (legend 'DE PI') and solid lines (legend 'DE PID') respectively. For comparison the simulation results with CPSO optimized PI controller are also shown in Figs. 3-11 with dotted lines

(legend 'CPSO PI'). Critical analysis of the dynamic responses clearly reveals that dynamic performance of DE PI controller is better than CPSO PI controller and the best performance is obtained with DE PID controller.

To show the robustness of the control strategy optimized by DE algorithm, controller parameters are tuned at +25%, + 50%, —25% and —50% changes in the load demand. As the power exchange between control areas is minimized with the decrease in settling times, the proposed objective function

Time (sec)

Figure 6 Change in frequency of area-1 for 1% step load increase in area-1 with ITSE objective function.

Time (sec)

Figure 7 Change in frequency of area-2 for 1% step load increase in area-1 with ITSE objective function.

0 5 10 15 20 25

Time (sec)

Figure 7 Change in frequency of area-2 for 1% step load increase in area-1 with ITSE objective function.

Time (sec)

Figure 8 Change in tie line power for 1% step load increase in area-1 with ITSE objective function.

J3 is used due to its better settling time. The tuned parameters are shown in Table 5. The settling times and its percentage improvements compared to the CPSO technique [25] are given in Table 6. It is clear from Table 6 that settling time is less with DE PI compared to CPSO PI at all the loading conditions and minimum settling times are obtained with DE PID. Figs. 12-17 show the dynamic response of the system under the above load demand variations. It is clear from Figs. 12-17 that the

designed controllers are robust and perform satisfactorily when load demand changes.

5.3. Extension to multi-source system

To get an accurate insight into the AGC topic, it is essential to include the important inherent requirement and the basic physical constraints in the system model. The important constraints

Time (sec)

Figure 9 Change in frequency of area-1 for 1% step load increase in area-1 with the proposed objective function.

0.005 0

-0.005 -0.01 -0.015 -0.02 -0.025 -0.03 -0.035

■^V......

Time (sec)

Figure 10 Change in frequency of area-2 for 1% step load increase in area-1 with the proposed objective function.

Time (sec)

Figure 11 Change in tie line power for 1% step load increase in area-1 with the proposed objective function.

Table 5 Tuned controller parameters for different loadings.

Objective function/controller parameters + 25% + 50% -25% -50%

PI controller Proportional gain (KP) -0.502 -0.5552 -0.5i89 -0.5235

Integral gain (Kj) 0.2479 0.2i5i 0.2255 0.25i6

PID controller Proportional gain (KP) 0.i229 0.5602 0.i823 0.3599

Integral gain (Kj) 0.796i 0.9628 0.8998 0.9349

Derivative gain (KD) 0.4699 0.6882 0.5098 0.6364

Table 6 Settling times at different loadings.

Parameters DE optimized PI controller DE optimized PID controller CPSO optimized PI controller [25]

Value Improvement (%) Value Improvement (%)

Ts (s) at + 25% Dfi 12.14 18.68 6.24 58.2 14.93

A/2 9.34 29.82 4.6 65.44 13.31

DPTie 10.66 28.55 5.64 62.19 14.92

Ts (s) at + 50% A/1 11.05 7.68 6.63 44.61 11.97

A/2 11.36 2.9 5.09 56.49 11.7

APTie 11.86 0.16 7.06 40.57 11.88

Ts (s) at -25% A/i 10.23 11.04 5.89 48.78 11.5

A/2 9.43 28.18 4.19 68.08 13.13

APTie 9.97 15 5.38 54.13 11.73

Ts (s) at -50% A/i 8.97 13.15 5.61 45.74 10.34

A/2 8.58 6.02 3.96 56.62 9.13

APTie 9.87 2.75 4.73 53.34 10.15

Time (sec)

Figure 12 Change in frequency of area-1 for increase in load demands ( + 25% to +50%) in area-1.

0 -2 -4 -6 -8 -10 -12 -14

+ 25 % DE PI + 25 % DE PID + 50 % DE PI + 50 % DE PID

Time (sec)

Figure 13 Change in frequency of area-2 for increase in load demands ( + 25% to +50%) in area-1.

which affect the power system performance are boiler dynamics for thermal plants, Generation Rate Constraint (GRC), and Governor Dead Band (GDB) nonlinearity [27]. In view of the above, the study is further extended to a more realistic network of two-area six unit system with different power gen-

erating units considering the above physical constraints as shown in Fig. 18. In the first area thermal, hydro and wind generating units are considered and in the second area thermal, hydro and diesel generating units are assumed. The transfer function model of wind and diesel generating units is adopted

Time (sec)

+ 25 % DE PI + 25 % DE PID

- + 50 % DE PI

- + 50 % DE PID

Figure 14 Change in tie line power for increase in load demands ( + 25% to +50%) in area-1.

-0.005

-0.015

-0.025

- 25 % DE PI

- 25 % DE PID

- 50 % DE PI

- 50 % DE PID

Time (sec)

Figure 15 Change in frequency of area-1 for decrease in load demands (—25% to —50%) in area-1.

■ 25% DE PI 25% DE PID

■ 50% DE PI 50% DE PID

Time (sec)

Figure 16 Change in frequency of area-2 for decrease in load demands (—25% to —50%) in area-1.

from [28]. The transfer function model of wind turbine system with pitch control is shown in Fig. 18. The model consists of a hydraulic pitch actuator, data fit pitch response and blade characteristics. The diesel unit is represented by a transfer function as shown in Fig. 18. Each unit has its regulation parameter and participation factor which decide the contribu-

tion to the nominal loading, summation of participation factor of each control being equal to l. Participation factors for thermal and hydro are assumed as 0.575 and 0.3 respectively. For wind and diesel same participation factors of 0.125 are assumed. The nominal parameters of the system under study are given in Appendix B.

0.005 0

-0.005 -0.01 -0.015 -0.02 -0.025

- 25 % DE PI

- 25 % DE PID

- 50 % DE PI

- 50 % DE PID

Time (sec)

Figure 17 Change in tie line power for decrease in load demands (—25% to —50%) in area-1.

1 + ítgi

AREA-1

A controller I

Boiler dynamics

l+sKnTn 1 — GRC

1+57V, 1+iTn

gÏÏGDB^

1+sTgl

1 + sri

l + sr2

l + .5srv

Hydro Area

Kpsi 1+Tps

T - * y fc S-2(1 + S7>i) 1 K,

1 + i 1 +STP2 1 + Í

AREA-2

-gH GDB

1 + STgi

Controller

t^-_ Boi

Boiler dynamics

1 + sKnTn

l+sTr..

1+sTr,

l+srg2

I + ÍTi

1 + sTi

l + .5srw

K,hr,ri 0 ~ '0

Hydro Area

1 + ÏH

Diesel power plant

Figure 18 Multi-area multi source power system with nonlinearities.

To include the effect of the boiler dynamics for thermal units, the detailed configuration shown in Fig. 19 [29] is considered. This model considers the long-term dynamics of fuel and steam flow on boiler drum pressure as well as combustion

controls. Governor dead band is defined as the total amount of a continued speed change within which there is no change in valve position. Steam turbine dead band is due to the backlash in the linkage connecting the servo piston to the camshaft.

Figure 19 Boiler dynamics configuration.

Much of this appears to occur in the rack and pinion used to rotate the camshaft that operates the control valves. Due to the governor dead band, an increase/decrease in speed can occur before the position of the valve changes. The speed governor dead band has a great effect on the dynamic performance of electric energy system. The backlash non-linearity tends to

produce a continuous sinusoidal oscillation with a natural period of about 2 s. For this analysis, in this study backlash non-linearity of 0.05% for the thermal system and 0.02% for the hydro system is considered. In a power system, power generation can change only at a specified maximum rate known as Generation Rate Constraint (GRC). In the present study, a GRC of 3% per min is considered for thermal units. The GRC's for hydro unit are 270% per minute for raising generation and 360% per minute for lowering generation are considered [15]. As the areas are assumed unequal, different PID controllers are considered for each generating unit. To investigate the effect of wind and diesel generation on the system performance, two cases are considered i.e. Case-A: System with thermal and hydro generating units (without wind and diesel units) and Case-B: System with thermal, hydro, wind and diesel generating units. When wind and diesel units are not considered in the system model, the participation factors of thermal unit are increased to 0.695(0.57 + 0.125). The same procedure as described in Section 5.1 is followed to optimize the PID controller parameters of each generating unit in each case. In all the cases, the proposed objective function J3 given by Eq. (10) is used due to its better performance. The final controller parameters are given in Table 7. A step increase in demand of 1% is applied at t = 0 s in area-1 and the system responses are shown in Figs. 20-22. The settling times and various error criteria for the above case are provided in Table 8. It is clear from Figs. 20-22 and Table 8 that, when the physical constraints are included in the system model, the system performance degrades for Case A i.e. the system with thermal

Table 7 Tuned controller parameters for of multi-source power system with physical constraints.

Parameters/ Case A: System with thermal and hydro units Case B: System with thermal, hydro, wind and

generating units (no wind and diesel) diesel units

KP K KD KP KI Kd

Area-1

Thermal 0.7050 0.4381 0.9681 0.5303 0.9272 0.2099

Hydro 0.7238 0.6568 0.5148 0.9820 0.4164 0.2687

Wind - - - 0.7307 0.5341 0.3751

Area-2

Thermal 0.3110 0.1947 0.1510 0.3824 0.2172 0.8451

Hydro 0.9014 0.0707 0.9584 0.0717 0.3625 0.1096

Diesel - - - 0.9136 0.0405 0.2279

0.02 0

X -0.02 <i

-0.04 -0.06 -0.08

0 10 20 30 40 50 60 70 80 90 100

Time (sec)

: A ------Case A: without wind and diesel units " -Case B: with wind and diesel units

' \ v -----

r, i i ! ri \-1 •'< i -VV

^ -0.02 <1

-0.04 -0.06 -0.08

0 10 20 30 40 50 60 70 80 90 100

Time (sec)

Figure 21 Change in frequency of area-2 for 1% change in area-1 for multisource system with physical constraints.

Case A: without wind and disel units Case B: with wind and disel units

Case A: without wind and diesel units Case B: with wind and diesel units

40 50 60

Time Csec)

Figure 22 Change in tie-line power for 1% change in area-1 for multisource system with physical constraints.

Table 8 Error criterion and settling times for multi-source power system with physical constraints.

Performance/case Error criterion Settling times TS (s)

ISE (x10~3) ITSE (x10~3) IAE (x10_1) ITAE A/1 A/2 A PTe

Case A 162.574 1209.718 17.596 20.082 40.57 40.51 60.77

Case B 3.0141 5.554 1.528 1.321 19.68 21.93 25.89

0.04 0.02 0

-0.02 -0.04 -0.06 -0.08

40 50 60 Time (sec)

£ -0.02 ч-f <1

-0.04 -0.06 -0.08

0 10 20 30 40 50 60 70 80 90 100

Time (sec)

Figure 24 Change in frequency of area-2 for 1% change in area-2 for multisource system with physical constraints.

Case A: without wind and diesel units Case B: with wind and diesel units

Case A: without wind and diesel units Case B: witht wind and diesel units

! Ï I и

"¡¡I

f1 '*, « i1 ' Il i

HI'! fi

0 10 20 30 40 50 60 70 80 90 100

Time (sec)

Figure 25 Change in tie-line power for 1% change in area-2 for multisource system with physical constraints.

and hydro units. It is also clear from Figs. 20-22 and Table 8 that the system performance improves with the inclusion of wind and diesel units. The improvements in the system response in Case B are due to the absence of physical constraints for wind and diesel units and they can quickly pick up the additional load demand thus stabilizing the system more quickly. For completeness, a 1% step increase in load demand is applied at t = 0 s in area-2 and the system responses are shown in Figs. 23-25. It is clear from Figs. 23-25 that the designed controllers are robust and perform satisfactorily when the location of disturbance changes. It is clear from Figs. 20-25 and Table 8 that the proposed approach can be applied to interconnected power systems with different sources of generation and different PID controllers for each generating unit.

6. Conclusion

This study presents the design and performance evaluation of Differential Evolution (DE) optimized Proportional-Integral (PI) and Proportional-Integral-Derivative (PID) controllers for Automatic Generation Control (AGC) of an interconnected power system with governor dead-band nonlinearity. For the optimization of controller parameters, selection of

suitable objective function is very important. Conventional objective functions used in the literature are Integral of Time multiplied by Squared Error (ITSE), Integral of Squared Error (ISE), Integral of Time multiplied by Absolute Error (ITAE) and Integral of Absolute Error (IAE). Three different objective functions are used for the design purpose in the present paper. The results obtained from the simulations show that the proposed control strategy optimized with a new objective function achieves better dynamic performances than the standard objective functions. The superiority of the proposed approach has been shown by comparing the results with a recently published Craziness based Particle Swarm Optimization (CPSO) technique for the same interconnected power system. It is observed that the proposed DE optimized PI controller outperforms the CPSO optimized PI controller and the best performance is obtained with DE optimized PID controller. Finally, the study is extended to a more realistic network of two-area six unit system with different power generating units considering physical constraints such as boiler dynamics for thermal plants, Generation Rate Constraint (GRC) and Governor Dead Band (GDB) nonlinearity. It is observed that the proposed approach can be applied to interconnected power systems with diverse sources of generation with different PID controllers for each generating unit.

Appendix A

B1, B2 = 0.425 p.u. MW/Hz; R1 = R2 = 2.4 Hz/p.u.; TG1 = -

TG2 = 0.2 s; TT1 = TT2 = 0.3 s; KPS1 = KPS2 = 120 Hz/p.u.

MW; TPS1 = TPS2 = 20 s; T12 = 0.0707 p.u.; a12 = -1.

Appendix B

B1 = B2 = 0.425 p.u. MW/Hz; R1 = R2 = 2.4 Hz/p.u.;

TG1 = 0.2 s; TT1 = 0.3 s; TG2 = 48.7 s; T1 = 0.513 s;

T2 = 10 s; Tw =1 s; Tr = 10 s; Kr = 0.333; K1 = 0.85,

K2 = 0.095, K3 = 0.92, cb = 200, Td = 0, T/ = 10,

kib = 0.03, Tib = 26, Trb = 69; K2 = 1.25; TP2 = 0.041 s;

K3 = 1.4; TP1 = 0.6 s; T1 = 0.025 s; KPC = 0.8;

Kdiesel = 16.5; T12 = 0.0866 p.u.

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