Fractional order PID controller for load frequency control

Energy Conversion and Management 85 (2014) 343–353

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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Fractional order PID controller for load frequency control Swati Sondhi ⇑, Yogesh V. Hote Department of Electrical Engineering, Indian Institute of Technology Roorkee, India

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 11 February 2014 Accepted 24 May 2014

Load frequency control (LFC) plays a very important role in providing quality power both in the case of isolated as well as interconnected power systems. In order to maintain good quality power supply, the LFC should possess robustness toward the parametric uncertainty of the system and good disturbance rejection capability. The fractional order controller has the properties such as, eliminating steady state error, robustness toward plant gain variations and also good disturbance rejection. This makes the fractional order PID (FOPID) controller quite suitable for the LFC. Therefore, in this paper a FOPID is designed for single area LFC for all three types of turbines i.e., non-reheated, reheated and hydro turbines. It is observed that the FOPID controller shows better robustness toward ±50% parametric uncertainty and disturbance rejection capability than the existing techniques. Finally, the optimization of controller parameters and robustness evaluation of the control technique is done on the basis of the integral error criterion. Ó 2014 Elsevier Ltd. All rights reserved.

Keywords: Fractional order PID (FOPID) controller Single area load frequency control (LFC) Non-reheated turbine Reheated turbine Hydro turbine Robustness

1. Introduction In modern times the main sources of electrical energy are the kinetic energy of water and the thermal energy of fossil fuels and nuclear fission. This energy is converted into mechanical energy by means of prime movers and further to electrical energy by the synchronous generators. The prime mover governing systems provide a way of controlling power and frequency. This operation is commonly termed as Load Frequency Control (LFC) or Automatic Generation Control (AGC) [1]. A complete power system comprises of the generation, transmission and distribution units individually located in different areas connected through transmission lines called tie-lines. In such a system, the area frequency and the tie line power fluctuations occur very frequently. Therefore, the power system needs to be stable so that the prescribed voltage levels and synchronization is maintained during any transient disturbance. This responsibility is shouldered by the LFC. Thus, LFC is considered to be an important component of the power system design, capable of supplying reliable electric power of good quality. The state of the art LFC systems have evolved from the early analog systems to the present day digital systems. The result is, a simple, yet robust, decentralized system that is capable to control a complex, highly nonlinear and continuously changing power system. An additional insight into the requirements and performance of AGC can be seen ⇑ Corresponding author. Tel.: +91 9897152605. E-mail addresses: (Y.V. Hote).

[email protected]

(S.

Sondhi),

http://dx.doi.org/10.1016/j.enconman.2014.05.091 0196-8904/Ó 2014 Elsevier Ltd. All rights reserved.

[email protected]

in [2,3]. They provide a good review of the properties of the conventional AGC systems. A power system may be an interconnected system of multiple areas or an isolated system comprising of single service area. The LFC plays an important role in both types of power systems. A single area power system is the one which comprises of a single generator supplying power to a single service area. The function of LFC in a single area power system is to restore the frequency to the specified nominal value in case of any fluctuation. However, in case of an interconnected power system, two or more independently controlled areas are connected together. In such systems, along with frequency, generation within each area also has to be controlled. This is required to maintain the scheduled power interchange. So, the main aim of the load frequency control in multi area power systems is to regulate the frequency to the specified nominal value and to maintain the interchange power between areas at the scheduled values. In such power systems, the basic idea is to restore balance between each area load and generation. However, in case of an isolated power system, preservation of interchange power is not an issue. So, the function of LFC is just to restore frequency to the specified nominal value. To achieve this there is a need of a robust and efficient control algorithm. The Load Frequency Controller needs to play the following important roles for a power system [4]: (a) Maintain zero steady state errors in case of frequency deviation. (b) Load disturbance rejection.

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(c) Minimization of the unscheduled tie line power flow between neighboring areas. (d) Handling modeling uncertainties and system nonlinearities. Hence, the problem of load frequency control is that of objective optimization and robust control. To perform these tasks, conventionally the LFC in a single area power system uses an integral controller [5]. But the major disadvantage of this is that the system performance is limited by the integral gain of the controller. A high gain may lead to large oscillations and instability in such type of controllers. The increasing complexity of the present day power systems has provoked researchers to explore appropriate control algorithms for LFC. Many advanced control methods have been proposed in the literature so far like PI/PID control [6], optimal control [7], variable structure control [8], adaptive and self tuning control [9,10], artificial intelligence control [11–17], robust control [18–23], etc. Researchers are also working exhaustively for developing the control algorithm of LFC for both single area [24–28] as well as multi area systems [29–45]. Recently, the concepts of reduced order modeling and internal model control have also been illustrated in the literature [5,46,47]. However, all these techniques experience certain limitations, especially in the case of uncertain environment, therefore there is still lot of scope for improvement in this area. In recent years, the fractional order control has emerged as a very efficient algorithm for the systems working in uncertain environment. Therefore, in this paper a load frequency control method has been suggested for single area based on the concept of fractional order control. This paper comprises of ten sections. Section 2 of the paper states the motivation for working on this technique, Section 3 gives the objectives of the paper, in Section 4 the preliminaries of fractional order control and hardware implementation techniques of the FOPID controller is given, while in Section 5 a brief introduction of the single area load frequency control is given. Section 6 illustrates the FOPID controller design technique, Section 7 gives the simulation results while Section 8 shows the performance analysis of the proposed designed technique. In Section 9 the idea of implementing this technique to multi area LFC is stated and the conclusion of the paper is given in Section 10.

2. Motivation As mentioned in the earlier section, there are many techniques available in the literature for designing the robust controller for the LFC. The conventional PID controllers have been widely used for this problem. However, the parameter values of the various generating units of the power system like generators, turbines, governors etc., keep altering depending upon the power flow conditions. Hence, it is essential that the control strategy applied, should have good capability of handling uncertainties in the system parameters and also good disturbance rejection but the conventional PID controllers are not efficient enough to handle these issues. Moreover, in spite of the fact that there are many control techniques available in the literature, parameter uncertainty and disturbance rejection have always remained important issues and constant efforts are made to formulate new design techniques which can give better disturbance rejection and better performance in the uncertain environment. A recent development in this direction is the formulation of fractional order controllers. According to the recent literature available, the fractional order controller is known to have an excellent capability of handling parameter uncertainty. It also possess the properties of excellent disturbance rejection, robustness to high frequency noise and elimination of steady state errors. It also gives better stability in case of nonlinear

systems. The various advantages of the fractional order control can be easily found in the literature [48]. All these properties make the fractional order control a very adaptable and desirable control strategy. Application of this technique has proved beneficial in various areas like robotics, power electronics, and process control [49–51]. The versatility and efficient performance of this technique in different areas motivated us to investigate its implementation to the problem of LFC. We observed that the fractional order controller ensured good stability and dynamic performance of LFC even under uncertain environment. 3. Objectives This paper presents the application of fractional order control technique for the design of a load frequency control system. The basic objectives of this paper can be stated as:  A FOPID controller is proposed for the load frequency control of single area power system.  The parameters of the proposed controller are optimized for LFC using the integral error criterion.  The robustness of the proposed controller is investigated by inserting 50% uncertainty in each parameter of the LFC system.  Finally, the optimal performance and robustness of the proposed controller is evaluated on the basis of integral error criterion. 4. Preliminaries of fractional order control Before defining the FOPID controller, it is important to understand the fractional order integral and fractional order derivative operators. The mathematical details of these operators can be referred from [52–54]. Once the fractional operator was mathematically defined, the use of fractional calculus in control systems became widespread. In 1999, a FOPID controller was proposed. Mathematically, it is of the form [48]:

CðsÞ ¼ kp þ

ki þ k d sl : sk

ð1Þ

where k and l can take any value in the range ð0; 2Þ. If l P 2 or k P 2 the controller is transformed to a higher-order structure which is of different form in comparison to the conventional PID controller. The fractional order controller described in (1) may be regarded as the general case of the conventional PID controller. In the field of control systems any control algorithm is practically useful only if the mathematically designed controller can be converted into a hardware circuit. Hence, the analog realization of the controller is a very important aspect for any control algorithm. Therefore, the hardware implementation technique of the FOPID controller given in the literature is illustrated here in brief. A FOPID controller can be realized in the form of circuit using a combination of resistors and capacitors. Let us consider a fractional order integrator, of the form given in (2), having the frequency band of practical interest (xL, xH),

C I ðsÞ ¼

1 : sm

ð2Þ

where m is the fractional power. The form given in (2) is modified to (3) using the fractional power pole method (FPP) [52], N1 Y

  i 1 þ s=z0 ðabÞ

KI i¼0 C I ðsÞ ¼   m ¼ K I N    : Y i 1 þ xsc 1 þ s=p0 ðabÞ i¼0

ð3Þ

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h

i

y

I(s)

y y where a ¼ 10 ð10ð1mÞÞ ;b ¼ 10½10m ; p0 ¼ xc 10½20m ; z0 ¼ ap0 ;   2 3 log xpmax 0 5 þ 1; xmax ¼ 100xH ; K I ¼ 1 ; N ¼ integer 4 logðabÞ xmc

R0

Rp

y ¼ approximation error of FPP in decibels;usually assumed 1db; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   y xc ¼ xL 10ð10mÞ  1 and xc is the corner frequency of the FPP:

R1

Rn

V(s) C0

Cn

C1

ð4Þ Fig. 2. Analogue realization of the fractional differentiator.

The fractional order integrator given in (3) can be realized into an analog circuit as shown in Fig. 1 [55]. In Fig. 1, I(s) is the current entering the integrator, V(s) is the voltage across the integrator, R0, R1, . . ., Rn denote resistors and C0, C1, . . ., Cn denote capacitors, such that

KI Ri ¼

N 1h Y



1

ðabÞðijÞ a

i

j¼0 N h Y

ðijÞ

1  ðabÞ

i

; Ci ¼

1 i

ðabÞ p0 Ri

;

i ¼ 0; 1; 2; . . . ; n:

ð5Þ

j¼0

C D ðsÞ ¼ sm ;

ð6Þ

where m is the fractional power.

C D ðsÞ ¼ K D

N    Y i 1 þ s=z0 ðabÞ

m

s

¼ KD

xc

i¼0 N  Y

ð7Þ

  ; i 1 þ s=p0 ðabÞ

i¼0 y where p0 ¼ az0 ; z0 ¼ xc 10½20m ;

and K D ¼ xm c :

ð8Þ

The fractional order differentiator in (7) can be realized into a circuit as shown in Fig. 2 [55]. In Fig. 2, I(s) is the current entering the differentiator, V(s) is the voltage across the differentiator, Rp, R0, R1, . . ., Rn denote resistors and C0, C1, . . ., Cn denote capacitors, such that N h  i Y ðijÞ 1  aðabÞ

Ci ¼

KD

j¼0 i

p0 ðabÞ

N h Y

1  ðabÞ

ðijÞ

i

; Rp ¼ K D

1 i

ðabÞ p0 C i

N 1 N      1 Y Y i i 1 þ s=z0 ðabÞ 1 þ s=z0 ðabÞ C B   B C TI i¼0 C: CðsÞ ¼ kp B þ ½T D sK D i¼0 B1 þ s K I Y N  N     C Y @ A i i 1 þ s=p0 ðabÞ 1 þ s=p0 ðabÞ i¼0

i¼0

The structure of the FOPID controller is as shown in Fig. 3. To design a FOPID controller the proportional element, the fractional integrator and the fractional differentiator are individually designed and then placed in the structure shown in Fig. 3. 5. Model of single area load frequency control The power system is usually a large scale system having very complex nonlinear dynamics. In case of the load frequency control, the power system taken into consideration is subjected to small changes in load. Therefore, it can be appropriately represented by the linear model, linearized around the operating point. This model of LFC consists of a governor, a turbine, load and machine, and droop characteristics. The droop characteristics is a type of feedback gain used to improve the damping properties of the power system. Here we consider the case of single service area where the power is supplied by a single generator. The linear model of such a system can be represented as shown in Fig. 4 [4]. The nomenclature of the different parameters used in the system is given in Table 1. Mathematically, the overall system can be represented in transfer function form as [4,56],

GðsÞ ¼

and

Gp ðsÞGt ðsÞGg ðsÞ : 1 þ Gp ðsÞGt ðsÞGg ðsÞ=R

ð12Þ

where

j¼0

Ri ¼

0

ð11Þ

Similarly if we consider a fractional order differentiator of the form in (6), with the frequency band of practical interest (xL, xH), it can be modified to the form (7) using the fractional power zero (FPZ) technique [52]



is a fractional order differentiator with 0 < mD < 1. The controller in (10) can be approximated as (11) using FPP and FPZ techniques.

i ¼ 0; 1; 2; . . . ; n:

ð9Þ

1 ¼ droop characteristics; R

ð13Þ

Now consider the FOPID controller represented in (1). This controller can be represented in the form

Gg ðsÞ represents the governor dynamics : Gg ðsÞ ¼

     TI 1 þ ½T D sðsmD Þ : CðsÞ ¼ kp 1 þ m s s I

Gt(s) represents the turbine dynamics:

1 : TGs þ 1

ð10Þ



 where TsI is a first order integrator, sm1I is a fractional order integrator with 0 < mI < 1, [TDs] is a first order differentiator and, ðsmD Þ R0

R1

kp

Rn

E(s)

I(s)

+ k i /s λ

U(s) +

V(s)

C0

C1

Fig. 1. Analogue realization of the fractional integrator.

Cn

k d /s µ Fig. 3. Structure of PIk Dl controller.

+

ð14Þ

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Droop Characteristics

Pd

u

-

-

PG

XG Governor

Turbine

f Load & Machine

Fig. 4. Linear model of a single area power system.

Table 1 Nomenclature of power system parameters.

DP d Kp Tp TT TG R Tr c Tw Df DP G DX G

Load disturbance (p.u. MW) Electric system gain Electric system time constant (s) Turbine time constant (s) Governor time constant (s) Speed regulation due to governor action(Hz/p.u. MW) Constant of reheat turbine Percentage of power generated in the reheat portion Time constant of hydro turbine Incremental frequency deviation (Hz) Incremental change in generator output (p.u. MW) Incremental change in governor valve position

1 : TT s þ 1

ð15Þ

cT r s þ 1 : ðT r s þ 1ÞðT T s þ 1Þ

ð16Þ

 In case of Non-reheated turbine : Gt ðsÞ ¼

 In case of Reheated turbine : Gt ðsÞ ¼

1  Tws  In case of Hydro Turbine : Gt ðsÞ ¼ : 1 þ 0:5T w s

1 : TPs þ 1

Hence, the model can be represented as:

¼

Gp ðsÞGt ðsÞGg ðsÞ 1 þ Gp ðsÞGt ðsÞGg ðsÞ=R KP : ðT P s þ 1ÞðT T s þ 1ÞðT G s þ 1Þ þ K P =R

250 : s3 þ 15:88s2 þ 42:46s þ 106:2

ð20Þ

5.1.2. Reheated turbine In this type of turbine, the steam, on leaving the high pressure section, returns to the boiler, where it is passed through a re-heater before returning to the intermediate pressure section. The parameter values for the case of reheated turbine are [46]: KP = 120, TP = 20, TT = 0.3, TG = 0.08, R = 2.4, Tr = 4.2, c = 0.35. The plant transfer function for the power system using reheated turbine with droop characteristics, represented by GR, is,

ð17Þ

ð18Þ

GðsÞ ¼

GNR ðsÞ ¼

GR ðsÞ ¼

Gp ðsÞ represents the power systemðload &machineÞdynamics : Gp ðsÞ ¼

turbine, the various parameter values are [46]: KP = 120, TP = 20, TT = 0.3, TG = 0.08, R = 2.4. Hence, plant model of a power system using non-reheated turbine with droop characteristics, represented by GNR, becomes,

87:5s þ 59:52 : s4 þ 16:12s3 þ 46:24s2 þ 48:65s þ 25:3

ð21Þ

5.2. Hydro turbine [1] The performance of this type of turbine depends on various properties of the water column feeding the turbine, like water inertia, water compressibility and pipe wall elasticity in the penstock. In case where the hydro turbine is used the parameter values are [46]: KP = 1, TP = 6, Tw = 4, TG = 0.2, R = 0.05. Therefore, the plant model for the case of power system using hydro turbine with droop characteristics, represented by GH is,

ð19Þ GH ðsÞ ¼

1:667s þ 0:4167 : s3 þ 5:667s2  29:92s þ 8:75

ð22Þ

In the model given above, the turbine dynamics depends on the type of turbine used in the power system. The turbine used in the system may be a steam turbine or a hydro turbine. The mathematical modeling of the power system using different turbines is given below.

The models given in (20)–(22) are used for the FOPID controller design in the later section.

5.1. Steam turbines [1]

Table 2 Solution values of controller parameters.

A steam turbine converts stored energy of high pressure and high temperature steam into rotating energy, which in turn is converted into electrical energy by the generator. They can further be of two types: non-reheated turbine and reheated turbine. 5.1.1. Non-reheated turbine A non-reheated turbine is the one which does not have an intermediate pressure section or a re-heater. In this case the steam directly passes to the low pressure section. For the non-reheated

ki

kd

k

l

Non-reheated turbine FOPID 2 PID 0.4036

3 0.6356

0.4 0.1832

0.9 1

1.15 1

Reheated turbine FOPID 6 PID 2.7935

4 1.2735

1 0.7866

0.9 1

1.2 1

Hydro turbine FOPID 18.5 PID 18.85

0.19 0.152

2.8 1.8124

1.02 1

1.1 1

kp

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The real and the imaginary parts of (28) are obtained as: Real Part:

6. FOPID design for single area load frequency control In this section the FOPID controller design technique is illustrated. This technique involves defining a stability region in the ðkP  ki Þ plane for a fixed value of kd for all k 2 ð0; 2Þ and l 2 ð0; 2Þ and then optimizing the values of kp ; ki ; k and l on the basis of the integral error criterion [57]. The design of FOPID controller for all three types of turbines is illustrated here. 6.1. Non-reheated turbine

n   po 3þk Cos ð1 þ kÞ x þ 42:46x1þk 2 n  po  15:88x2þk þ xk ð106:2 þ 250kP Þ þ Cos k 2 n  po þ 250ki þ 250kd xkþl Cos ðk þ lÞ 2 ¼ 0:

ð30Þ

Imaginary Part: For the case of non-reheated turbine consider the plant model in (20) and the FOPID controller as given in (1). The characteristic equation of the control system becomes,

Pðs; kp ; ki ; kd ; k; lÞ ¼ 1 þ GNR ðsÞCðsÞ ¼ 0:

ð23Þ

Substituting (1) and (20) in (23) we get,

 250 Pðs; kp ; ki ; kd ; k; lÞ ¼ 1 þ 3 s þ 15:88s2 þ 42:46s þ 106:2

 ki  kp þ k þ kd sl ¼ 0: s

ð24Þ

 Pðs;kp ; ki ;kd ; k; lÞ ¼ s3þk þ 15:88s2þk þ 42:46s1þk þ 106:2 þ 250kp sk þ 250kd skþl þ 250ki ¼ 0:

ð25Þ

n   po 3þk Sin ð1 þ kÞ x þ 42:46x1þk 2 n  po  þ Sin k 15:88x2þk þ xk ð106:2 þ 250kP Þ 2 n  po þ 250kd xkþl Sin ðk þ lÞ 2 ¼ 0:

ð31Þ

Observe that here two equations are obtained in terms of three variables. So there are two possibilities for calculating the expressions of kP, ki and kd i.e.,  either fix the value of kd arbitrarily and plot the CRB and the RRB in the (kP  ki) plane for various k 2 ð0; 2Þ and l e (0, 2) or,  fix the value of ki and plot the CRB and RRB in the (kP  kd) plane for various k 2 ð0; 2Þ and l e (0, 2).

The closed-loop system expressed by (23) is stable if (25) has none of the roots in the right half of the s-plane. The stability domain, say s0 , can be defined as the region for which, if the parameters ðkp ; ki ; kd ; k; lÞ 2 s0 , then no roots of (25) lie in the right half s-plane. Since roots of the equation obtained in (25) are real as well as complex. Therefore, the stability region s, for a FOPID controller is marked by the intersection of the real root boundary (RRB) and the complex root boundary (CRB). The real root boundary is the boundary where the real roots cross the imaginary axis while the complex root boundary is where the pair of complex roots cross the imaginary axis. Mathematically, these boundaries are described as

Here, we fix an arbitrary value of kd = 0.4. In the next step substituting kd = 0.4 and solving (30) and (31) simultaneously we get the expressions of kp and ki as, n   1 p p  p x3þk Cos ð1 þ kÞ þ 42:46x1þk Cos ð1 þ kÞ kp ¼ k 2 2 250x Cos k 2  p  p  p 2þk k kþl 15:88x Cos k þ 106:2x Cos k þ 100x Cos ðk þ lÞ 2 2 2 ð32Þ þ250ki g;

RRB : Pð0; kp ; ki ; kd ; k; lÞ ¼ 0; for x 2 ð0; 1Þ;

ki ¼

CRB : Pðjx; kp ; ki ; kd ; k; lÞ ¼ 0; for x 2 ð0; 1Þ:

The expression for kP and ki are obtained in terms of k, l and x. For (32) and (33) if k and l is fixed from within the set (0, 2) and x is varied from (0, 1), the curve traced in (kP  ki) plane is called the CRB. The region obtained by the intersection of the CRB and the RRB in the (kP  ki) plane is called the global stability region for that combination of k and l i.e., for that combination of k and l all the values of kP and ki within this region are capable of

Substituting s = 0 in (25), the RRB is obtained as ki = 0. To obtain the CRB, replace s = jx in (25), we get, 3þk

Pðjx; kp ; ki ; kd ; k; lÞ ¼ ðjxÞ

2þk

þ 15:88ðjxÞ

þ 42:46ðjxÞ

k

1þk

þ ð106:2 þ 250kp ÞðjxÞ þ 250kd ðjxÞ þ 250ki ¼ 0:

kþl

ð26Þ

Solve (26) using mathematical identity (27),

kp kp k j ¼ cos þ j sin : 2 2

x3þk þ 42:46x1þk þ 100xkþl Sin  250Sin k p2

 p l 2

ð33Þ

:

90 80

ð27Þ

70

Thus we get,

Express (28) in the form (29) and equate the real and imaginary parts to 0:

Pðx; kP ; ki ; kd ; k; lÞ ¼ RefPðx; kP ; ki ; kd ; k; lÞg þ jImfPðx; kP ; ki ; kd ; k; lÞg ¼ 0:

60

Global

Stability

50

Region

Ki

n    p po 3þk Cos ð1 þ kÞ þ jSin ð1 þ kÞ x þ 42:46x1þk 2 2 n  p  po  þ jSin k 15:88x2þk þ xk ð106:2 þ 250kP Þ þ Cos k 2 2 n   p po þ jSin ðk þ lÞ þ 250ki ¼ 0: þ 250kd xkþl Cos ðk þ lÞ 2 2 ð28Þ

40 Kp=2 Ki=3

30 20 10 0 -50

-40

-30

-20

-10

0

10

Kp

ð29Þ

Fig. 5. Stability region obtained for k ¼ 0:9 and

l ¼ 1:15.

20

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stabilizing the system. Hence, by varying k 2 ð0; 2Þ and l e (0, 2), we can obtain the global stability regions for all possible combinations ofk and l. The combination of k and l that gives the largest stability region is selected. Once the stability region is obtained, the values of kP and ki from within this region that give the most optimum performance are chosen using the integral error criterion. The stability region obtained in this case is shown in Fig. 5. The controller parameters in this case are kP = 2, ki = 3, kd = 0.4, k ¼ 0:9 and l = 1.15. The point kP = 2 and ki = 3 is shown in the Fig. 5. by ‘+’. Hence for the case of non-reheated turbines the controller is obtained as:

ki ¼



 p þ 16:12x3þk sin ð3 þ kÞ 2 2   p p 2þk 1þk þ 46:24x sin ð2 þ kÞ þ x sin ð1 þ kÞ 48:65 þ 87:5kp 2 2  p  p þ xk sin k 25:3 þ 59:52kp þ 87:5kd x1þkþl sin ð1 þ k þ lÞ 2 2   p þ 59:52kd xkþl sin ðk þ lÞ þ 87:5ki x ¼ 0: ð38Þ 2

x4þk sin ð4 þ kÞ

p

Solving (37) and (38) simultaneously, kP and ki are obtained as,

      87:5x5þ2k þ 4046x3þ2k  2213:75x1þ2k þ 7656:25kd x2þ2kþl sin l p2 þ 5208kd x1þ2kþl sin ðl  1Þ p2  959:4625x3þ2k þ 2895:648x1þ2k þ 5208kd x1þ2kþl sin ð1 þ lÞ p2 þ 3542:6304kd x2kþl sin l p2     

; 87:5x 87:5x1þk cos ð1 þ kÞ p2 þ 59:52xk cos k p2 þ 5208x1þk sin ð1 þ kÞ p2 þ 3542:6304xk cos k p2

ð39Þ

 kp ¼































x4þk cos ð4 þ kÞ p2 þ 16:12x3þk cos ð3 þ kÞ p2 þ 46:24x2þk cos ð2 þ kÞ p2 þ 48:65x1þk cos ð1 þ kÞ p2 þ 25:3xk cos k p2 þ 87:5kd x1þkþl cos ð1 þ k þ lÞ p2 þ 59:52kd xkþl cos ðk þ lÞ p2 þ 59:52ki    : 87:5x1þk cos ð1 þ kÞ p2 þ 59:52xk cos k p2

ð40Þ

CðsÞ ¼ 2 þ

3 þ 0:4s1:15 : s0:9

ð34Þ

Note: Note that here the value of kd is arbitrarily chosen as 0.4 and can be chosen other than 0.4 also. It is also possible to fix an arbitrary value of ki instead of kd and then calculate the values of kP and kd using the same procedure. The combination of k and l that gives the largest stability region is selected because it provides the largest number of possible values of kP and ki which are capable of stabilizing the system. The optimization of the values of kP and ki from within the obtained stability region, is done on the basis of the integral error criterion i.e., the combination of kP and ki values that gives the least error is chosen. 6.2. Reheated turbine The plant model for the case of reheated turbine power system given in (21) and the FOPID controller in (1) is now considered. The same controller design procedure is followed as described for the case of non-reheated turbines. The characteristic polynomial becomes,

  ki P s;kp ;ki ;kd ; k; l ¼ 1 þ kp þ k þ kd sl s

 87:5s þ 59:52  4 ¼ 0: ð35Þ 3 2 s þ 16:12s þ 46:24s þ 48:65s þ 25:3

Pðs; kp ; ki ; kd ; k; lÞ ¼ s

4þk

þ 16:12s k

3þk

þ 46:24s

þ 25:3s þ 87:5kp s

1þk

2þk

þ 48:65s

þ ki 87:5s þ 87:5kd s1þkþl

Separating the real and imaginary parts of (36) by substituting s ¼ jx, we get, Real Part:



p

 p þ 16:12x3þk cos ð3 þ kÞ 2 2   p p 48:65 þ 87:5kp þ 46:24x2þk cos ð2 þ kÞ þ x1þk cos ð1 þ kÞ 2 2  p  p 25:3 þ 59:52kp þ 87:5kd x1þkþl cos ð1 þ k þ lÞ þ xk cos k 2 2  p ð37Þ þ 59:52kd xkþl cos ðk þ lÞ þ 59:52ki ¼ 0: 2 Imaginary Part:

CðsÞ ¼ 6 þ

4 þ s1:2 : s0:9

ð41Þ

6.3. Hydro turbine When a hydro turbine is used, the plant model of the power system is as given in (22). From Eqs. (22) and (1) the characteristic equation of the hydro turbine power system becomes as given in (42)

  ki P s; kp ; ki ; kd ; k; l ¼ 1 þ kp þ k þ kd sl s

 1:667s þ 0:4167 ¼ 0:  3 s þ 5:667s2  29:92s þ 8:75

ð42Þ

 P s;kp ; ki ; kd ; k; l ¼ s3þk þ 5:667s2þk  29:92s1þk þ 8:75sk  1:667kp s1þk þ 0:4167kp sk  1:667ki s þ 0:4167ki  1:667kd s1þkþl þ 0:4167kd skþl ¼ 0:

ð43Þ

1þk

þ 59:52kp sk þ 59:52ki þ 59:52kd skþl ¼ 0: ð36Þ

x4þk cos ð4 þ kÞ

By tracing the global stability region using the method explained previously, the controller parameters are obtained as: kP = 6, ki = 4, kd = 1, k ¼ 0:9 and l = 1.2. i.e.,

Separating (43) into real and imaginary parts by substituting s ¼ jx, Real Part:



 p þ 5:667x2þk cos ð2 þ kÞ 2 2  p 1þk 29:92 þ 1:667kp  x cos ð1 þ kÞ 2  p þ xk cos k 8:75 þ 0:4167kp þ 0:4167ki 2  p  1:667kd x1þkþl cos ð1 þ k þ lÞ 2  p kþl þ 0:4167kd x cos ðk þ lÞ ¼ 0: 2

x3þk cos ð3 þ kÞ

p

Imaginary Part:

ð44Þ

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  p p x3þk sin ð3 þ kÞ þ 5:667x2þk sin ð2 þ kÞ 2 2  p 1þk 29:92 þ 1:667kp  x sin ð1 þ kÞ 2  p 8:75 þ 0:4167kp  1:667ki x þ xk sin k 2  p  1:667kd x1þkþl sin ð1 þ k þ lÞ 2  p þ 0:4167kd xkþl sin ðk þ lÞ 2 ¼ 0:

controller has a good capability to handle the parameter uncertainties and disturbance rejection. 7. Simulation results

ð45Þ

Solving (44) and (45) simultaneously we get, From Eqs. (46) and (47), the controller parameters obtained are:

ki ¼

In this section, the simulation results obtained by the implementation of the fractional order controller, designed in the previous section, are presented. The response of the FOPID controller is shown for both nominal as well as uncertain cases. It is observed that the FOPID controller gives better response for the nominal as well as the uncertain case for all the three types of turbines. Fig. 7a shows the response obtained by implementing the FOPID controller to the nominal non-reheated turbine power sys-

               0:4167x3þ2k  12:4676x1þ2k þ 0:6946kd x1þ2kþl sin ð1  lÞ p2 þ 0:1736kd x2kþl sin l p2 1:667x4þ2k sin ð3 þ kÞ p2 cos ð1 þ kÞ p2  9:4468x3þ2k sin ð2 þ kÞ p2 cos ð1 þ kÞ p2 þ 48:876x2þ2k sin ð1 þ kÞ p2 Þcos ð1 þ kÞ p2  14:586kd x1þ2k sin k p2 cos ð1 þ kÞ p2 þ 2:7788kd x2þ2kþl sin ð1 þ k þ lÞ p2 cos ð1 þ kÞ p2 0:6946kd x1þ2kþl sin ðk þ lÞ p2 cos ð1 þ kÞ p2    

; 1:667x 1:667x1þk cos ð1 þ kÞ p2 þ 0:4167xk cos k p2 þ 0:1736xk sin k p2

ð46Þ

 kp ¼

























x3þk cos ð3 þ kÞ p2 þ 5:667x2þk cos ð2 þ kÞ p2  29:92x1þk cos ð1 þ kÞ p2 þ 8:75xk cos k p2 þ 0:4167ki  1:667kd x1þkþl cos ð1 þ k þ lÞ p2 þ 0:4167kd xkþl cos ðk þ lÞ p2    : 1:667x1þk cos ð1 þ kÞ p2 þ 0:4167xk cos k p2 ð47Þ

kP ¼ 18:5; ki ¼ 0:19; kd ¼ 2:8; k ¼ 1:02 and CðsÞ ¼ 18:5 þ

0:19 þ 2:8s1:1 : s1:02

l ¼ 1:1: i:e:; ð48Þ

The values of the controller parameters are given in Table 2. 6.4. Robustness In the modern day complex power systems, the parameter uncertainty is a major issue. Therefore, it is very essential that the control technique being applied to the LFC should be robust toward the parametric uncertainty of the system. So, once the controller is designed, it is also important to check whether the controller has the capability to deal with the parameter uncertainties or not. To investigate the robustness of the controller the power system model is considered as an uncertain model as shown in Fig. 6 [5]. For this, a ± 50% lower and upper bound uncertainties in the plant parameters are considered i.e.,  For non-reheated turbines: Kp e [60, 180], TP e [10, 30], TT e [0.15, 0.45], TG e [0.04, 0.12] and R e [1.2, 3.6] .  For reheated turbines: Kp e [60, 180], TP e [10, 30], TT e [0.15, 0.45], TG e [0.04, 0.12], Tr e [2.1, 6.3], c e [0.175, 0.525] and R e [1.2, 3.6].  For hydro turbines: Kp e [0.5, 1.5], TP e [3, 9], Tw e [2, 6], TG e [0.1, 0.3] and R e [0.025, 0.075]. The proposed FOPID controller is implemented to the above described uncertainties of the power system. The response of the FOPID controller obtained for all three types of turbines is shown in Figs. 7a–9c and the comparison of its performance to the existing techniques is shown in Tables 3–5. The details of the responses of FOPID and the performance evaluation is given in Section 7 and 8 respectively. From the results obtained, it is observed that the proposed FOPID controller gives better results, in case of parameter uncertainties and shows better disturbance rejection, than the IMC-PID designed in [46,47]. This highlights that the FOPID

tem. In this case a disturbance of 0.01 is added to the system at time 2 s. From Fig. 7a it is observed that the FOPID controller proposed here gives the fastest disturbance rejection as compared to the techniques illustrated in [46,47] i.e., the FOPID proposed here gives faster disturbance rejection than the internal model control based PID suggested by Tan (Tan’s IMC-PID), the controller suggested by Tan using the second order plus delay time model (Tan’s SOPDT), and the controllers designed using the model order reduction techniques like Pade approximation (Liu Pade), Routh approximation(Liu Routh) and second order plus delay time(Liu SOPDT). Hence it is evident from the figure that the FOPID controller performs much better than the techniques proposed by Tan [46] and Saxena [47]. Figs. 7b and 7c show the response of the FOPID controller in case of ±50% parametric uncertainty. In these cases parametric uncertainty is added to the system, along with a disturbance of 0.01 at time 2 s. It can be seen from Figs. 7b and 7c that in case of ±50% parametric uncertainty also, the proposed FOPID controller gives better disturbance rejection than the existing techniques. It is clear from the figures that the capacity of the proposed FOPID to handle disturbances under uncertain environment is also much better than the IMC-PID proposed in [46,5] and the controller designed using the reduced order concept in [47]. For the reheated as well as the hydro turbine power systems also similar results are observed. The responses obtained for the reheated turbine and hydro turbine power systems for the nominal as well as the uncertain cases are illustrated in Figs. 8a–8c and Figs. 9a–9c, respectively. From Fig. 8a–8c, it can be observed that the concept of reduced order modeling suggested by Saxena in [47] gives a large steady state error in both nominal as well as the ±50% uncertain cases. The responses in Figs. 8a–8c show that the FOPID controller has a much better disturbance rejection capability than the techniques suggested in [46,47], hence it is the most suitable controller for the case of reheated turbine power systems. In case of the hydro turbine power systems, it is seen that the transfer function model has unstable poles. So, in such cases, the controller suggested in [47] is not applicable. Moreover, the IMC-PID technique proposed in [46] is capable of stabilizing the system but from

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1 RTG

δ5 + u

1 TG

-

1 s

+

-

1 TT

δ4

+

1 s

Kp Tp

δ3 Governer

+

1 s

-

1 Tp

δ2 Power System

Turbine

+

δ1 Fig. 6. Linear model of power system with uncertain parameters.

Table 3 Comparison of the performance indices for the non-reheated turbine power system. Design method

50% Uncertainty case Nominal

FOPID (proposed approach) Tan’s IMC-PID[46] Liu SOPDT[47] Liu Routh[47] Liu Pade[47]

Lower bound

Upper bound

ISE

IAE

ITAE

ISE

IAE

ITAE

ISE

IAE

ITAE

1.364  105 0.0001377 0.0008232 0.0008702 0.0008499

0.003877 0.01573 0.08061 0.08166 0.08183

0.0121 0.046 0.4808 0.4832 0.484

2.221  105 0.0001931 0.0008662 0.0009179 0.0009027

0.005813 0.01968 0.08247 0.08329 0.08336

0.01804 0.06174 0.4857 0.4871 0.4871

8.981  106 0.0000959 0.0008091 0.0008959 0.0008468

0.003828 0.0157 0.07825 0.08118 0.07998

0.01252 0.05188 0.4738 0.4815 0.4794

ISE = Integral squared error, IAE = Integral absolute error, ITAE = Integral time-Weighted absolute error.

Table 4 Comparison of the performance indices for the reheated turbine power system. Design method

50% Uncertainty case Nominal

FOPID (proposed approach) Tan’s IMC-PID[46] Liu SOPDT[47] Liu Routh[47] Liu Pade[47]

Lower bound

Upper bound

ISE

IAE

ITAE

ISE

IAE

ITAE

ISE

IAE

ITAE

1.58  105 6.17  105 7.735  106 2.52  106 4.467  106

0.0040 0.01119 0.003 0.0044 0.0066

0.013 0.04252 0.02158 0.03 0.04969

1.513  105 4.842  105 1.487  107 8.644  107 1.294  106

0.004462 0.007893 0.001312 0.002261 0.003196

0.01589 0.0232 0.01045 0.01238 0.02026

1.55  105 4.312  106 7.083  107 2.032  106 3.966  106

0.003545 0.011 0.003 0.0044 0.007

0.011 0.04874 0.02396 0.03205 0.05374

Table 5 Comparison of the performance indices for the hydro turbine power system. Design method

50% Uncertainty case Nominal

FOPID (proposed approach) Tan’s IMC-PID[46] Liu SOPDT[47] Liu Routh[47] Liu Pade[47]

Lower bound

Upper bound

ISE

IAE

ITAE

ISE

IAE

ITAE

ISE

IAE

ITAE

0.0002045 0.0002774 NA NA NA

0.05109 0.06574 NA NA NA

0.4404 0.6523 NA NA NA

0.0001324 0.000193 NA NA NA

0.0497 0.06565 NA NA NA

0.552 0.8451 NA NA NA

0.0004703 0.0005289 NA NA NA

0.07054 0.08234 NA NA NA

0.6012 0.7932 NA NA NA

Figs. 9a–9c it can be seen that the proposed FOPID controller gives better performance than the IMC-PID controller suggested in [46]. Hence, from Figs. 7a–9c it is clear that the disturbance rejection capability of the proposed FOPID controller, in the nominal as well as the uncertain operating conditions, is better than the existing techniques for all three types of turbine systems. This property of the FOPID controller makes it very suitable for the LFC because the main task of the LFC is to handle frequency

fluctuations occurring in the power system even when the parameters of the system are not constant. The disturbances introduced into the system here represent the fluctuations occurring in the power system while the ±50% variation in the various parameters represents the parametric uncertainty occurring in the system. Hence, from the obtained results it is observed that the FOPID gives better performance than the existing control techniques.

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0.01

x 10

Liu SOPDT Liu Routh Liu Pade Tan's SOPDT FOPID Tan's IMC-PID

Tan's IMC-PID

12 10

FOPID

Amplitude

8

0 FOPID

-0.01

Amplitude

14

6

Steady State Error Tan's IMC-PID

-0.02 -0.03

4 -0.04 2

-0.06

-2 -4

FOPID Tan's IMC-PID Liu Pade Liu Routh Liu SOPDT

-0.05

0

0

2

4

6

8

10

12

14

16

18

20

Time 0

1

2

3

4

5

6

7

8

9

10

Time

Fig. 8a. Responses of a reheated turbine power system using various controllers for nominal parameters.

Fig. 7a. Responses of a non-reheated turbine power system using various controllers for nominal parameters. 0.005 0

0.01

FOPID Steady State Error

-0.005

Amplitude

Amplitude

0.005

0

LIU SOPDT Liu Routh Liu Pade Tan's SOPDT FOPID Tan's IMC-PID

Tan's IMC-PID

-0.01

-0.02

FOPID

0

1

2

3

4

5

6

7

8

9

-0.035 0

2

4

6

8

10

12

14

10

20

0.01

0

-3

FOPID

Amplitude

0 -2 -4

Steady State Error Tan's IMC-PID

-0.02

-0.03

-6

FOPID Tan's IMC-PID Liu Pade Liu Routh Liu SOPDT

-0.04

-8 FOPID

Liu Routh Liu SOPDT Liu Pade Tan's SOPDT FOPID Tan's IMC-PID

-10 Tan's IMC-PID

-12 -14

18

Fig. 8b. Responses of a reheated turbine power system using various controllers for lower bound parameters.

-0.01

Amplitude

16

Time

Fig. 7b. Responses of a non-reheated turbine power system using various controllers for lower bound.

x 10

FOPID Tan's IMC-PID Liu Pade Liu Routh Liu SOPDT

-0.03

Time

2

Tan's IMC-PID

-0.015

-0.025

-0.005

-0.015

-0.01

0

1

2

3

4

5

6

7

8

9

-0.05 0

2

4

6

8

10

12

14

16

18

20

Time Fig. 8c. Responses of a reheated turbine power system using various controllers for upper bound parameters. 10

Time Fig. 7c. Responses of a non-reheated turbine power system using various controllers for upper bound.

8. Performance analysis In order to analyze the performance of the proposed controller quantitatively, the various integral error indices are calculated.

These indices help in making a clear comparative study of the performance of the proposed technique with that of the existing techniques. The responses presented in the previous section show that the FOPID controller gives a much better disturbance rejection than the one designed in [5,46,47]. Further, the performance of the controllers can be quantitatively measured using the various integral error criterion like Integral Absolute Error, Integral Squared

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1

x 10

-3

0

Amplitude

-1 -2 FOPID

-3 -4 Tan's IMC-PID

-5 -6 -7

0

5

10

15

20

25

30

35

40

45

50

suggested in [46,47], with the proposed technique, for the nonreheated turbine is given in Table 3. The performance analysis for the case of the reheated and hydro turbine are given in Tables 4 and 5, respectively. The comparison indicates that the proposed FOPID controller performs better than the existing methods in nominal as well as uncertain conditions. Therefore, on comparing these results, it is observed that the FOPID gives better disturbance rejection as well as better control under uncertain operating environment in all the three turbine cases. From the results obtained in this section, it is seen that the controller designed using the proposed approach is more robust in nature toward disturbances than the existing controllers, in both nominal as well as uncertain operating conditions. Thus, the main objective of designing a controller i.e., delivering optimum performance under uncertain environment, has been successfully achieved.

Time Fig. 9a. Responses of a hydro turbine power system using various controllers for nominal parameters.

0.5

x 10

-3

0 -0.5

Amplitude

-1 -1.5

ACEi ¼ DPtiei þ Bi Dfi

ð49Þ

where Bi is the frequency bias setting. The feedback control of the area i takes the form,

-2.5

ui ¼ K i ðsÞACEi

-3 Tan's IMC-PID

-3.5

0

5

10

15

20

25

30

35

40

45

50

Time Fig. 9b. Responses of a hydro turbine power system using various controllers for lower bound parameters.

x 10

ð50Þ

A decentralized controller can be tuned assuming that there is no tie-line exchange power, i.e., DPtiei = 0. In this case the local feedback control will be

-4

2

The design of FOPID load frequency controller proposed in this paper can be extended to multi-area power systems also. The difference between LFC of multi-area and single area is that for multi area systems the net interchange through the tie-line should return to the scheduled values and the frequency of each area should return to its nominal value. So a composite measure, called area control error (ACE), is used as the feedback variable. For area i, the ACE is defined as

FOPID

-2

-4.5

9. Multi-area load frequency control

ui ¼ K i ðsÞBi Dfi

So it is clear that to tune a decentralized load frequency controller, one just needs to multiply the plant model by the local bias coefficient, and then follow the same procedure as in single-area LFC-PID tuning, i.e., tune FOPID controller for

-3

Pi ðsÞ ¼

0

ð51Þ

Ggi ðsÞGti ðsÞGpi ðsÞ Bi 1 þ Ggi ðsÞGti ðsÞGpi ðsÞ=Ri

ð52Þ

Thus the fractional order controller for load frequency controller for each area can be tuned independently using the proposed approach.

Amplitude

-2

10. Conclusion

FOPID

-4

-6 Tan's IMC-PID

-8

-10

-12 0

5

10

15

20

25

30

35

40

45

50

Time Fig. 9c. Responses of a hydro turbine power system using various controllers for upper bound parameters.

Error and Integral Time-Weighted Absolute Error. The comparison of the performance indices, for the controller design techniques

With the ever increasing demand of electric power there is a great need to have an efficient LFC system which can handle the system parameter uncertainty. In this paper, a Load Frequency Control design technique based on the concept of fractional order control for all the three types of turbines i.e., non-reheated, reheated and the hydro turbines has been presented. Further, the robustness of the FOPID controller for the parametric uncertainty has also been evaluated. Finally, the performance of the proposed approach is compared to that of the IMC-PID controller proposed recently. It is seen that the performance of the FOPID controller is much better than the IMC-PID controller as well as than the controllers designed using the concept of reduced order modeling. The results also show that the proposed FOPID controller has a good capability of disturbance rejection and handling parametric uncertainty and hence is very suitable for the problem of LFC.

S. Sondhi, Y.V. Hote / Energy Conversion and Management 85 (2014) 343–353

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