978-1-4244-4547-9/09/$26.00 ©2009 IEEE TENCON 2009

A Novel Robust Load Frequency Controller for a Two Area Interconnected Power System using LMI

and Compact Genetic Algorithms

Chamnan Koisap Electrical Engineering Department

Faculty of Engineering, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand

Somyot Kaitwanidvilai Center of Excellence for Innovative Energy Systems King Mongkut's Institute of Technology Ladkrabang,

Bangkok 10520, Thailand drsomyotk@gmail.com

Abstract— This paper proposes a new technique for designing a fixed-structure robust load frequency controller for a two area interconnected power system. The proposed technique uses Linear Matrix Inequality (LMI) method to form an initial solution, and then the global search algorithm, Compact Genetic Algorithm (cGA), is adopted for evaluating the final solution. By combining both techniques, LMI and cGA, a better solution in terms of robust performance can be achieved. Infinity norm from disturbances to states are formulated as the cost function in our optimization. The performance of the designed system is investigated in comparison with the conventional H infinity loop shaping controller, the robust controller designed by LMI method and the reduced order robust controller by Hankel norm model reduction method. As results indicated, stability margin of our proposed controller is better than that of the static controller designed by LMI method and the reduced order robust controller. In addition, order of the proposed controller is much lower than that of conventional robust loop shaping controller, making it easy to implement in practice.

Keywords- Fixed-structure robust loop shaping control; LMI approach; Compact Genetic algorithms; H infinity loop shaping control

I. INTRODUCTION At present, stability analysis is an important issue in the design of power system stabilizer. Normally, in a power system, uncertainties are occurred by various sources, for example, parametric uncertainty, disturbances, suddenly switching, load changing, faults, etc. In a two or more area interconnected power system, analysis of stability and dynamic performance is not easy, but it can be carried out by using control system theory. To design an effective controller, many techniques such as fuzzy logic control in [1, 2], PI control in [3], GA based fuzzy control in [4], etc. have been proposed and designed for many kinds of power systems. In [1], a fuzzy logic was adopted to design a controller for a two area interconnected power system. As shown in their results, the fuzzy logic can efficiently control the system at any load changing conditions. However, the uncertainty criterion is not considered in their paper. Mathur and Manjunath [2] studied the dynamic performance of thermal units with asynchronous tie-line using fuzzy controller. They verified the performance

of the proposed system when the load frequency was changed by their simulation results. Kumar [3] studied dynamic performance of a power system with a synchronous tie-lines considering parameter uncertainties. PI control scheme was adopted in his paper and GA was used to tune the fuzzy controller for a two area interconnected power system [4]. However, all techniques mentioned above do not include the system uncertainty or robust criterion into their designs. Robust control is a well-known technique to design a high performance controller for a system under uncertainty and disturbance conditions. In this control scheme, uncertainty can be modeled by many kinds of models such as, multiplicative uncertainty model (in Mixed-Sensitivity method), co-prime factor uncertainty model (in H infinity loop shaping method [5]), etc. Many robust control techniques have been investigated in power system control such as LMI based H infinity loop shaping in [6], loop shaping control in [7], inverse additive perturbation in [8], etc. As shown in previous research works, robust H infinity control is one of the most popular techniques for designing an effective controller under uncertainty and disturbance conditions. One of the most popular techniques is H infinity loop shaping which is an intuitive and simple method. In addition, classical loop shaping which is a sensible method is incorporated in the design. However, the resulting controller obtained from this approach has high order, making it difficult to implement in practical work. To overcome this problem, this paper focuses on the design of structure-specified controller which can guarantee the robust performance by maximizing the stability margin of the controlled system. The technique used in this paper is the combination of LMI approach and cGA. LMI is used to find an initial controller which then be used for specifying the ranges of controller parameters in cGA. Then, cGA is adopted to find the final controller. As shown in the results, our proposed technique gains better stability margin compared to the conventional LMI technique [6] and the reduced order controller. In addition, the order of the proposed controller is much lower than that of the full order robust loop shaping controller.

The remainder of this paper is shown as follows. Section II illustrates the conventional robust loop shaping. Section III

1

describes the proposed technique. Simulations and results are shown in Section IV, and Section V concludes the paper.

II. H∞ LOOP SHAPING CONTROL AND STATIC H∞ LOOP SHAPING CONTROL

A. Conventional H∞ Loop Shaping Control Η∞ loop shaping control was first introduced by [9]. In this design, desired open loop shape in frequency domain is specified by shaping the open loop of the system, G, with weighting functions, pre-compensator (W1) and post-compensator (W2). The uncertainty model of the system is formulated as normalized co-prime factors that divide the shaped plant (Gs) into nominator factor (Ns) and denominator factor (Ms). Consequently, the shaped plant can be written as:

1 2sG W GW= (1)

1( )( )s s Ns s MsG N M −= + Δ + Δ (2) where ∆Ns and ∆Ms are the uncertainty transfer functions in the nominator and denominator factors, respectively. ||∆Ns, ∆Ms||∞ ≤ ε, where ε is the stability margin. The determination of the normalized co-prime and the solving of the Η∞ loop shaping control can be seen in [5]. The design steps can be briefly described as follows: 1. Specify the pre- and post-compensator weights for achieving the desired loop shape. 2. Find the optimal stability margin (εopt) by solving the following equation.

1 1 1( )infopt opt s sstab K

II G K M

Kγ ε − − −

∞

⎡ ⎤= = −⎢ ⎥

⎣ ⎦ (3)

If εopt is too low, then go to Step 1 to select new weights. 3. Select the stability margin (ε<εopt) and then synthesize the controller, K∞, by solving the following inequality.

[ ]

1 1

1 1

( )

( )

zw s s

s s

IT I G K M

K

II G K I G

Kε

− −∞∞

∞ ∞

− −∞

∞ ∞

⎡ ⎤= −⎢ ⎥

⎣ ⎦

⎡ ⎤= − ≤⎢ ⎥⎣ ⎦

(4)

4. Final controller (K) is 1 2K W K W∞= (5)

B. Static H∞ Loop Shaping Control Static H∞ loop shaping control [6, 10] is a technique for designing a robust loop shaping controller in which the structure of the final controller is the same as weighting function. In this technique, the controller is designed using the law u = KSTy, where u is input, y is output, and KST is the static controller (constant). The advantage of this type of control is that it is not necessary to have the sensor feedback of all states. In addition, this technique reduces the complexity of the resulting controller designed by conventional H∞ loop shaping control. Followings describe the design of a static H∞ loop shaping control [6, 10].

Steps 1 and 2 are the same as conventional H∞ loop shaping control. Step 3: Solve LMI system in (6) and (7) ( ) ( ) 0TA LC R R A LC+ + + < (6)

0

T T T

T

AR RA BB RC LCR I I

L I I

γγ

γ

⎛ ⎞+ − −⎜ ⎟− <⎜ ⎟⎜ ⎟− −⎝ ⎠

(7)

where A, B, C, and D are the state space elements of the shaped plant Gs, 1( )T TL BD ZC R−= − + , TR I DD= + , Z ≥ 0 is a unique positive define solution to the following algebraic Riccati equation:

1 1 1 1( ) ( ) 0T T T T TA BS D C Z Z A BS D C ZC R CZ BS S− − − −− + − − + =where TS I D D= + . Check the feasibility of the LMI system. If the solution is not feasible, then go to Step 1 to change weights.

Step 4: Determine the Lyapunov matrix R>0 and γ by solving the LMI in (7). Then, the controller (K) which is a constant matrix can be determined by solving the following inequality.

0T T TA BKC C K B+ + < (8)

where

00 0 0

, ,0

00

T T

T

BAR RA RC LII

A BDCR I I

L I I

γγ

γ

⎛ ⎞+ − ⎛ ⎞⎜ ⎟ ⎜ ⎟−⎜ ⎟ ⎜ ⎟= =⎜ ⎟ ⎜ ⎟−⎜ ⎟ ⎜ ⎟⎜ ⎟− − ⎝ ⎠⎝ ⎠

( )0 0C CR I=

Step 5: The final controller (KST) is

KST = W1KW2 (9)

As shown in [11], the controller K is structured as a constant matrix; then the final controller’s structure is the product of the specified weights. More details of designing a static Η∞ loop shaping control can be seen in [6, 10]. In addition, the method in [11] enhances the computational time by reducing the iteration in the design and can achieve the same resultant stability margin.

III. FIXED-STRUCTURE OUTPUT FEEDBACK ROBUST LOOP SHAPING CONTROL

The advantages of simple structure, guaranteed existence of a solution and output feedback can be obtained by static Η∞ loop shaping control [6, 10]; however, the stability margin obtained by this technique is much lower than that of the conventional robust loop shaping technique. This is undesirable when a high stability margin is required. To enhance the ability of the design, we propose a cGA based fixed-structure robust loop shaping control to design a fixed-structure robust loop shaping controller. The proposed technique can be described as follows.

2

1. Specify weighting functions and design the static Η∞ loop shaping control by adopting the technique described in Section II, B. 2. Assume the structure of final controller as: K = K(p) W2 Then, 1

1 ( )K W K p−∞ = . (10)

Assign the resulting controller from static Η∞ loop shaping control as the initial chromosomes of cGA. In this paper, by substituting (10) into (4), the infinity norm of transfer function from disturbances to states, subjected to being minimized, can be written as:

[ ]

cos

1 111

1

1

( ( ))( )

t zw

s s

J T

II G W K p I G

W K p

γε ∞

− −−

∞

= = =

⎡ ⎤= −⎢ ⎥

⎣ ⎦

(11)

Consequently, it is reasonable to set the fitness function of cGA as:

[ ] 11 111

1

( ( ))( ) ( ) s s

I I G W K p I Gfitness fs if K stabilizes the plantW K p

−− −−

∞

⎡ ⎤−⎢ ⎥= ⎣ ⎦

= 0.00001 otherwise (12) 3. Use cGA to find the optimal parameter, p*. cGA was presented in [12] and showed a faster convergence of solution compared to simple GA. The followings describe the steps of cGA adopted in the proposed design. Step 1: Specify the controller structure. In this paper, the structure of controller is selected as the pre-compensator weight. The optimal controller parameter is unknown parameters that cGA attempts to evaluate. Set the controller achieved by Steps 1 and 2 as the initial chromosome. Step 2: Initialize the probability vector (pb). The number of members in vector pb is calculated from the number of unknown parameters and the number of bits per unknown. For example, assume that number of unknown parameters is 9 and the number of bits per unknown is 8. Then, the length of probability vector (m) is 9×8 = 72. The initial probability, pb, for all elements is set to be 0.5. Step 3: Generate s individuals from the vector, where s is defined as the tournament selection of size s. In this paper, s is selected as 10 and S means the unknown controller parameter vector.

for i=1 to s do S[i] = generate(pb)

where generate means the individual generation procedure that computes the new individuals based on the probability vector pb.

Step 4: Use (12) to compute the fitness value of each S. Keep S that has the maximum fitness value as the winner, and S that has the minimum fitness value as the loser.

winner, loser = compete(S[1], S[2]…., S[s])

where compete means the comparison procedure.

Step 5: Update the probability vector pb from the winner and loser. The following pseudo code is used to describe the updating algorithm.

for i=1 to m do if winner[i] ≠ loser[i] if winner[i] = 1 then pb[i] = pb[i] + (1/n) else pb[i] = pb[i] – (1/n) if pb[i] > 1 means probability = 1 if pb[i] < 0 means probability = 0

where n is the population size, and m is the chromosome length. Step 6: Check the convergence by for i=1 to m do if pb[i] > 0 and pb[i]<1 then return to step 2 If the solution is converged, pb is the optimal solution. More details of compact genetic algorithm can be seen in [12].

IV. SIMULATIONS AND RESULTS Load frequency control of a two area connected power system is shown in Fig. 1. As seen in this figure, the two area power systems are connected with a tie bus no.B7and no.B8 which is used for controlling the load frequency. The dynamic model of this system [13] can be written as:

x Ax Buy Cx

•= +=

where

1 1

1 1 1

1 1

1 1 1

12 12

2 212

2 2 2

2 2

2 2 2

1

2

1 0 0 0 0

1 10 0 0 0 0

1 10 0 0 0 0

,0 0 0 0 010 0 0 0

1 10 0 0 0 0

1 10 0 0 0 0

10 0 0 0 0 0,

10 0 0 0 0 0

P P

P P P

T T

G G

P P

P P P

T T

G G

T

G

G

K KT T T

T T

R T TA T T

K KaT T T

T T

R T T

TB

T

⎡ ⎤− −⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥⎢ ⎥⎢ ⎥= −⎢ ⎥⎢ ⎥− −⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

1 0 0 1 0 0 00 0 0 1 1 0 0

C⎡ ⎤

= ⎢ ⎥−⎣ ⎦

1 1

2 1

3 1

4 1

5 2

6 2

7 2

G

E

tie

G

E

x fx Px X

X x Px fx Px X

Δ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥Δ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥Δ⎢ ⎥ ⎢ ⎥= ⇒ Δ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥Δ⎢ ⎥ ⎢ ⎥

Δ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥Δ⎣ ⎦⎣ ⎦

3

1 2

1 2 1 2 1 2

1 2 12 12

where at the nominal condition, , 20 sec,, 120, , 0.3sec, , 2.4 / ,

, 0.08 sec 0.545, 1

P P

P P T T Z

G G

T TK K T T R R H puMWT T T a

== = =

= = = −

Figure 1 The two area connected power system.

In the system, the control input is the speed changer (control vector) of each area. In this paper, the speed changers in area 1 and 2 are called as Channel 1 and 2, respectively. In the design of H infinity loop shaping control, we can select the performance weights as [6]:

1

( ) ( ),( ) ( )

a s b a s bW diags s c s s c

⎛ ⎞+ += ⎜ ⎟+ +⎝ ⎠, W2 = I (13)

Where a, b, and c are the proper constants. As the guidelines in [6, 10], the weights can be selected for 4 cases. Details of the constant values in each case can be seen in the first column of Table 2. In the proposed technique, the structure of final controller, K(p), is the same as weight’s structure. Thus, the final controller can be written as:

3 41 2

9 9

5 6 7 8

9 9

( ) ( )( )

( ) ( )

p s pp s ps s p s s p

K pp s p p s ps s p s s p

++⎡ ⎤⎢ ⎥+ +⎢ ⎥=⎢ + + ⎥⎢ ⎥+ +⎣ ⎦

(14)

For simplicity, this paper illustrates the design example of case IV. By using LMI approach mentioned in Section II-B, the robust controller for the case IV can be synthesized as:

2 2

2 2

0.9672 0.2902 0.3958 0.11874 4( )

0.3890 0.1167 0.9628 0.28884 4

Initial

s ss s s sK p

s ss s s s

− − − −⎡ ⎤⎢ ⎥+ += ⎢ ⎥

− − − −⎢ ⎥⎢ ⎥+ +⎣ ⎦

(15)

The stability margin obtained from this controller is 0.2748. To enhance the stability margin of this system, the proposed technique mentioned in Section 3 is adopted. By considering the controller in (15), the controller parameter range (chromosome ranges) are chosen as p1-8 ∈ [-10, 10] and p9 ∈ [0, 20]. When running cGA for 67 generations, the optimal solution is obtained as:

2 2*

2 2

0.5689 2.3270 1.7652 2.553814.639 14.639( )

1.5533 0.4913 1.7722 1.289514.639 14.639

s ss s s sK p

s ss s s s

− − − −⎡ ⎤⎢ ⎥+ += ⎢ ⎥

− + − −⎢ ⎥⎢ ⎥+ +⎣ ⎦

(16)

Fig. 2 shows the convergence of solution by cGA. Stability margin obtained from the proposed controller in (16) is 0.4374. Clearly, the proposed controller is better than the static H infinity loop shaping in [6] in terms of stability margin. To verify the effectiveness of the proposed technique, the conventional H infinity loop shaping and reduced-order controller are designed.

0 10 20 30 40 50 60 700.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

Generations

Sta

bili

ty M

argi

n ( ε

)

← KInitial

(p)

Figure 2. Convergence of solution.

The resulting full order controller provides the stability margin of 0.6389. This value is very high compared to all techniques; however, the order of this controller is 14. It is difficult to implement this controller in practical work. Normally, the reduced order control technique is adopted for reducing the controller order. Table 1 shows the stability margins obtained by some reduced order controllers. As seen in this table, stability margin is decreased when the order of controller is decreased. As shown in the table, the stability margin obtained from the 6th order reduction controller is only 0.1764.

TABLE I. STABILITY MARGIN OBTAINED FROM THE REDUCED ORDER CONTROLLERS.

Order Stability margin (ε )

14

10

8

6

0.6389

0.5959

0.4453

0.1764

Performance and stability margins of the robust controllers for other cases (I-III) are also investigated in this paper. As seen in Table II, the stability margin obtained from the proposed technique is much better than that of the static H infinity loop shaping controller in [6] and the reduced order controller by Hankel norm model reduction. In Fig.3, time domain responses of all controllers when the input disturbance is occurred are shown. Fig. 3 (a) shows the response when the input disturbance is occurred at channel 1, and Fig. 3(b) shows the response when the input disturbance is occurred at Channel 2. Clearly, the response from the proposed controller is better than that of the controller in [6] and the reduced order controller in terms of small oscillation and fast settling time.

4

0 10 20 30 40 50 60 70 80 90 100-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

Time (sec)

Δf1

Controller in [6]

Proposed Controller

H∞ loop shapingReduced Order

(a)

0 10 20 30 40 50 60 70 80 90 100-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Time (sec)

Δf2

Controller in [6]

Proposed Controller

H∞ loop shapingReduced Order

(b)

Figure 3. (Case IV, a = 2, b = 0.3, c = 4). The output of systems at nominal condition when there is the disturbance (0.1u(t)) entered in (a) channel 1 (b)

channel 2 of the system.

Figs. 4 and 5 show the responses when the parameters of the system are changed. In Fig. 4, the parameters of nominal plant are changed to the following parameters:

1 2 1 2 1 2

1 2 1 2

12 12

, 20sec, , 140, , 0.01sec,, 2.4 / , , 0.1 sec,

0.545, 1

P P P P T T

Z G G

T T K K T TR R H puMW T TT a

= = == =

= = −

As seen in this figure, our proposed controller performs better settling time than other structured robust controllers. In Fig. 5, the parameters of nominal plant are changed to

1 2 1 2 1 2

1 2 1 2

12 12

, 20sec, , 140, , 0.3sec,, 2.4 / , , 0.08 sec,

0.545, 1

P P P P T T

Z G G

T T K K T TR R H puMW T TT a

= = == =

= = −

TABLE II. SIMULATION RESULTS OF STABILITY MARGIN OBTAINED FROM VARIOUS ROBUST CONTROLLERS.

Parameters involved in pre compensator

Stability margin (ε ) Static Η∞ loop shaping

[4th order]

Proposed technique

[4th order]

Reduced order Full order [15th order]

Case I (a=3, b=0.2, c=5)

Case II (a=2, b=0.12, c=4)

Case III (a=1.5, b=0.35, c=4)

Case IV (a=2, b=0.3, c=4)

0.2286

0.2807

0.3395

0.2748

0.3788

0.4526

0.4473

0.4374

0.2644 [9th order]

0.2582 [7th order]

0.3301 [6th order]

0.1764 [6th order]

0.6455

0.6763

0.6830

0.6389

0 10 20 30 40 50 60 70 80 90 100-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

Time (sec)

Δf1

Controller in [6]

Proposed Controller

H∞ loop shapingReduced Order

(a)

0 10 20 30 40 50 60 70 80 90 100-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Time (sec)

Δf2

Controller in [6]

Proposed Controller

H∞ loop shapingReduced Order

(b)

Figure 4. (Case IV, a = 2, b = 0.3, c = 4). The output of systems at the perturbed condition when there is the disturbance (0.1u(t)) entered in (a)

channel 1 (b) channel 2 of the system.

5

0 10 20 30 40 50 60 70 80 90 100-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Time (sec)

Δf1

Controller in [6]Proposed Controller

H∞ loop shapingReduced Order

(a)

0 10 20 30 40 50 60 70 80 90 100-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Time (sec)

Δf2

Controller in [6]

Proposed Controller

H∞ loop shapingReduced Order

(b)

Figure 5. (Case IV, a = 2, b = 0.3, c = 4). The output of systems at the perturbed condition when there is disturbance (0.1u(t)) entered in (a) channel

1 (b) channel 2 of the system.

As seen in this figure, under the perturbed condition and disturbances, our proposed controller performs better performance in terms of fast settling time, low oscillation compared to the conventional structured robust controllers. Clearly, the responses of our proposed technique are much better than those of all structured controllers.

V. CONCLUSIONS This paper proposes a new technique to design a fixed-

structure robust controller for a two area connected power system. The proposed technique uses LMI approach for designing an initial controller and for selecting the appropriate controller parameter ranges. cGA is then adopted to find the final optimal controller. As seen in the results, our proposed controller gains better robust performance (measured by an index, stability margin) compared to the conventional static H infinity loop shaping [6], reduced order controller, and the order of the proposed controller is much lower than that of the conventional H infinity loop shaping controller. The effectiveness of the proposed controller is verified by the time

domain responses under the conditions of both parametric uncertainties and input disturbance.

ACKNOWLEDGEMENT This research work was funded by the King Mongkut’s Institute of Technology Ladkrabang Research fund.

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