Automatic generation control using two degree of freedom fractional order PID controller

Electrical Power and Energy Systems 58 (2014) 120–129

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Automatic generation control using two degree of freedom fractional order PID controller Sanjoy Debbarma a,⇑, Lalit Chandra Saikia b,1, Nidul Sinha b,1 a b

Department of Electrical & Electronics Engineering, National Institute of Technology Meghalaya, Shillong, India Department of Electrical Engineering, National Institute of Technology Silchar, Assam, India

a r t i c l e

i n f o

Article history: Received 15 March 2013 Received in revised form 30 December 2013 Accepted 6 January 2014

Keywords: Automatic generation control Firefly algorithm Fractional order controller Integer order controller 2-DOF-FOPID controller

a b s t r a c t In this paper, Two-Degree-of-Freedom-Fractional Order PID (2-DOF-FOPID) controller is proposed for automatic generation control (AGC) of power systems. Proposed controller is tested for the first time on a three unequal area thermal systems considering reheat turbines and appropriate generation rate constraints (GRCs). The simultaneous optimization of several parameters of the controllers and speed regulation parameter (R) of the governors is done by a recently developed metaheuristic nature-inspired algorithm known as Firefly Algorithm (FA). Investigation clearly reveals the superiority of the 2-DOF-FOPID controller in terms of settling time and reduced oscillations. Present work also explores the effectiveness of the Firefly algorithm based optimization technique in finding the optimal parameters of the controller and selection of R parameter. Further, the convergence characteristics of the FA are compared to justify its efficiency with other well established optimization technique such as PSO, BFO and ABC. Sensitivity analysis confirms the robustness of the 2-DOF-FOPID controller for different loading conditions and wide changes in inertia constant (H) parameter. Furthermore, the performance of proposed controller is tested against higher degree of perturbation and random load pattern. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Automatic generation control (AGC) deals with the controlling of active power, that is, generator output in response to changes in system frequency and tie-line power interchange. Increased in the complexity of the modern power system has necessitated the use of advanced intelligent control scheme. Various control methodologies, such as optimal control, variable structure control, adaptive and self-tuning control has been reported in the past to solve AGC problem and are available in [1–3]. Some authors have investigated fuzzy logic based control and ANN approaches for AGC [4,5]. The problems identified in fuzzy logic controller is that a considerable computational time is required for rules base to be examined and in case of artificial neural network (ANN), more time is required for the data base for training the neural network controller. Compare to the above mentioned approaches, classical based controller has been mostly used to suppress the oscillations due to their simplicity in execution. Authors in [6] suggested that classical integer order (IO) based integral plus double derivative (IDD) controller can outperform other conventional controller such as I, PI and PID controller. Although conventional IO controller are well ⇑ Corresponding author. Tel.: +91 9401154379; fax: +91 0364 250113. E-mail addresses: [email protected] (S. Debbarma), lcsaikia@yahoo. com (L.C. Saikia), [email protected] (N. Sinha). 1 Tel.: +91 9435173835; fax: +91 3842 233797. 0142-0615/$ - see front matter Ó 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2014.01.011

known for its simplicity and widely used in AGC for controlling purpose, but there is no guarantee that such controller would provide the best dynamic response under realistically constrained conditions. It has been noticed that the basic approaches of IO based classical controller are not effective in achieving good dynamic performances when subjected to wide changes in magnitude of step load perturbation (SLP). To overcome this problems, authors in [7] has introduced fractional order (FO) based classical controller to solve multi-area AGC problem under deregulated environment and their investigation reveals that FOPID controller is far better than IO controller. The main advantages associated with FOPID controller is its two extra tuning knobs (parameters) known as k (non-integer order of integrator) and l (non-integer order of differentiator) that provides more flexibility for adjustment of system dynamics. Surprisingly, owing to this advantage of FO controller, most of the past researches were focused only on using IO based classical controller and their optimization using various optimization techniques and very less effort has been made to design and apply robust FO based AGC controller. References [8,9] used advanced control algorithm with two degree of freedom (2DOF) concept to enhance the control performances of an IO based PI and PID controller. The flexibility of 2-DOF over single degree of freedom is from the point of view of achieving high performance in set-point tracking and the regulation in the presence of disturbance inputs. To provide this additional flexibility, authors in [10] and [11] attempted 2-DOF internal model control (IMC) based

S. Debbarma et al. / Electrical Power and Energy Systems 58 (2014) 120–129

121

Nomenclature f i Pri DPDi D Di Ri Kri Tti Tpi KIi KDi

li Dfi bo DPtie

i-j

nominal system frequency (Hz) subscript referred to area i (1, 2, 3) rated power of area i (MW) incremental load change in area i (p.u) DPDi/Dfi (p.u MW/Hz) speed regulation parameter of area i (Hz/pu MW). steam turbine reheat coefficient of area I steam turbine time constant of area i (s) 2Hi/f Di (s) integral gain of controller in area i derivative gain of controller in area i order of derivative gain of controller in area i incremental change in frequency of area i (Hz) initial attractiveness of a firefly incremental change in tie line power connecting between area i and area j (p.u)

controller and parallel 2-DOF-PID controller to solve the load frequency control (LFC) problem respectively. However their investigations have not dealt with fractional order controller and were limited to 2-DOF integer order (IO) controller. The superiority and advantage of fractional order controller along with two-degree-of-freedom concept is yet to be explored in the field of AGC. Also, no literature in the past compared and studied the performance of several fractional order (FO) controllers in AGC of three unequal area thermal systems which needs further investigations. Literature survey shows that in the past many researchers have used different heuristic optimization techniques to deal with the AGC problem. The difficulties in AGC are not only designing of a robust controller but also to optimize its corresponding parameters effectively for optimal solution. To achieve optimal solution, many intelligent optimization approaches such as genetic algorithm (GA), particle swarm optimization (PSO), bacterial foraging optimization (BFO) and artificial bee colony (ABC) are successfully applied to solve the AGC problems and are available in the literatures [12–15]. Some authors have applied GA based optimization in [12]. Although GA has shown their effectiveness and dominancy over classical approach in some of the work pertain to AGC but recent research has identified some deficiencies in GA performance like premature convergence which may degrades its efficiency and reduces the search capability [13]. Authors in [13,4] clearly proved the superiority of the BFO over the classical technique, GA and PSO in terms of convergence, robustness and precision. Like GA, PSO is also less susceptible to getting trapped on local optimum. Gozde et al. [15] discussed that ABC can produce more optimal solutions than PSO technique in the field of AGC. The complexity of AGC problems and its optimal optimization reveals the necessity for development of more efficient algorithms in order to accurately minimize the ACE signal to zero. Recently, a new metaheuristic nature-inspired algorithm so called Firefly Algorithm (FA) based on the flashing light of fireflies has been successfully applied to solve different engineering problem [16–18]. Although the FA has got many similarities with other algorithms, which are based on the so-called swarm intelligence, such as the famous PSO, and BFO, it is indeed much simpler both in concept and implementation. The idea behind this algorithm is that the social behaviour and especially the flashing light of fireflies can be easily formulated and associated with the objective function of a given optimization problem [16]. FA seems very promising for dealing with optimization problem, but has been rarely reported. Recent analysis also identified the characteristic feature of the firefly algorithm is the fact that it simulates a parallel independent

T ⁄ Hi Pgi Tij Tgi Tri Bi Kpi KPi ki bi

a c

simulation time (s) superscript denotes optimum value inertia constant of area i (s) incremental generation change in area 1 (p.u) synchronizing coefficients steam governor time constant of area i (s) steam turbine reheat time constant of area i (s) frequency bias constant of area i (p.u MW/Hz) 1/Di. (Hz/pu) proportional gain of controller in area i order of derivative gain of controller in area i area frequency response characteristics of area i randomization parameter absorption coefficient

run strategy, where in every iteration, a swarm of n fireflies has generated n solutions. Each firefly works almost independently and as a result the algorithm, will converge very quickly with the fireflies aggregating closely to the optimal solution. Authors in [17] proved that firefly algorithm is far superior to both PSO and GA in terms of both efficiency and success rate. Thus to justify the effectiveness of FA in the area of power system, authors in [18] analysed economic dispatch (ED) problems using FA technique and the performances are compared with those available in the literature such as GA, PSO, BFO, and biogeography-based optimization (BBO). Subsequently Debbarma et al. [19] explored the robustness of firefly algorithm in finding optimal solution for the controllers. The proposed optimization technique is found to be very efficient and outperforms the other techniques thus encouraging further researches for complex problems. It is known that with only primary control (i.e. secondary control absent) the smaller the governor droop the smaller the steady state error in frequency but in the presence of secondary control there is nothing to be sacrosanct to use a small governor droop (of the order of 4–6% used in practice) as any large but credible value of R can also guarantee zero steady state error in frequency. Higher value of R results into easy realisation and economical governor [6,7]. In view of the above, the objectives of the present work are: (a) To design and apply a new classical controller named as Two-Degree-of-Freedom-Fractional Order Proportional plus Integral plus Derivative (2-DOF-FOPID) controller and compare its performance with several FO based controllers as well as integer order (IO) based controllers. (b) To compare the convergence characteristics of FA with other algorithms such as PSO, BFO and ABC. (c) To apply Firefly Algorithm (FA) based optimisation technique for simultaneous optimization of several parameters. (d) To study the performance of 2-DOF-FOPID Controller under random load pattern and higher magnitude of SLP. (e) To check the robustness of the optimum parameters value of 2-DOF-FOPID through sensitivity analysis. 2. Description of the system model The simulation based investigations have been carried on a three unequal area thermal system having area1: 2000 MW, area2: 4000, area3: 8000 MW provided with appropriate GRC of 3% per minute and reheat turbine in all the areas. The transfer function

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Fig. 1. Transfer function model of three area thermal system with integral controller.

model of three area thermal system with integral controller is shown in Fig. 1. The nominal system parameters are depicted in Appendix. In three unequal AGC systems, per unit values of different parameters of the unequal areas are considered to be same on their respective MW capacity bases. Hence, during modeling three unequal capacities, the quantities a12 = Pr1/Pr2, a23 = Pr2/Pr3, a13 = Pr1/Pr3 are considered. Firefly Algorithm (FA) is used for simultaneously optimization of several parameters available in the system and the control design is based on integral square error (ISE), Integral of Time multiplied Absolute Error (ITAE) criterion and figure of demerits (FD) given by Eqs. (1)–(3) respectively.

J 1 ¼ ISE ¼

Z

J 2 ¼ ITAE ¼

T

n o 2 ðDfi Þ þ ðDP tieij Þ2 dt

ð1Þ

0

Z

T

n

o jDfi j2 þ jDPtieij j2 t:dt

ð2Þ

0

the performance of set-point is not taking into consideration and thus cannot provide additional flexibility to the control system design like 2-DOF. The 2-DOF controller generates an output signal based on the difference between a reference signal and a measured system output. It computes a weighted difference signal for each of the proportional, integral, and derivative actions according to the set-point weights. The controller output is the sum of the proportional, integral, and derivative actions on the respective difference signals, where each action is weighted according to the chosen gain parameters. No doubt, that 2-DOF PID controller can actively control and improve the dynamic performances and can deals with any complex control problems. However, it is well known fact that the introduction of fractional order calculus idea to conventional integer order controller design extends the opportunity of added performance improvement [7]. The most commonly used Riemann–Liouville (R–L) definition for fractional derivative is given by Eq. (4) n

2

2

J 3 ¼ FD ¼ ðPOÞ þ ðUSÞ þ ðSTÞ

2

a

ð3Þ

aDt

f ðtÞ ¼

1 d Cðn  aÞ dtn

Z

t

ðt  sÞna1 f ðsÞds

ð4Þ

a

MATLAB software has been used to obtain dynamic responses for Dfi and DPtie i–j for different load perturbation.

where n  1 6 a < n, n is an integer and U () is the Euler’s gamma function. And the definition for fractional integral is given by Eq. (5)

3. Design of Two-Degree-of-Freedom – FOPID controller

a a D t f ðtÞ

The concept of Two-Degree-of-Freedom (2-DOF) based controller has recently attracted the attention in different areas of control engineering because of their better control quality for both smooth set point variable tracking and good disturbance rejection [8,9]. While in a conventional Single-DOF based controller, tuning is performed only on the basis of a load-disturbance specification and

¼

1 CðaÞ

Z

t

ðt  sÞa1 f ðsÞds

ð5Þ

a

where a D at is the fractional operator. The Laplace transformation of Riemann–Liouville definition (3) for the fractional derivative is given by Eq. (6)

Lfa D at f ðtÞg ¼ sa FðsÞ 

n1 X ska Dtak1 f ðtÞjt¼0 k¼0

ð6Þ

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S. Debbarma et al. / Electrical Power and Energy Systems 58 (2014) 120–129

D(s) R(s)

F(s)

+ -

C(s)

+

+

P(s)

Y(s)

Fig. 2. Two-degree-of-freedom control scheme.

Firefly Algorithm Objective function f(x), x = (x 1,… xd)T Initialize a population of fireflies x i (i = 1, 2,….., n) Define light absorption coefficient γ while (t < MaxGeneration) for i = 1: n all n fireflies for j = 1: i all n fireflies Light intensity Ii at xi is determined by f(xi) if (Ij > Ii) Move firefly i towards j in all d dimensions end if Attractiveness varies with distance r via exp [−γ r2] Evaluate new solutions and update light intensity end for j end for i Rank the fireflies and find the current best end while Postprocess results and visualization Fig. 3. Pseudo code of the firefly algorithm.

for n  1 6 a 6 n where Lff ðtÞg indicates the normal Laplace transformation. This means that when zero initial conditions are assumed, the systems with dynamic behaviour described by differential equations including fractional derivatives give rise to transfer functions with fractional order of s. The fractional derivative or integral sa can be approximated by a well-known transfer function proposed by Oustaloup in the prespecified frequency range ½xl ; xh  using a recursive distribution of poles and zeros is given by Eq. (7)

sa ¼ K

N Y 1 þ ðs=xz;n Þ n¼1

1 þ ðs=xp;n Þ

ð7Þ

‘K’ is an adjusted gain so that if K = 1 then the gain is zero db for a 1 rad/s frequency. The number of poles and zeros ‘N’ is chosen in advance: low values resulting in simpler approximations but cause ripples in both gain and phase behaviours. Such ripples can be removed by increasing the values of N but it will make the approximation complex. Frequencies of poles and zeros are given by Eqs. (8)–(12)

pffiffiffi

xz;1 ¼ xl n

ð8Þ

xp;n ¼ xz;n e; n ¼ 1 . . . N

ð9Þ

pffiffiffi ¼ xp;n g;

ð10Þ

xz;nþ1

n ¼ 1...N  1

e ¼ ðxh =xl Þm=N

ð11Þ

g ¼ ðxn =xl Þð1v Þ=N

ð12Þ

Fig. 4. Comparison of performance of FOI, FOPI, FOPID and 2-DOF-FOPID controllers in three unequal area thermal systems at nominal loading. (a) Df2 = f(t), (b) DPtie13 = f(t).

The most common form of FO controller is the PIkDl where k and l are the non-integer order of integrator and differentiator respectively and can be any real numbers. The transfer function of a single degree of freedom FOPID controller has the form given by (13)

Gc ðsÞ ¼ K p þ

Ki þ K d sl sk

ð13Þ

The use of PID controllers not only improves stability but also helps to achieve fast response in the system compared to PI controller. However, the derivative operator(s) in the PID controller will boost disturbances drastically which causes system instability. Therefore, a filter is added to the fractional order (FO) derivative term to reduce the detrimental effect of the high-frequency measurement noise. Fig. 2 shows the control structure of the

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S. Debbarma et al. / Electrical Power and Energy Systems 58 (2014) 120–129

Fig. 5. Comparison of performance of I, PI, PID and 2-DOF-FOPID controllers in three unequal area thermal systems at nominal loading. (a) Df1 = f(t), (b) Df2 = f(t), (c) DPtie1-2 = f(t), (d) DPtie13 = f(t).

Table 1 Peak Overshoots (POs) and Settling Times (STs). Fig. No.

Responses

Controllers I

Fig. Fig. Fig. Fig.

5(a) 5(b) 5(c) 5(d)

Df1 Df2 DPtie12 DPtie13

PI

PID

ST

PO

ST

PO

ST

PO

ST

0.011550 0.005733 0.003475 0.001860

98.81 94.27 87.34 90.28

0.009735 0.005733 0.003169 0.001498

98.76 88.23 87.33 90.28

0.008474 0.005118 0.001597 0.000796

72.92 63.19 70.43 65.03

0.004341 0.000025 0.000 0.000

22.12 19.76 39.37 39.37

proposed parallel 2-DOF-FOPID controller, where R(s) represents reference signal, Y(s) represents the feedback from measured system output, F(s) acts as a pre-filter on the reference signal, C(s) represents single degree-of-freedom controller and D(s) implies load disturbance. The transfer function of C(s) and F(s) from Fig. 2 can be written as

CðsÞ ¼ K P þ

KI K D sl þ l k s 1 þ KKD sN

ð14Þ

P

FðsÞ ¼ K P b þ

KI K D sl þ ld k s 1 þ KKD sN

2-DOF-FOPID

PO

ð15Þ

P

where N represents the derivative filter coefficient and b and d represents weightings that influence the set- point response or set- point weight parameters. Thus the output of the proposed controller, U(s) can be finally obtained as

UðsÞ ¼ K P fbRðsÞ  YðsÞg þ

KI K D sl fdRðsÞ  YðsÞg fRðsÞ  YðsÞg þ l sk 1 þ KKD sN P

ð16Þ The parameters of the proposed parallel 2-DOF-FOPID controller such as K Pi , K Ii , K Di , ki , li , bi , di , N i are optimized using firefly algorithm based optimization technique. The details pertaining to Firefly Algorithm (FA) is discussed in the next section. The design problem can be formulated as the following constrained optimization problem, where the constraints are the 2-DOF-FOPID controller parameter bounds given below: min Minimize J; Subject to K min 6 K Pi 6 K max 6 K Ii 6 K max Pi Pi ; K Ii Ii max and K min Di 6 K Di 6 K Di

kmin 6 ki 6 kmax and i i

lmin 6 li 6 lmax i i

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S. Debbarma et al. / Electrical Power and Energy Systems 58 (2014) 120–129

Fig. 7. Comparison of convergence characteristics of FA with PSO, BFO and ABC.

Fig. 6. Performance comparison of I, PI, PID and 2-DOF-FOPID controller. (a) Df2 = f(t), (b) DPtie1-2 = f(t).

bmin 6 bi 6 bmax and dmin 6 di 6 dmax ; Nmin 6 Ni 6 Nmax i i i i i i

ð17Þ

min min max max max where J is the objective function, K min Pi , K Ii , K Di , K Pi , K Ii , K Di are the minimum and maximum value of the controller gains respectively, kmin , lmin , kmax , lmax are the minimum and maximum value i i i i of order of integral and derivative gains, bmin , dmin , ðbmax , dmax are i i i i the minimum and maximum value of proportional and derivative set- point weights, Nmin , N max are the minimum and maximum value i ii of derivative filter coefficient. The minimum and maximum bounds for controller gains are chosen as 0 and 1 and for k and l, minimum and maximum values are also chosen as 0 and 1. The bounds for d and b are taken as 0 and 4 while for derivative filter coefficient (N) it is chosen as 0 and 100.

result, it seems more effective in multi-objective optimization. As per recent bibliography, the statistical performance of the firefly algorithm was measured against other well-known optimization algorithms using various standard stochastic test functions [17]. A more detailed description concerning theoretical and implementation feature of the proposed method is provided in [18]. It was based on the following three idealized behaviour of the flashing characteristics of fireflies: (1) all fireflies are unisex so that one firefly is attracted to other fireflies regardless of their sex; (2) attractiveness is proportional to their brightness, thus for any two flashing fireflies, the less bright one will move towards the brighter one. The attractiveness is proportional to the brightness and they both decrease as their distance increases. If no one is brighter than a particular firefly, it moves randomly; (3) the brightness or light intensity of a firefly is affected or determined by the landscape of the objective function to be optimized. For a maximization problem, the brightness can simply be proportional to the objective function. Other forms of brightness can be defined in a similar way to the fitness function GA or BFO. Based on these three rules, the basic steps of the Firefly Algorithm (FA) can be summarized as the pseudo code shown in Fig. 3. 4.1. Attractiveness The form of attractiveness function of a firefly is the following monotonically decreasing function:

br ¼ b0 expðcrm Þ; with m P 1

ð18Þ

where r is the distance between any two fireflies, b0 is the initial attractiveness at r = 0, and c is an absorption coefficient which controls the decrease of the light intensity. 4.2. Distance and movement

4. Firefly Algorithm based optimization technique

The distance between any two fireflies i and j at xi and xj, respectively, is the Cartesian distance

The Firefly Algorithm (FA) is a metaheuristic, nature-inspired, optimization algorithm which is based on the social (flashing) behaviour of fireflies, or lighting bugs, in the summer sky in the tropical temperature regions. This algorithm was developed and introduced by Yang [16]. Its main advantage is the fact that it uses mainly real random numbers, and it is based on the global communication among the swarming particles (i.e., the fireflies), and as a

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u d uX rij ¼ kxi  xj k ¼ t ðxi;k  xj;k Þ2

ð19Þ

k¼1

where xi;k is the kth component of the spatial coordinate xi of ith firefly and d is the number of dimensions. In 2-D case i.e. for d = 2, we have

Table 2 Performance indices values. Sl. No.

1

Case

Nominal

ISE

ITAE

FD

2-DOF-FOPID

PID

PI

I

2-DOF-FOPID

PID

PI

I

2-DOF-FOPID

PID

PI

I

0.0017

0.0045

0.0097

0.098

0.0071

0.0100

0.0184

0.0188

625.05

2025.02

5625.01

5776.01

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S. Debbarma et al. / Electrical Power and Energy Systems 58 (2014) 120–129

rij ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxi  xj Þ2 þ ðyi  yj Þ2

ð20Þ

However, the calculation of distance r can also be defined using other distance metrics, based on the nature of the problem, such as Manhattan distance or Mahalanobis distance. The movement of a firefly i is attracted to another more attractive (brighter) firefly j is determined by

xi ¼ xi þ b0 e

cr2ij

  1 ðxj  xi Þ þ a rand  2

ð21Þ

where the first term is the current position of a firefly, the second term is due to the attraction while the third term is randomization with coefficient a being the randomization parameter and rand is a random number generator uniformly distributed in [0,1]. In this FA based optimization, the parameters of FA technique are tuned for optimal performance and their tuned values are b0 ¼ 0:4; a ¼ 0:5 and c ¼ 0:5; firefly = 6, maximum generation = 100. 5. Results and analysis 5.1. Simultaneous optimization of FO controller parameters and governor speed regulation parameters Ri

Fig. 8. (a) Random load pattern and (b) frequency deviation in area2 obtained using PID and 2-DOF-FOPID controller.

Proposed work is tested on a three unequal area thermal system followed by 1% step load perturbation (SLP) in area1 keeping frequency bias (Bi) fixed at area frequency response characteristics (bi). Several fractional order controllers, such as, Fractional order Integral (FOI), Fractional order Proportional -Integral (FOPI), Fractional order Proportional- Integral -Derivative (FOPID) and proposed 2-DOF-FOPID controller have been considered separately in the system. Firefly algorithm based optimization technique is used for simultaneously optimization of several parameters of the FO controllers and ‘R’ parameters to achieve optimal solution. Integral of squared error (ISE) have been used as performance index to design the optimal controller. The Matlab simulations for

Table 3 Optimum parameter values of 2-DOF-FOPID controller at different system conditions, parameters and magnitude of SLP. Optimum parameters

K P1 K I1 K D1 k1

l1

b1 d1 N 1 K P2 K I2 K D2 k2

l2

b2 d2 N 2 K P3 K I3 K D3 k3

l3

b3 d3 N 3 R1 R2 R3

50% Loading

0.2537 0.2580 0.8162 0.9409 0.0086 2.0280 2.2121 90.000 0.3116 0.1745 0.1727 0.9062 0.0076 1.0044 0.0116 90.000 0.4404 0.1225 0.1854 0.9462 0.0686 0.6297 0.0048 97.000 7.1 7.7 6.2

Loading

Inertia Constant (H)

5% SLP

+20%

-20%

+20%

20%

0.2691 0.2508 0.7662 0.9510 0.0091 1.9833 2.0011 90.000 0.3275 0.1953 0.2154 0.9362 0.0172 1.1002 0.2611 90.000 0.4328 0.1378 0.2045 0.9202 0.0731 0.3361 0.0278 80.000 7.2 6.6 7.8

0.2701 0.2619 0.7580 0.9420 0.0091 1.9562 1.0052 89.000 0.2976 0.1589 0.2175 0.9091 0.0087 1.1562 1.0011 93.000 0.4694 0.1605 0.2274 0.9481 0.0791 0.3090 0.0037 80.000 7.4 7.5 7.0

0.2398 0.2489 0.7956 0.9488 0.0093 2.1987 2.1161 87.000 0.3276 0.1871 0.1961 0.9109 0.0058 1.1001 0.0489 90.000 0.4268 0.1472 0.1769 0.9462 0.0686 0.4371 0.0176 75.000 7.3 7.1 6.8

0.2689 0.2373 0.7969 0.9452 0.0106 2.1026 1.8761 91.000 0.3192 0.1845 0.1999 0.9207 0.0077 2.1541 0.0233 90.000 0.4300 0.1423 0.1629 0.9399 0.0686 0.2976 0.0070 81.000 6.8 7.5 8.1

0.3564 0.4987 0.0744 0.8790 0.9087 0.1178 0.2177 100.00 0.4765 0.2945 0.6678 0.9062 0.4001 0.1889 1.0078 92.000 0.7409 0.0927 0.0899 0.8962 0.0686 0.0577 0.1167 80.000 7.3 6.7 6.3

S. Debbarma et al. / Electrical Power and Energy Systems 58 (2014) 120–129

Fig. 9. Comparison of dynamic response of Df2 = f(t) for 70% loading with K Pi , K Ii , K Di , ki , li , bi , di , N i and Ri corresponding to 70% and 50% loading.

127

Fig. 12. Comparison of dynamic response of Df1 = f(t) for H = 4 s with K Pi , K Ii , K Di , ki , li , bi , di , Ni and Ri corresponding to H = 4 s and H = 5 s.

Fig. 10. Comparison of dynamic response of Df2 = f(t) for 30% loading with K Pi , K Ii , K Di , ki , li , bi , di , N i and Ri corresponding to 30% and 50% loading.

Fig. 13. Comparison of dynamic response considering 5% SLP for integral and 2DOF-FOPID controller. (a) Df1 = f(t), (b) DPtie1-2 = f(t).

Fig. 11. Comparison of dynamic response of Df1 = f(t) for H = 6 s with K Pi , K Ii , K Di , ki , li , bi , di , Ni and Ri corresponding to H = 6 s and H = 5 s.

all FO controllers are carried out taking N = 3, xl = 0.01 rad/s and xh = 40 rad/s. When the system is provided with FOI controllers, the obtained optimum parameters are K I1 ¼ 0:2201, K I2 ¼ 0:1612, K I3 ¼ 0:1365, k1 ¼ 0:9109, k2 ¼ 0:9302, k3 ¼ 0:9350,    R1 ¼ 7:3 Hz=p:u MW, R2 = 7.1 Hz/p.u MW, R3 = 7.2 Hz/p.u MW.

For FOPI, the optimum parameters obtained are K P1 = 0.0019, K P2 ¼ 0:0034, K P3 ¼ 0:0001, K I1 ¼ 0:1802, K I2 ¼ 0:1300, K I3 ¼ 0:1401, k1 ¼ 0:9211, k2 ¼ 0:9231, k3 ¼ 0:9365, R1 = 7.4 Hz/p.u MW, R2 = 7.2 Hz/p.u MW, R3 = 7.1 Hz/p.u MW. While for FOPID controller, optimal values are K P1 = 0.0001, K P2 ¼ 0:0025, K P3 ¼ 0:0050, K I1 ¼ 0:2302, K I2 ¼ 0:2606, K I3 ¼ 0:2826, K D1 ¼ 0:4366, K D2 ¼ 0:2621, K D3 ¼ 0:0691, k1 ¼ 0:9091, k2 ¼ 0:9582,     k3 ¼ 0:9155, l1 ¼ 0:9509, l2 ¼ 0:7522, l3 ¼ 0:1385, R1 = 7.1 Hz/

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p.u MW, R2 = 7.4 Hz/p.u MW, R3 = 5.6 Hz/p.u MW. The optimal parameters obtained for 2-DOF-FOPID controller are tabulated in Table 3. The dynamic responses of the system corresponding to the optimum values for each controller are obtained and compared in Fig. 4. Critical examination reveals that performances of 2-DOFFOPID are far better than aforesaid FO controllers in terms of less settling time, undershoots and reduced oscillations. 5.2. Performance comparison of 2-DOF-FOPID controller with conventional IO controllers In this section, several IO based controllers are considered for investigations and the corresponding dynamic responses are compared with the proposed 2-DOF-FOPID controller. ISE criterion have been used as objective function and provided in Eq. (1). When the system is provided with IO based Integral controllers, the obtained optimum parameters are K I1 ¼ 0:2877, K I2 ¼ 0:1502, K I3 ¼ 0:3511, R1 = 8.2 Hz/p.u MW, R2 = 7.1 Hz/p.u MW, R3 = 7.6 Hz/p.u MW. For PI controller, optimum values are K P1 ¼ 0:0066, K P2 ¼ 0:0198, K P3 ¼ 0:0105, K I1 ¼ 0:2311, K I2 ¼ 0:1611, K I3 ¼ 0:1032, R1 = 7.8 Hz/p.u MW, R2 = 7.3 Hz/p.u MW, R3 = 7.9 Hz/p.u MW and for PID controller, values are found to be K P1 ¼ 0:1202, K P2 ¼ 0:1188, K P3 ¼ 0:1392, K I1 ¼ 0:1902, K I2 ¼ 0:2000, K I3 ¼ 0:1571, K D1 ¼ 0:0009, K D2 ¼ 0:1003, K D3 ¼ 0:0910, R1 = 8.2 Hz/p.u MW, R2 = 7.1 Hz/p.u MW, R3 = 7.6 Hz/p.u MW. Corresponding dynamic responses are obtained and compared with 2-DOF-FOPID controller to assess the best one and are depicted in Fig. 5. The settling time and peak overshoots of Df1, Df2, DPtie12 and DPtie13, are noted and listed in Table 1. Observing the data in Table 1, it is clearly seen that the responses for I and PI are practically same with more settling time, peak overshoots and oscillations while the responses for PID controllers are somewhat better than I and PI. Whereas the performance of proposed 2-DOF-FOPID controller remarkably improves the settling time with less overshoots and oscillations compared to I, PI and PID and satisfies the robust performance of the system. Again to evaluate the dynamic performances of the proposed controller, Integral of Time multiplied Absolute Error (ITAE) criteria have been used as performance indices. The responses of the controllers optimized with ITAE for Df2 = f(t) and DPtie1-2 = f(t) are shown in Fig. 6(a) and (b). It is observed that the 2-DOF-FOPID controller is indeed superior compared to the conventional controllers. Further studies are carried out considering figure of demerits (FD) as performance index and the corresponding numerical values are reported with ISE and ITAE in Table 2. It can be observed that proposed 2-DOF-FOPID controller provides better results than conventional controllers with all three performance indices. Therefore, it can be concluded from the analysis, that the addition of concept of two-degree-of-freedom to fractional order PID controller has resulted into much better controller performance. It is also to be appreciated that firefly algorithm has optimized efficiently twenty-seven numbers of parameter simultaneously in a system. The convergence characteristics of Firefly Algorithm (FA) is compared with PSO, BFO and ABC and depicted in Fig. 7. The robustness and convergence efficiency of a FA based optimization technique can be clearly concluded from this comparison. To illustrate the superiority of the proposed method, a random load pattern as shown in Fig. 8(a) is applied to the control area1 and under such condition the performance of the proposed 2DOF-FOPID controller is examined. To obtain the optimal parameters of the controller, the firefly algorithm is executed for several runs. The result obtained after simulation are depicted in Fig. 8(b), where performance of conventional PID are compared with the proposed 2-DOF-FOPID controller. From dynamic responses, it can be observed that the proposed 2-DOF-FOPID controller achieves the best performance following random load change.

5.3. Sensitivity analysis Sensitivity analysis is carried out to check the robustness of the optimum 2-DOF-FOPID controller parameters such as K Pi , K Ii , K Di , ki , li , bi , di , N i and Ri obtained at nominal condition for wide changes in system loading condition and inertia constant (H) by ±20% from its nominal 50%. The corresponding optimum values are tabulated in Table 3. The dynamic responses for each changed loading conditions with their corresponding K Pi , K Ii , K Di , ki , li , bi , di , N i and Ri are compared with the responses for changed loading conditions with K Pi , K Ii , K Di , ki , li , bi , di , N i and Ri obtained at nominal condition. Critical examination of the frequency responses depicted in Figs. 9 and 10 clearly reveals that responses are more or less same. Thus the obtained values corresponding to nominal 50% are quite robust and need not be reset for wide changes in the system loading. Same observation is also seen in Figs. 11 and 12 for wide change in inertia constant (H). 5.4. Performance of 2-DOF-FOPID controller under higher degree of SLP A case study is added to show how the proposed controller behaves and maintain the stability when subjected to large disturbances i.e. SLP of higher magnitudes. The optimum values of R parameters and integral controller gains so obtained are K I1 ¼ 0:2051, K I2 ¼ 0:2217, K I3 ¼ 0:2922, R1 = 7.4 Hz/p.u MW, R2 = 7.7 Hz/p.u MW, R3 = 8.6 Hz/p.u MW, while the optimum values of 2-DOF-FOPID controller is tabulated in Table 3. It has been seen from the obtained results given in Fig. 13 that proposed FA based 2-DOF-FOPID controller provides much better control performances than the conventional controller even when the degree of SLP is extends from 1–5%. 6. Conclusion Proposed Two-Degree of Freedom - Fractional Order PID (2DOF-FOPID) Controller is applied for the first time in AGC of a three unequal area thermal systems considering appropriate GRC and single stage of reheat turbine. A new metaheuristic nature-inspired algorithm, so called Firefly Algorithm (FA), has been applied for simultaneous optimization of several parameters of the controllers and R parameters of the governor. Performances of several IO and FO controllers are studied and corresponding responses are compared with proposed 2-DOF-FOPID controller. Critical examination reveals that 2-DOF-FOPID controller provides more effective and promising results. Simultaneous optimization of R parameters with several controller parameters in a large multi area thermal system using powerful FA techniques also explores the use of much higher values of R as higher value of ‘R’ results into easy realisation and economical governor. Sensitivity analysis confirms the robustness of the proposed controller subjected to changing loading condition and inertia constant (H). Proposed controller also performed well against higher degree of step load perturbation and random load pattern. Appendix A Nominal parameters of the system are f = 60 Hz, Tgi = 0.08 s, Tri = 10 s, Hi = 5 s, Tti = 0.3 s, Kr = 0.5, Di = 0.00833 p.u MW/Hz, Tpi = 20 s, Kpi = 120 Hz/p.u MW, Initial loading = 50%, Tij = 0.545. References [1] Yamashita K, Taniguchi T. Optimal observer design for load frequency control. Int J Electr Power Energy Syst 1986;8:93–100.

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