Scholarly article on topic 'Application of backtracking search algorithm in load frequency control of multi-area interconnected power system'

Application of backtracking search algorithm in load frequency control of multi-area interconnected power system Academic research paper on "Electrical engineering, electronic engineering, information engineering"

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{"Load frequency control" / "Backtracking search algorithm" / "Boiler dynamics" / "Governor dead band" / "Sensitivity analysis" / "Transient analysis"}

Abstract of research paper on Electrical engineering, electronic engineering, information engineering, author of scientific article — Dipayan Guha, Provas Kumar Roy, Subrata Banerjee

Abstract This paper introduces a new powerful evolutionary algorithm called backtracking search algorithm (BSA) for solving load frequency control (LFC) problem in power system. Initially, two-area non-reheat thermal power plant is considered and gains of PI/PID controllers are optimized using BSA. This paper compares BSA’s effectiveness in solving LFC problem with the performances of other optimization techniques reported in the literature. Nonlinearities of power system such as reheater, governor dead band, boiler dynamics and generation rate constraint are included in the system modeling to identify the system stability and its performance is compared with craziness based PSO technique. Additionally, two more test systems namely three-area and four-area hydro-thermal plant with nonlinearity are considered to demonstrate the efficiency of proposed algorithm. The comparative analysis of the performances indicates that the proposed controller gives better results than other techniques available in the literature. Sensitivity analysis showed robustness of proposed controller under loading and parameter uncertainty.

Academic research paper on topic "Application of backtracking search algorithm in load frequency control of multi-area interconnected power system"

Ain Shams Engineering Journal (2016) xxx, xxx-xxx

Ain Shams University Ain Shams Engineering Journal

www.elsevier.com/locate/asej www.sciencedirect.com

ELECTRICAL ENGINEERING

Application of backtracking search algorithm in load frequency control of multi-area interconnected power system

Dipayan Guhaa,% Provas Kumar Royb, Subrata Banerjeec

a Department of Electrical Engineering, Dr. B.C.Roy Engineering College, Durgapur, West Bengal, India b Department of Electrical Engineering, Jalpaiguri Government Engineering College, Jalpaiguri, West Bengal, India c Department of Electrical Engineering, NIT Durgapur, Durgapur, West Bengal, India

Received 19 September 2015; revised 27 December 2015; accepted 16 January 2016

KEYWORDS

Load frequency control; Backtracking search algorithm;

Boiler dynamics; Governor dead band; Sensitivity analysis; Transient analysis

Abstract This paper introduces a new powerful evolutionary algorithm called backtracking search algorithm (BSA) for solving load frequency control (LFC) problem in power system. Initially, two-area non-reheat thermal power plant is considered and gains of PI/PID controllers are optimized using BSA. This paper compares BSA's effectiveness in solving LFC problem with the performances of other optimization techniques reported in the literature. Nonlinearities of power system such as reheater, governor dead band, boiler dynamics and generation rate constraint are included in the system modeling to identify the system stability and its performance is compared with crazi-ness based PSO technique. Additionally, two more test systems namely three-area and four-area hydro-thermal plant with nonlinearity are considered to demonstrate the efficiency of proposed algorithm. The comparative analysis of the performances indicates that the proposed controller gives better results than other techniques available in the literature. Sensitivity analysis showed robustness of proposed controller under loading and parameter uncertainty.

© 2016 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

* Corresponding author at: Dr. B.C.Roy Engineering College, Fuljhore, Jemua Road, Durgapur 713206, West Bengal, India. Tel.: + 91 9674336951.

E-mail address: guha.dipayan@yahoo.com (D. Guha). Peer review under responsibility of Ain Shams University.

The successful operation of an interconnected power system network requires balance between total power generation and total load demands plus losses incurred in the system. The high complexity and nonlinearity together with time-varying nature of power system network cause variations of an operating point w.r.t time which may result in variations of frequency and net interchanging power between neighboring control areas. Large variations of frequency and tie-line power may yield instability to the system [1]. Classical control

http://dx.doi.org/10.1016/j.asej.2016.01.004

2090-4479 © 2016 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Figure 1 Block diagram of two-area interconnected non-reheat thermal-thermal power system (Test system-1).

scheme via flywheel governor of synchronous machine is no longer able to compensate these variations and hence, a supplementary control scheme using load frequency controller (LFC) is incorporated into the system. LFC not only maintains system stability with variation of load, but also satisfies other system constraints such as security and reliability.

The LFC problem of an interconnected power system network is enlarged by the several researchers from time to time over last few decades in order to present better dynamic responses. Several control mechanisms have been proposed in the literature to solve LFC problem, such as classical controller [2-10], sliding mode controller (SMC) [11], robust controller [12], non-integer controller [13-15], controller [16], and intelligent controller [17-20]. In [11], authors presented a new discrete time SMC using full state feedback for an interconnected four area power system network. More recently authors of [13-15] proposed non-integer controller to solve LFC problem for an interconnected multi-area power system network and showed better system dynamics than the results available in the literature. Several intelligent controllers based on fuzzy logic [17], neuro-fuzzy [18], advance fuzzy [19], artificial neural network [20] were presented in the area of LFC and demonstrated their applicability to improve dynamic performances of the same. Moreover, these advanced techniques are complex and require sound knowledge about their internal structure, thus reducing their applicability in practical field. On the other hand, classical PID-controller and its variant are most popularly used in practice because of its simple structure, robustness, reliability, and easily understood, and gives better system performances despite the wide variations of system parameters [7]. Operating points of the power system network are continuously changing due to sudden load perturbation

(SLP); hence, selections of optimal values of the designed controller is a major issue in the study of LFC problem. Different population based optimization techniques such as differential evolution (DE) [7,21], genetic algorithm (GA) [22], particle swarm optimization (PSO) [23], artificial bee colony algorithm (ABCA) [24], bacteria foraging optimization algorithm (BFOA) [3], hybrid bacteria foraging optimization algorithm - particle swarm optimization (hBFOA-PSO) [2], craziness based particle swarm optimization (CRPSO) [4], gravitational search algorithm (GSA) [8], and firefly algorithm (FA) [9,13] were proposed in this area to select optimum gains of designed controller. However, recent research identifies some drawbacks of these methods. Nearly all the said algorithms depend on the proper initialization of input parameters and slow convergence rate toward the global solutions. Moreover, they sometimes produce local solution instead of the global ones. Among the well popular optimization techniques, DE has a very simple algorithmic structure that can be effortlessly applied to LFC problem. The DE adopts mutation, crossover and selection strategies like GA. Though the mutation process involved in DE is stronger than that of GA, the search ability of DE is highly affected by the choice of mutation factor and crossover rate. PSO is another popularly used optimization algorithm in the optimization field, but it is highly susceptible to the initial values of weighting factor of the cognitive and social components and weighting strategy of the velocity vector. The success of ABCA is highly dependent on the limit value, number of employed bees. In the line of ''no free lunch" algorithm, there is no meta-heuristic optimization scheme which is well suited for all optimization problems, i.e., a particular evolutionary algorithm gives better results on a set of problems, but same algorithm may also show poor performance on a different

Table 1 Nominal values of system parameters.

Parameters Values Parameters Values Parameters Values

Test system-1: Two-area non-reheat thermal-thermal power system (without GDB)

f 60 Hz Ttl = Tt2 0.3 s Ri = R2 2.4 Hz/p.u MW

Prl = Pr2 2000 MW Tsgi = Tsg2 0.03 s T12 0.545 p.u

II 120 Hz/pu MW Tpsl Tps2 20 s Bi = B2 0.425 pu MW/Hz

Test system-1: Two-area non-reheat thermal-thermal power system (with GDB)

f 60 Hz Ttl = Tt2 0.3 s Ri = R2 2.4 Hz/p.u MW

Prl = Pr2 2000 MW T = T 1 sgi 1 sg2 0.2 s T12 0.0707 p.u

Kpsi = Kps2 l20 Hz/pu MW Tpsl = Tps2 20 s Bi = B2 0.425 pu MW/Hz

Test system-2: Three-area thermal-thermal-hydro power system

f 60 Ttl = Tt2 0.3 s Bi = B2 = B3 0.425 pu MW/Hz

Prl = Pr2 = Pr3 2000 MW Tsgi = Tsg2 0.08 s Ri = R2 = R3 2.4 Hz/p.u MW

T12 = T23 = T3l 0.086 pu T ± w 1 s Tri = Tr2 i0 s

Krl = Kr2 0.5 Tpi = Tp2 = Tp3 20 s Kpi = 2pK = pK 3 i20 Hz/pu MW

Nominal values of coal fired, well tuned boiler

Td 40 Tf 25 KIB 0.020

Tib 90 TRB 69 Ki 0.85

K2 0.095 K3 0.92 Cb 200

Test system-3: four-area hydro-thermal power plant [32]

Thermal unit

f 60 Hz R 2.4 Hz/p.u Tti 0.3 s

Bi 0.425 p.u. MW/Hz Tgl 0.08 s tPI 20 s

Kpi 120 Hz/p.u. MW T12 = T13 = T23 0.707 p.u i i, 2, 3

Hydro unit

kP 80 Hz/p.u. MW Tw 1 s B4 0.425 p.u. MW/Hz

tp 13 s R 2.4 Hz/p.u Ti4 0.0707 p.u

Figure 2 Block diagram of drum type coal fired well tuned boiler.

set of problems. Thus, there remains a need for efficient and effective optimization technique to solve nonlinear optimization problems. Apart from the aforesaid optimization techniques, other evolutionary algorithms (EA) namely biogeography based optimization (BBO) [25], krill herd algorithm (KHA) [26], chemical reaction optimization (CRO)

Figure 3 General flowchart of BSA.

[27], teaching learning based optimization (TLBO) [28], and differential search algorithm (DSA) [29] were successfully designed and implemented in the other fields of power system.

Having knowledge of all the aforesaid, an attempt has been made in this paper to design and implement a relatively new optimization technique called backtracking search algorithm

Table 2 Optimum gains of controller parameters for test system-1.

Controller gains Evolutionary Algorithms

Proposed Controllers Ref. [2] Ref. [8]

BSA: PI BSA: PID hBFOA-PSO PSO GA BFOA Conventional GSA with PI

Kii 1.1996 1.9657 0.4741 0.4756 0.2662 0.4741 0.4741 0.6179

Ki2 0.1851 0.0164 0.4741 0.4756 0.2662 0.4741 0.4741 0.6179

Kpi 0.3873 1.0663 -0.3317 -0.3597 -0.2346 -0.3317 -0.3317 -0.1880

Kp2 1.9909 1.5606 -0.3317 -0.3597 -0.2346 -0.3317 -0.3317 -0.1880

Kd1 - 0.3462 - - - - - -

Kd2 - 0.4259 - - - - - -

Objective function 0.3510 0.1288 1.1865 1.2142 2.7475 1.8379 3.5795 0.6659

Damping ratio 0.0971 0.0269 0.1795 0.1887 0.2359 0.174 0.0206 0.1144

Bold values shows the best results.

Table 3 Comparative performance between different algorithms in terms of settling time of transient responses for test system-1.

Parameters Proposed controllers Ref. [2] Ref. [7] Ref. [8]

BSA: PI BSA: PID hBFOA-PSO PSO GA BFOA Conventional DE GSA

Area-1 5.11 1.39 7.39 7.37 10.59 5.52 45 8.96 12

Area-2 6.31 3.30 7.65 7.82 11.39 7.09 45.01 8.16 11.90

Tie-line power 4.58 3.48 5.73 5 9.37 6.35 28.27 5.75 8.90

Bold signifies best results.

(BSA) to solve real-time LFC problem of an interconnected power system. Unlike other EA's, BSA is the most effective algorithm in terms of CPU running times and has a single control parameter. The design problem can be treated as an optimization problem and BSA is used to tune the settings of classical controllers. The dynamic responses are studied with step load perturbation (SLP) in area-1 considering overshoot, settling time of frequency and tie-line power deviations as performance indices. By minimizing time multiplied integral absolute error fitness function, stability of the interconnected system is improved. To present the test system be more realistic and to verify the algorithm's effectiveness in nonlinear environment, power system nonlinearities such as reheater, governor dead band, boiler dynamics and generation rate constraint are considered during system modeling. Dynamic performances of BSA optimized PI/PID controller are compared with other algorithms available in the literature to prove its superiority. Finally, a realistic random load pattern is considered to demonstrate the robustness of the proposed method.

The rest of the paper is organized as follows: Section 2 describes mathematical model of test system. Nonlinearities of power system networks are explained in Section 3. The proposed BSA optimization algorithm is briefly explained in Section 4. The controllers' structure with objective function is illustrated in Section 5, followed by the implementation of BSA in LFC problem in Section 6. Transient analysis of the test systems with simulation results are presented in Section 7 and finally, Section 8 gives conclusion.

2. Mathematical modeling of test system

Before designing a controller structure, it must be ensured that mathematical model of test system must be valid for time

domain simulation. The power system under investigation consists of an interconnected two-area non-reheat thermal-thermal power system network, which is widely used in the literature for design and analysis of LFC problem [2]. Each area has a rating of 2000 MW with nominal loading of 1000 MW. The linearized model of governor, turbine used for simulation and LFC study of the power system are shown in Fig. 1. The different parameters used in Fig. 1 are as follows: Tg is the time constant of speed governor, Tt is the time constant of steam turbine, Kps is the gain of power system unit, Tps is the time constant of power system unit, R1 and R2 are the speed regulation parameter of speed governor in area-1 and area-2, respectively. T12 is the synchronizing time constant of tie-line, B1 and B2 are the frequency bias parameter of the respective areas, APD1 and APD2 are the load disturbances in area-1 and area-2, respectively, Af and Af2 are deviation of frequency in area-1 and area-2, respectively. Nominal values of system parameters are taken from [2] and illustrated in Table 1.

3. Nonlinearities of power system networks

A physical system is a collection of several physical components connected together to serve a specific task. In practice, no physical system can be identified by all its components. But to make the system be realistic, we need to consider all the possible nonlinearities of the system during the modeling of same. To get an accurate imminent of LFC problem, it is essential to include the important inherent requirements and include them into modeling. The basic sources of nonlinearities in power system network such as governor dead band (GDB), boiler dynamics (BD) and generation rate constraints (GRC) are considered in this paper to study the effect of the same

time in sec

BSA hBFOA-PSO PSO GA_

-0.1 -0.12 -0.14

10 15 20

time in sec

■ji -0.01

| -0.02

o a. d)

=F -0.03

hBFOA-PSO

time in sec

Figure 4 Comparative analysis between different EA's for test system-1 equipped with PI-controller (a) changes of frequency in area 1, (b) changes of frequency in area 2, (c) changes of tie-line power.

on the power system dynamics. This section describes the basic sources of nonlinearities in power system.

Gdb(s) — 0.8 - — s p

3.1. Governor dead band (GDB)

The speed governor is a device that varies the governor controlled valves to adjust energy input levels to maintain a uniform speed of a rotating machine. Governor dead band (GBD) [6] is defined as the total magnitude of sustained speed changes within which there are no resulting changes in valve positions. The hysteresis type of nonlinearity is used to define GDB and its maximum value is 0.06% [6]. Describing function approach is used to find the linear model of GDB and is defined as follows:

3.2. Boiler dynamics (BD)

The boiler is meant for producing steam under pressure. Fig. 2 shows the block diagram of coal fired well-tuned drum type boiler [6]. This includes the long-term dynamics of fuel and steam flow on boiler drum pressure. Nominal values of all parameters are taken from [6] and given in Table 1. The T.F models of pressure control unit and fuel system are defined in (2) and (3), respectively.

KIB{1 + sT1B)(1 + sTrB )

Gn(s) —;

s( 1 + 0.1sTrb)

Table 4 Optimum controller settings, minimum damping ratio, fitness function value, settling time for test system-1 with GDB.

Parameter BSA: PI BSA: PI BSA: PID BSA: PID hBFOA-PSO:PI hBFOA-PSO:PI CRPSO

(ISE) (ITAE) (ISE) (ITAE) (ISE) [2] (ITAE) [2] (ISE) [4]

Optimum values of K,i 0.4068 0.4774 i.9702 i.9979 0.3673 0.3260 0.249i

controller gains Ki2 0.0069 0.2072 i.0279 0.3420 0.3673 0.3260 0.249i

Kpi 0.4233 0.i423 i.6838 i.5838 0.2i86 0.4253 0.25i4

Kp2 0.6985 0.5534 0.540i 0.59i2 0.2i86 0.4253 0.25i4

Kdi - - i.62i2 i.42i2 - - -

Kd2 - - 0.2904 0.3904 - - -

Objective function 2.64 x i0~4 0.0586 2.39 x i0~5 0.0i22 20.5i x i0~4 0.3948 22.4i x i0~4

Minimum damping ratio 0.0i7i 0.0364 0.0992 0.0939 0.0434 0.i084 0.06i2

Settling time (s) Dfi i2.5 ii.5 5.9i 5.49 37.02 i4.22 23.84

A/2 20.7 i6.7 6.ii 6.12 37.02 i5.32 23.83

APtie 20.8 i6.3 7.0i 4.31 25.35 ii.05 i7.57

Bold signifies best results.

time in sec

-BSA:PI

-hBFOA-PSO:PI

PSO:PI

-BSA:PID

-CRPSO:PI

o a. e ni

time in sec

BSA:PI

hBFOA-PSO:PI PSO:PI CRPSO:PI BSAPID

Figure 5 Comparative analysis between different EA's for test system-1 with GDB (a) changes of frequency, (b) changes of tie-line power.

Table 5 Optimum values of controller parameters using BSA.

Mode of operation Optimum values Objective function (J)

Kii Ki2 Kpi Kp2 Kdi Kd2

PID-controller with reheat turbine PID-controller with reheat turbine and GDB PID-controller with reheat turbine, GDB and BD PID-controller with reheat turbine, GDB, BD and GRC i.9872 i.9852 i.9984 i.9688 i.89i6 i.9960 i.9978 i.7968 i.8942 i.7658 i.8873 i.5778 0.8362 0.7926 0.8543 0.984i 0.3462 0.4967 0.4633 0.3824 0.7436 i.24i3 i.5844 i.3728 0.3350 0.4539 0.4732 0.5020

Table 6 Transient specifications of test system in terms of overshoot, undershoot and settling time.

Mode of operation Area-1 Area-2 Tie-line power

OS US ST OS US ST OS US ST

PID-controller with reheat turbine 0.0158 -0.1380 7.01 0.0081 -0.0926 10.8 0.0017 -0.0305 13.7

PID-controller with reheat turbine plus GDB 0.0197 -0.1665 6.29 0.0177 -0.1209 11.2 0.0024 -0.0372 14.2

PID with reheat turbine, GDB and BD 0.0184 -0.1738 6.64 0.0176 -0.1248 11.1 0.0022 -0.0390 13.5

PID with reheat turbine, GDB, BD and GRC 0.4103 -0.6451 19.6 0.4621 -0.6276 19.8 0.0107 -0.0875 26.7

OS = overshoot, US = undershoot, ST = settling time (s).

0.02 0

£ -0.04

ra -0.06 >

§ -0.1

- -0.12 -0.14 -0.16 -0.18

with GDB

■ with reheater and GDB with reheater, GDB and BD

time in sec

0.005 0

)-0.005 !±

(n -0.01 o ra

> -0.015 e

nie -0.025 ~ -0.03 -0.035 -0.04

with GDB

■with GDB and reheater with GDB, reheater and BD

time in sec

Figure 6 Effect of nonlinearities to the dynamics of test system-1 (a) changes of frequency, (b) changes of tie-line power.

0.24 r

0.16 0.14 0.12

0.0145 r

0.014 0.0135 0.013 0.0125 0.012 0.0115 0.011

40 60 80

Number of Generation

40 60 80

Number of Generation

2.8 2.75 2.7

-i 2.65

w 2 6 ^ m 26 œ

Ü 2.55 2.5 2.45

2.4 -2.35 -

0.33 0.32 0.31 0.3

ä 0.29 >

S 0.28

11 0.27 0.26 0.25 0.24 0.23

number of iteration

40 60 80

Number of generation

Figure 8 Convergence behavior of proposed BSA (a) Pi-controller (ITAE) with GDB, (b) Pi-controller (ISE) with GDB, (c) PIDcontroller (ITAE) with GDB, (d) 10% SLP in all areas of test system-1.

Table 7 Optimum values of controller parameters, ITAE value, maximum overshoot, settling time under different loading conditions.

Parameters Optimum values of controller parameters ITAE Maximum overshoot Settling time

Kn Ki2 Kp1 Kp2 Kd1 Kd2 value A/1 A/2 APtie A/1 A/2 APtie

10% SLP in both areas 20% SLP in area-1 20% SLP in both areas 1.9867 1.9905 1.9570 0.0109 1.8362 1.3475 1.8238 0.1037 0.8950 0.4057 0.8833 1.4925 0.2386 0.2965 8.76 > 5.15 > < 10-6 < 10—5 1.82 x 105 0 1.05 x 0 : 10—4 3.93 2.18 3.90 7.35 3.99 3.93

1.9632 1.9570 1.9274 1.9116 0.9127 0.9026 0.4784 0 0 0 4.19 4.19 5.04

Gi2(s) =

1 + sTf

where e~TDS can be approximated as in (4) using 2nd order Pade approximation technique. The signal flow graph (SFG) method is used to find the linear model of BD and its T.F model is defined in (5).

1 _ si _i_ siH —sTd_ 1_ 2 ^ 2

1 + si +

G(S)=-

K3(1 + G1G2)

1 — K1K2K3G1G2 + G1G2 + K1K3G2 where G1 = G11 * G12.

10% SLP at areal only 10% SLP at both areas 20% SLP at areal only 20% SLP at both areas

time in sec

œ 5 o ü

10% SLP at area1 only -10% SLP at both areas 20% SLP at area1 only 20% SLP at both areas

46 time in sec

Figure 9 Transient responses of test system-1 under different loading conditions with PID-controller (a) changes of frequency, (b) changes of tie-line power.

Table 8 Optimum values of controller parameters of test system-1 using BSA with 10% SLP in area-1 under wide variation of system parameters.

Parameter variation

% of change Controller parameters tuned by ITAE

Controller parameters tuned by ISE

Kii Ki2 Kpi K,

Nominal

Parameter variation

No change 1.9657 0.0164 1.0663 1.5606 0.3462 0.4259 1.9528 0.7063 1.7938 0.6093 1.8656 1.4221

+ 50% + 25% -25% -50%

+ 50% + 25% -25% -50%

+ 50% + 25% -25% -50%

1.9846 1.9857 1.9765 1.9749

1.9621 1.9682 1.9852 1.8795

1.9785 1.9759 1.9478 1.9657

0.5474 0.0313 1.8741 0.0129

0.8706 0.0862 0.0310 1.7737

1.5618 0.0438 1.5474 0.2154

1.1300 1.0351 1.2290 1.2272

1.0811 1.2109 0.9703 0.6259

0.8991 0.9650 1.7945 1.9719

1.5317 1.4739 1.9093 0.9430

1.9537 1.9758 1.4020 1.2365

1.8902 1.8679 1.7687 0.0112

0.3657 0.3258 0.3505 0.3828

0.3813 0.4510 0.2650 0.0716

0.2929 1.3349 0.5222 0.3398

0.2714 0.2179 0.7880 1.6243

0.8389 0.7183 0.0764 0.0691

0.8745 1.0249 0.1558 0.9870

1.9574 1.9784 1.9829 1.8246

1.8673 1.9150 1.9590 1.9122

1.9723 1.9723 1.8957 1.8990

0.1504 0.2759 0.4535 0.0230

0.6525 0.7279 0.3168 0.0287

0.3218 0.3218 0.0033 0.0659

1.5407 1.5397 1.7399 1.5118

1.7203 1.2926 1.9712 1.9923

1.8217 1.8217 1.5372 1.9052

0.0219 0.7348 0.3656 0.2296

1.6982 0.4569 0.3724 0.0899

0.1343 0.1343 0.1679 0.9574

1.7650 1.7833 1.9419 1.6966

1.5987 1.8150 1.7738 1.8205

1.8976 1.8976 1.8157 1.7893

% of change Controller parameters tuned by ITSE

Controller parameters tuned by IAE

Ki1 Ki2 Kp1 K,

1.3702 1.8713 1.9828 1.5718

1.0196 1.9816 1.0796 0.9909

1.4334 1.4334 1.6913 1.4903

Nominal

No change + 50% + 25% -25% -50%

+ 50% + 25% -25% -50%

+ 50% + 25% -25% -50%

1.9081 1.9684 1.8908 1.9978 1.9847

1.9725 1.8819 1.9607 1.9810

1.8951 1.9527 1.9834 1.8921

0.3076 0.0227 0.0375 0.3311 0.1972

0.2618 0.4024 0.0772 0.0898

0.5432 0.0427 0.0801 0.5087

1.8653 1.8792 1.8086 1.8594 1.9414

1.8349 1.7529 1.8680 1.9862

1.6704 1.9941 1.9642 1.8105

1.4879 0.7064 1.4987 1.4817 1.7578

1.4632 1.9055 1.9724 1.3523

1.2541 1.3135 0.2943 1.8639

0.6591 0.6404 0.6143 0.5405 0.4725

0.8718 0.5807 0.3654 0.2016

0.4061 0.5780 0.5122 0.5549

1.5743 1.9768 0.6908 1.7802 1.0403

1.4592 1.6708 1.2398 1.3527

1.6688 1.2936 1.8325 1.0555

1.9869 1.8942 1.9938 1.9666 1.9281

1.9784 1.9814 1.8839 1.9591

1.9565 1.9474 1.9646 1.9856

0.0921 1.6943 1.4094 1.1842 1.7555

1.9792 0.5031 0.8720 1.9458

1.5953 1.9171 0.0543 0.0237

1.9750 1.9088 1.5755 1.9712 1.8632

1.8062 1.8674 1.8367 1.5623

1.8472 1.7768 1.2635 1.5706

1.7348 0.3503 1.2917 1.3527 1.6305

1.6729 1.9271 0.0268 1.0445

1.7753 1.7653 1.8952 0.2646

0.3957 0.6923 0.6840 0.7443 0.8132

0.6374 0.8601 0.8914 0.4358

0.8696 0.8209 0.3314 0.3836

1.8652 0.1733 0.4296 0.6168 0.7210

0.8377 1.9893 0.0535 1.1210

0.2063 0.8360 0.0629 0.8863

Table 9 Performance indices, settling time of test system 1 with 10% SLP in area- 1 under wide variation of system parameters.

Evolutionary algorithm % of change Performances indices Settling time in s (ITAE)

ITAE ISE ITSE IAE f Df2 DPtie

BSA No change 0.1288 0.0034 0.0029 0.1431 1.39 3.30 3.48

hBFOA-PSO [2] 1.4933 0.1324 0.1650 0.9251 5.17 6.81 4.59

Change of turbine time constant, Tt in the range of ±50%

BSA + 50% 0.1408 0.0043 0.0031 0.1438 2.95 3 3.12

+ 25% 0.1309 0.0045 0.0032 0.1445 1.11 3.29 3.52

-25% 0.1269 0.0030 0.0027 0.1518 1.13 3.57 3.48

-50% 0.1391 0.0029 0.0025 0.1457 1.98 2.88 3.59

hBFOA-PSO [2] + 50% 2.2629 0.1634 0.222 1.112 9.17 10.04 7.55

+ 25% 1.7288 0.1463 0.1875 0.988 7.84 7.09 7.35

-25% 1.4075 0.1214 0.1515 0.8898 6.32 5.5 5.02

-50% 1.4542 0.114 0.1448 0.8807 6.77 4.94 5.63

Change of .speed governor time constant, TG in the range of ±50%

BSA + 50% 0.1286 0.0038 0.0028 0.1501 2.36 3.55 3.56

+ 25% 0.1270 0.0037 0.0030 0.1448 2.43 2.87 3.47

-25% 0.1326 0.0034 0.0027 0.1446 2 3.74 3.94

-50% 0.1332 0.0040 0.0026 0.1475 1.17 3.77 3.78

hBFOA-PSO [2] + 50% 1.573 0.1398 0.1746 0.9509 7.6 6.85 4.46

+ 25% 1.5251 0.1359 0.1694 0.9364 7.53 6.82 4.52

-25% 1.4725 0.1293 0.1615 0.9159 5.32 6.77 4.68

-50% 1.4614 0.1264 0.1588 0.9088 5.48 6.62 4.76

Change of synchronizing time constant of tie-line, T12 in the range of ±50%

BSA + 50% 0.1073 0.0032 0.0029 0.1454 1.05 2.55 2.77

+ 25% 0.1164 0.0032 0.0026 0.1461 1.39 2.86 2.96

-25% 0.1462 0.0040 0.0028 0.1508 1.42 3.93 4.46

-50% 0.1469 0.0037 0.0035 0.1940 1.17 5.65 5.71

hBFOA-PSO [2] + 50% 1.3702 0.126 0.1561 0.9254 6.61 7.19 5.13

+ 25% 1.4946 0.1286 0.1595 0.9288 7.06 6.22 5.35

-25% 1.5043 0.1386 0.1746 0.89209 5.51 7.37 5.02

-50% 1.7231 0.1491 0.1935 0.9698 6.26 6.77 5.81

Table 10 Optimum values of controller variables for test system-2.

Parameters Algorithms Kii Ki2 Ki3 Kp1 Kp2 Kp3 Kd1 Kd2 Kd3 J

Without GRC BSA: PI 1.6638 0.0831 0.0683 1.8898 0.0788 0.0673 - - - 0.0578

BSA: PID 1.9667 0.8788 0.0116 1.8810 0.0265 0.0622 1.0067 1.9784 0.1919 0.0475

With GRC BSA: PI 1.1131 0.3241 0.1227 1.9416 0.7227 0.0661 - - - 0.0674

BSA: PID 0.3885 0.2405 0.2219 1.9106 0.3749 0.2658 1.3317 0.6157 0.0949 0.0853

hBFOA-PSO 0.0952 0.1199 0.0210 0.1502 0.1202 0.0399 - - - NA

3.3. Generation rate constraint (GRC)

It is mainly caused by mechanical and thermodynamics constraints of steam turbine. A typical GRC is represented by saturation type nonlinearity and considered as limiter with steam turbine, since power generation can only be changed at a maximum specified rate. If this constrain is not considered during modeling of power system, the system is likely to chase large turbulence thus resulting into wear and tear of controllers. In general GRC [5] of hydro unit is larger than that of thermal unit. In hydro unit, the value is 270%/min for raising genera-

tion and 360%/min for lowering generation, whereas the same for thermal unit is 3%/min.

4. Backtracking search algorithm (BSA)

Optimization is a very important and popular tool for the researchers over past few decades in the area of power system operation and control. Its aim is to find the best values of system parameters through the determination of objective function in order to extract maximum benefits from the system. BSA is one of the new population based EA's introduced by

Table 11 Output specifications of test system-2 in terms of settling time and overshoot without GRC.

Algorithms A/1 A/2 A/3 APfiei APtie2 APtie3

OS ST OS ST OS ST OS ST OS ST OS ST

BSA: PI 0.0101 13.6 0.0025 23.4 0.0048 27.4 0 23.9 0.0013 20.1 0.0011 29.8

BSA: PID 0.0020 8.68 0.0021 25.8 0.0044 29.7 0 22.5 0.0010 27.5 0 33.9

5 -0.005

-0.015

-0.025

time in sec

time in sec

Figure 10 Transient performances of test system-2 without GRC equipped with BSA optimized PID-controller (a) changes of frequency, (b) changes of tie-line power.

Civicioglu. It is fast and capable of solving multimodal problems. It has simple structure and unique mechanism for generating trial individuals enables it to solve nonlinear, dynamic optimization problem successfully and rapidly. BSA can be explained by dividing its evolutionary mechanism into five parts as follows: initialization, selection-I, mutation, crossover and selection-II [30,31].

4.1. Iwitw/featww

Input:N, D, low, up Output: pop for/= 1 : N forj = 1 : D

pop(/,j) = rand * (upj) — low(j)) + lowj);

BSA initially scatters the population (pop) in the solution space by using uniform random distribution as defined in (6).

P0Pij ~ U(/owj up) (6)

where i — 1,2,3,----, N,j = 1,2, 3,----,D, U is the

uniform distribution function, low and up are defined as lower and upper bounds of the solution space, respectively. The pseudo code of defining initial population, according to (6), is as follows:

where N and D are the population size and dimension of the problem (number of control variables), respectively.

4.2. Se/ectiow-I

Selection-I stage in BSA involves in determination of historical population (o/d_pop) for finding search direction.

Initially, historical population is defined randomly using (7)

time in sec

time in sec

Figure 11 Transient performances of test system-2 with GRC equipped with BSA optimized PID-controller (a) changes of frequency, (b) changes of tie-line power.

o/d-pop^ ~ U(/owj, up;) (7)

In BSA, fa < b, where a and b are two random numbers between (0,1), then historical population is updated using (8).

for i — 1 : N for; — 1 :D

o/d_pop(i,;)—pop(i,;); (8)

It is evident from (8) that BSA randomly selects population from the previous generation and remembers until it is changed. Basically, historical population is the swarm memory of BSA. After having historical population, (9) is used for shuffling the order of individual population.

o/d_pop :— permutiwg(o/d_pop) (9)

where permuting is a random shuffling function and ':—'is the update operation.

4.3. Mutation

4.4. Crossover

The final form of the trial population ( T) is generated by using crossover process of BSA. The initial value of the trial population is mMiaini, generated from mutation process. The pseudo code of crossover process in BSA is given below.

/«put :mutat«t, mixrate, N, D Output: Tria/popu/atio« (T) map(1:N,1:D) — 1 if ra«d < ra«d for i — 1 : N

u — rawdperm(D)

map(i, u(1 : cei/(mixrate * ra«d* D))) — 1 ewd e/se for i — 1 : N map(i, randi(D)) — 1 ewd ewd

for i — 1 : N for; — 1:D if map,;,- — 1 t/7 : — popij e«d ewd e«d

The mutation process in BSA is generated by using (10)

Mutatwt — pop + F.(o/d_pop — pop) (10)

where F controls the amplitude of search direction and this paper considered F — 4 * rawdg (Brownian-walk). rawdg represents random number chosen from gamma distribution with unit size and shape.

where ceil is the ceiling function. It mainly involves two processes: first strategy includes mixrate (binary integer-valued matrix) that indicates the individual T to be manipulated using individual values of pop. The second strategy allows only one random number to select from mutatwt in each trial. Line 25 and line 7-9 in algorithm show first and second strategy of crossover process in BSA, respectively.

■ hBFOA-PSO

30 40 time in sec

Figure 12 Transient performances of test system-2 with 1% SLP in all area (a) changes of frequency in area-1, (b) changes of frequency in area-2, (c) changes of frequency in area-3.

Table 12 Comparative analysis between performance indices obtained by different algorithms for test system-2 with 1% SLP in area-1 including GRC.

Optimization Algorithms Df Df2 Df, DPfiei DPtie2 DPtie3

OS ST OS ST OS ST OS ST OS ST OS ST

BSA: PID 0.0020 8.68 0.0021 25.8 0.0044 29.7 1.70 x 10-6 22.6 0.0010 27.5 7.41 x 10-6 33.9

BSA: PI 0.0079 17.2 0.0028 29.6 0.0070 34.4 0 30.13 0.0016 26.5 0.0015 36.1

ISE [20] 3.5 68.2 3.266 61.3 3.225 69.4 0.018 96.1 0.715 106.7 0.625 113.4

ANFIS [20] 1.65 36.9 1.377 49.2 2.107 62.7 0 54.7 0.669 55.2 1.105 62.3

Bold signifies best results.

Figure 13 Schematic diagram of four-area hydro-thermal power system.

Some individual solution at the end of BSA crossover strategy may exceed the search space boundaries. In order to control the movement of the same, boundary control mechanism of BSA is employed and the pseudo-code of this strategy is as described below:

Input : T, lowj, up

Output : T

/or i = 1 : N

/or j = 1 : D

f'^ij < lowj)or(Tij > uj

Ty = randast(upj — lowj) + lowj

4.5. Selection-!/

In selection-II process of BSA, individual values of trial population (T) has better fitness value than individual corresponding population (pop) are used to update individual population (pop) using greedy selection process. If the best individual of pop has better fitness value obtained so far, then global minimum value is replaced by the best individual of population (pop). The detailed BSA structure is available in [30].

The general flowchart of BSA is illustrated in Fig. 3.

using proposed BSA to have minimum overshoot, settling time in area frequencies and power exchange between different control areas. Effectiveness of proposed optimization algorithm is compared with other EA's available in the literature. In this paper, ITAE criterion is used to minimize the objective function as defined in (13).

J = X(aès(A/1) + afos(A/2) + aès(APfte)) * t * AT (13)

where AT is small time interval in s, Tsim is the maximum simulation time in s. The design problem can be formulated as the following constrained optimization problem. Minimize J Subjected to:

Kmin 6 Ki 6 K^max

Kpmin 6 KP 6 Kpmax (14)

KD,min 6 KD 6 KD,max

where KP1D min, KP1D max are the minimum and maximum gains of PI/PID-controller, respectively. The minimum and maximum values of the controller gains are chosen as 0 and 2, respectively.

6. Implementation of BSA in LFC problem

In this article, BSA is employed to solve LFC problem of an interconnected multi-area power system network and to find optimal solutions satisfying all operating constraints. The algorithmic steps of the proposed BSA method applied to LFC system are enumerated below.

Step 1. Randomly initialize population (pop) within search space using following pseudo code and evaluate fitness function using (13).

/or i = 1 : pops'ze /or j = 1 : dim

pop(/, j) = rand * (up(j) — low j)) + low(j); end end

5. Controller structure and objective function

The error input to the controllers is the ACE of the respective areas as defined in (11) and the controlled output from the controller is defined in (12).

e:(t) = ACE = B1A/ + APtie e2(t)= ACE2 = £2 A/2 — APtie

M1 (t) = Kp1ACE1 + Ki1 R ACEdt + Kd1 ^At^

«2 (t) = K^ACE + Knf ACE2dt + Kd2 diA|E21

The primary objective of LFC system is to restore system frequency to its schedule value as quickly as possible and reduces tie-line power oscillations between neighboring control areas under small perturbation of load. In order to satisfy above conditions, settings of PI-control (Ki1; Ki2, Kp1, Kp2) and PID-controller (Ki1; K,2, Kp1, Kp2, Kd1, Kd2) are optimized

where popsize is the population size, dim is the number of control variables (4 for Pi-controller and 6 for PID-controller), up and /ow are the upper and lower bounds of control variables.

Step 2. Randomly initialize historical population (o/d_pop) with in the search space.

Input : pops'ze, low, up Output : old_pop /or i = 1 : pops'ze /or j = 1 : dim

old_pop(/, j) = rand * (up(j) -end end

low(j)) + low(j);

Step 3. Update historical population at the beginning of each iteration using following pseudo code.

if rand < rand for i = 1 : popsize for j = 1 : dim old-pop(i;j) pop(i,j'); end end

where old-pop: indicates updating of historical population by current population.

Step 4. Perform mutation strategy using (10). Step 5. Perform crossover strategy using following pseudo code.

if rand < rand for i = 1 : popsize

u = randperm(dim); map(i, u(1 : ceil(dim_rate * rand * dim)) = 1; end else

for i = 1 : popsize

map(i, randi(dim)) = 1; end end

for i = 1 : popsize for j = 1 : dim

offsprings(i,j) = pop(i,j) + F* (°ld_pop(i,j) —pop(i,j)); end

Step 6. Check, whether output solution (offsprings) goes beyond the search space or not using boundary conditions as described in Section 4.4.

Step 7. Select the fitter solution (fitnessoff springs) from the generated population as in step 7 and evaluate fitness value using (13).

Step 8. Check, whether fitnessoff springs gives better fitness value than that obtained so far using initial population or not. If fitnessoff springs gives better results, then update offsprings by fitnessoff springs, otherwise keep the previous one.

Step 9. Sort the current population from best to worst and use for the next generation.

Step 10. Go to step 3 until termination criterion is met.

7. Results and discussion

The main idea of performing the simulation study is to test the effectiveness of the proposed algorithm based PI/PID controller in multi-area power system networks. The model of test system under study has been developed in Matlab/Simulink environment and proposed BSA algorithm is written in (.m file). The objective function (time multiplied absolute value of ACE) as depicted in (13) is evaluated in (.m file) using BSA to optimize gains of controller parameters. The nominal values of system parameters are adopted from [2] and listed in Table 1.

For successful implementation of BSA, optimum gains of different parameters are required to be determined. Due to

the uncertain behavior of BSA, several trials have been made to select optimum values of input parameters of the same for successful optimization of controller parameters and following are found to be best for LFC problem:

Population size = 40, elitism parameter = 4, iteration cycle = 100, dimension of problem, i.e. number of control variables = 6 (for PID structure) and 4 (for PI structure).

7.1. Transient analysis of test system 1

7.1.1. Linear model

Initially, two-area interconnected identical linear non-reheat thermal-thermal power system shown in Fig. 1 is considered for investigation, with 10% SLP in area-1. Two numbers of PI-controllers are designed using BSA and included in power system modeling. Table 2 gives optimum values of controller settings at the end of optimization. With the proposed BSA method using ITAE as an objective function defined in (13), the minimum ITAE value obtained by BSA is 0.3510 and that of using hBFOA-PSO is 1.1865, using PSO is 1.2142, using BFOA is 2.7475, using GA is 1.8379, using ZN is 3.7568, using DE is 0.9911 and using GSA is 0.6659 for the same test system with similar controller structure. Improvement of ITAE value by BSA is 70.4% (hBFOA-PSO), 71.09% (PSO), 87.2% (BFOA), 80.9% (GA), 90.65% (ZN), 64.58% (DE) and 47.3% (GSA) for the similar test system with PI-controller. Hence, it clearly noted from the above discussion that minimum ITAE value is obtained by BSA for the same test system with similar controller structure. The settling time taken by the area frequencies and tie-line power oscillations by BSA techniques is 5.11 s (Afj), 6.31 s (Af2), 4.58 s (APtie), whereas, settling time for the area frequencies and tie-line power is 7.39 s, 7.65 s, 5.73 s using hBFOA-PSO [2], 7.37 s, 7.82 s, 5 s using PSO [2], 10.59 s, 11.39 s, 9.37 s using BFOA [2], 5.52 s, 7.09 s, 6.37 s using GA [2], 45 s, 45 s, 28 s using ZN [2], 8.96 s, 8.16 s, 5.75 s using DE [7], 12 s, 11.90 s, 8.90 s using GSA [8]. Improvement of settling time using BSA is 7.43% (A/1), 11% (Af2), 20.1% (APtie) w.r.t other EA's. The comparative transient analysis between BSA and other EA's is given in Table 3 and Fig. 4. It is clearly observed from Fig. 4 and Table 3 that BSA gives better transient responses than other EA's. Further, it is noted from Table 3 that ITAE value by BSA tuned PID-controller is 0.1288 and settling time for the area frequencies and tie-line power oscillations is 1.39 s, 3.30 s and 3.48 s, respectively. Improvement of ITAE value and settling time by BSA optimized PID-controller are 63.3%, 72.79%, 47.7%, 24.02%, respectively, than those of BSA optimized PI-controller.

7.1.2. Nonlinear model

The study is further extended by incorporating GDB nonlin-earity into the power system modeling. The nominal values of system parameters are taken from [2] and illustrated in Table 1. During this study, one more objective function based on ISE criterion, as defined in (15), is considered. 1% SLP in area 1 is considered to investigate the effect of GDB on the dynamics of power system.

J = X(/ + (A/2)2 + (APtie)2) * AT (15)

■ä -0.005

-0.015

-0.025

■BSA CLPSO

time in sec

TD >■

-0.005

-0.015

-0.025

■BSA CLPSO

time in sec

■BSA CLPSO

time in sec

time in sec

Figure 14 Transient performances of test system-3 with 1% SLP in area-1 (a) changes of frequency in area-1, (b) changes of frequency in area-2, (c) changes of tie-line power in area-1, (d) changes of tie-line power in area-2.

Table 13 Optimum values of PID-controller settings, fitness values and comparative system performances for test system-3 after 1% SLP in area-1.

Optimization Area-1 Area-2 Area-3 Area-4

techniques kpi ki1 kd1 kp2 ki2 kd2 kp3 ki3 kd3 kp4 ki4 kd4

CLPSO 0.9533 0.9790 0.3341 0.8919 0.9867 0.3114 0.7488 0.4464 0.6467 0.7589 0.1068 0.7082

BSA 0.7501 0.9950 0.3774 0.9721 0.9993 0.2847 0.7945 0.2329 0.7558 0.9680 0.3406 0.5066

Parameters IMC [32] QOHSA [32] CLPSO BSA

OS US ST OS US ST OS US ST OS US ST

(x10"3) (x10-3) (x 10"3) (x 10"3)

Dfi 16 - 11.14 9.5 - 10.86 9.2 0.0252 16.37 4.5 0.0125 15

D/2 3.20 - 24.82 0.87 - 29.45 4.9 0.0241 17.03 4 0.0124 14.76

/ 3.20 - 24.82 1.1 - 28.73 5.3 0.0213 17.77 3.33 0.0105 15.37

Df4 3.20 - 24.32 0.974 - 26.65 3.4 0.0246 16.07 3.2 0.0125 14.42

DPtie, 12 0.88 - 29.35 0.317 - 30.41 4.45 0.0155 11.49 1.09 0.0077 7.19

DPtie, 23 0.88 - 29.35 0.311 - 45.77 8.4 0.0011 20.05 4.3 7.52 x 10"4 16.99

DPtie, 14 1.10 - 24.04 0.357 - 55.29 5.4 0.0022 18.50 2.7 0.0010 13.33

ISE 5.26 x 10" 4 4.26 x 10" -5 2.1721 x 10"5 2.1952 x 10"5

ITSE 3.8 x 10"4 6.80 x 10" -5 1.7133 x 10"5 1.7255 x 10"5

ITAE 0.5924 0.5098 0.1181 0.1168

Bold signifies best results (for entry ''-" means not applicable).

Figure 15 Robustness analysis of test system-3 with random load pattern, (a) changes of frequency in area-1, (b) changes of tie-line power in area-1.

Optimum values of PI-controller, ITAE and ISE value, settling time of area frequencies and tie-line power, and minimum damping ration by BSA are given in Table 4. The ISE value using BSA based PI-controller is 2.64 x 10-4 and that of using hBFOA-PSO and CRPSO is 20.51 x 10—4 and 22.41 x 10—4, respectively. Improvement of ISE value by BSA is 87.1% as compared to hBFOA-PSO and 88.22% as compared to CRPSO. Similarly, ITAE value by BSA based PI-controller is 0.0586 and that of using hBFOA-PSO is 0.3948. Improvement of ITAE value by BSA is 85.16% than that of hBFOA-PSO. Comparative performances of transient behavior with GDB nonlinearity are depicted in Fig. 5. Additionally, the system is also investigated with PID-controller and at the end of optimization, controller gains are given in Table 4. The ISE and ITAE value with BSA based PID-controller are 2.39 x 10—5 and 0.0122, respectively. Improvement of fitness value with BSA optimized PID-controller is 90.9% (ISE) and 79.2% (ITAE). It is evident from Fig. 5 and Table 4 that BSA optimized PID-controller gives better results than other; thus, it may be accepted as better controller structure opti-

mized by BSA and consequent studies in this article are based on BSA based PID-controller.

Additionally, reheater, BD and GRC types of nonlinearities are taken into consideration to investigate their effects on the power system dynamics. Optimum settings of PID-controller using BSA are tabulated in Table 5 and output specifications in terms of overshoot, undershoot and settling time are given in Table 6. Fig. 6 shows the transient behavior of concerned power system network with GDB, BD and reheater. In addition to that, GRC is also incorporated into the system modeling and its effect with other nonlinearities is shown in Fig. 7. It is clearly evident from Figs. 6 and 7 that nonlinearities have prominent effect on the transient behavior of power system network.

7.1.3. Convergence behavior of proposed BSA The convergence curves of proposed algorithm for PI-controller structure with GBD using ITAE criterion, PI-controller structure with GBD using ISE criterion, PID-controller structure with GBD using ITAE criterion and

PI-controller with sudden load variation are shown in Fig. 8 (a)-(d) respectively. From the convergence graphs, it may be observed that the objective function value converges smoothly with the optimum value without any unexpected oscillations, thus ensuring convergence consistency of the proposed BSA.

7.1.4. Sensitivity analysis

Sensitivity analysis is performed to investigate robustness of proposed BSA based PID-controller with changing of loads and time constant of different system parameters such as speed governor, steam turbine, and synchronizing time constant of tie-line in the range of ±50%. Obtained results in terms of ITAE value, optimum values of controller parameters, maximum overshoot and settling time of area frequencies and tieline power under different loading conditions are given in Table 7.The dynamic responses under sudden load variation are depicted in Fig. 9. In addition to the objective functions as defined in (13) and (15), two more objective functions based on ITSE and IAE criterion are considered in this article to test robustness of proposed controller structure and they are defined in (16) and (17).

IAE — XX (abs(Afi) + abs(D/2) + abs(DPtle)) (16)

ITSE — X((4/i)2 + (A/2)2 + (DPtie)2) * t * DT (17)

Table 8 gives the optimum settings of PID-controller parameter during system parameter variation and it is done by taking 10% SLP in area-1. Comparative analysis between proposed BSA and hBFOA-PSO in terms of different cost functions (ITAE, ISE, IAE, ITSE) and settling times of area frequency and tie-line power oscillations under system parameter variations are illustrated in Table 9. It is seen from Table 9 that all variations of area frequencies and tie-line power to different values of time constant of system parameters are in the vicinity of the results obtained by nominal settings.

7.2. Transient analysis of test system 2

One more test system having three area hydro-thermal interconnected power system network equipped with PI/PID controller is considered to test effectiveness and further validation of proposed optimization technique. The T.F model of same is given in [18] and nominal values of system parameters are illustrated in Table 1. Thermal area is studied with single stage reheat turbine and hydro unit is equipped with electrical governor. The internal characteristics of hydro unit differ significantly from thermal area and for details readers are referred to [5].

7.2.1. Linear model

Initially, the system is studied with linear mode without any physical constraints and at the end of optimization through BSA, the controller parameters are given in Table 10. The ITAE value with BSA tuned PI-controller is 0.0578 and that of using BSA tuned PID-controller is 0.0475. Improvement of ITAE value with PID-controller is 17.82%. Output transient specifications in terms of overshoot and settling time are given in Table 11. The changes of frequency and tie-line

power with 1% SLP in area-1 using PID-controller are shown in Fig. 10.

7.2.2. Nonlinear model

GRC of steam turbine is incorporated into system modeling to make the system be realistic one. The optimum gains of BSA based PI/PID controllers with GRC are tabulated in Table 10 and transient responses in terms of changes of frequency and tie-line power are shown in Fig. 11. To investigate superiority of proposed BSA based controller, the performance of the same is compared with hBFOA-PSO and ANFIS techniques. Fig. 12 show comparative performances of BSA controller with hBFOA-PSO and Table 12 gives comparative transient performances between ANFIS and results obtained using ISE criterion. It is further observed form Table 12 and Fig. 12 that BSA tuned PID-controller gives better results than others.

7.3. Test system-3

To appraise the applicability and efficacy of the proposed BSA tuned PID-controller, the study is forwarded to another test system, namely four-area hydro-thermal power plant [32]. The schematic diagram of the concerned test system is shown in Fig. 13; however, for detail of transfer function model of the same, the readers are referred to [32]. It is viewed from Fig. 13 that areas 1, 2 and 3 are reheat type thermal unit while area 4 is a hydro power plant. The nominal values of system parameters are taken from [32] and listed in Table 1. In this phase of analysis, BSA is used to identify the optimal settings of PID-controller between [0, 1] employing (13), (16) and (17). The optimal controller gains and minimum fitness values (ITAE, ISE, and ITSE) are presented in Table 13. To show the advantage of the proposed algorithm, the controller is further designed by comprehensive learning particle swarm optimization (CLPSO) and the controller gains are specified in Table 13. For details of CLPSO, the readers are referred to [33]. The comparative dynamic behavior after 1% SLP in area-1 at t — 0 s is depicted in Fig. 14. Typical transient specifications such as peak overshoot, undershoot and settling time are noted down from Fig. 14 and displayed in Table 13. The system performances of the proposed method are compared with CLPSO, internal model control (IMC) [5], and quasi-oppositional harmony search (QOHS) [5] techniques to judge its superiority. Critical observation of Fig. 14 and Table 13 reveals that the proposed BSA-tuned PID-controller improves system dynamics remarkably as compared to others as listed in Table 13.

Furthermore, to affirm the robustness of proposed method, a random load pattern is considered and applied to area-1 of test system-3. In this phase of analysis, a real-time random load perturbation is considered for simulation. The profile of continuous load perturbation is depicted in Fig. 15 (blue color, solid line). The dynamic behavior of the concerned power system after load perturbation is depicted in Fig. 15. To demonstrate the superiority of proposed method, the dynamic responses are compared with those of CLPSO and the comparative responses are displayed in Fig. 15. It may be inferred from Fig. 15 that the proposed BSA tuned PID-controller yields robust and stable performance under the said disturbed condition.

8. Conclusion

This article presents the usage of BSA as new optimization algorithm to solve LFC problem for an interconnected power system network and gives a broad analysis of its tuning performances with its robustness. Initially, two-area interconnected non-reheat power system is considered and parameters of PI/ PID controller are optimized by the proposed algorithm. Standard time domain error criteria such as ITAE, ISE, IAE and ITSE are considered to minimize fitness function using proposed method. To reveal superiority of BSA, the performances of same are compared with other optimization techniques available in the literature for the similar test system with same controller structure and analysis shows good results in terms of fitness value, peak overshoot, settling time of frequency and tie-line power deviations. On the other hand, robustness of designed PI/PID controller is also proved by the applied method. Accordingly, the designed controller is not affected by the variation of loads and system parameters. Furthermore, the study is extended to test the effectiveness of proposed algorithm considering power system nonlinearities such as rehea-ter, GDB, BD and GRC of steam turbine and analysis shows that proposed algorithm ensures satisfactory results in the presence of all nonlinearities. Furthermore, a four-area hydro-thermal power plant is included in the simulation study to show the novelty of the proposed technique and a comparative study is made to appraise the acceptability and efficacy of the proposed BSA technique. Finally, it is concluded that proposed controllers tuned by BSA may effectively be applied in practical nonlinear system.

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Dipayan Guha was born in 1986 at Barasat, North 24 Parganas, West Bengal, India. He received the B.Tech degree in Electrical Engineering from Academy of Technology, Adisaptagram, Hooghly, India, in 2009; M. Tech degree in Electrical Engineering with specialization of Control and Automation from Asansol Engineering College, Asansol, Burdwan, India, in 2013 and pursuing PhD from NIT Durgapur. Presently he is working as an Assistant Professor in the department of Electrical Engineering, Dr.B.C.Roy Engineering College, Durgapur, India. His field of research interest includes FACTS, Automatic Generation Control, Power System Stabilizer and Evolutionary com-

Provas Kumar Roy was born in 1973 at Mejia, Bankura, West Bengal, India. He received the BE degree in Electrical Engineering from R. E. College, Durgapur, Burdwan, India, in 1997; ME degree In Electrical Machine from Jadavpur University, Kolkata, India, in 2001 and PhD from NIT Durgapur in 2011. Presently he is working as Professor in the department of Electrical Engineering, Dr. B. C. Roy Engineering College, Durgapur, India. He has published more than 15 research papers in international journals. His field of research interest includes

Economic Load Dispatch, Optimal Power flow, FACTS, Unit Commitment, Automatic Generation Control, Power System Stabilizer and Evolutionary computing techniques.

Subrata Banerjee received his Bachelors, Masters and PhD Degree in the year 1989, 1994 and 2004 respectively. Dr. Banerjee has about 25 years professional experience out of which Teaching: 19 Years, Research: 04 years, and Industry: 01 Year. He is presently working as professor in Electrical Engineering at NIT, Durgapur. He has published a number of research papers in National/International Journals and Conference Records. His area of interest includes Controller Design, Intelligent Control, optimization techniques, control of Switch-mode converters and non-linear dynamics of converters, Renewable Energy. He is a Fellow of Institute of Engineers (India), life member of Systems Society of India, and member of IEEE (USA). He has successfully completed some research and consultancy projects including one major from DST, Govt. of India. Prof. Banerjee has guided 03 PhD and 12 M. Tech students and some are pursuing their degree under his guidance. He has received few academic awards including 4 no. of Best papers and TATA RAO Prize in his credit. He has filed one Indian Patent in 2013. Prof. Banerjee is acting as regular reviewer of IEEE Transaction on PE, IEEE Transaction on IE, Electrical Power Components and Systems, etc.