Design of State Feedback Stabilizer for Multi Machine Power System

Design of State Feedback Stabilizer for Multi Machine Power System Using PSO Algorithm Abolfazl Jalilvand

Amin Safari

Reza Aghmasheh

Department of Electrical Engineering Zanjan University Zanjan, Iran [email protected]

Department of Electrical Engineering Zanjan University Zanjan, Iran

Department of Electrical Engineering Zanjan University Zanjan, Iran

Abstract— In this paper an optimal state feedback design as

a power system stabilizer (PSS) using particle swarm optimization (PSO) is presented. The problem of selecting the parameters of the state feedback PSS for a multi machine power system is converted to an optimization problem solved by PSO with the eigenvalue-based objective functions. Both the relative stability of lowfrequency modes and the practical implementation of PSSs as Considerations for a stable system are included in the constraints. The locally measured states are fed back at the AVR reference input of each machine after multiplication by suitable feedback gains. The obtained stabilizer is confirmed by eigenvalue analysis and simulation results of a multi machine power system under different operating conditions and exposed to small disturbances. Keywords- State feedback; eigenvalue analysis; Particle Swarm Optimization; Dynamic stability.

I.

INTRODUCTION

Stability of power systems is one of the most important aspects in electric system operation. This arises from the fact that the power system must maintain frequency and voltage levels, under any disturbance, like a sudden increase in the load, loss of one generator or switching out of a transmission line, during a fault [1]. Since the development of interconnection of large electric power systems, there have been spontaneous system oscillations at very low frequencies in order of 0.2 to 3.0 Hz. Once started, they would continue for a long period of time. In some cases, they continue to grow, causing system separation if no adequate damping is available. Moreover, low-frequency oscillations present limitations on the power-transfer capability. To enhance system damping, the generators are equipped with power system stabilizers (PSSs) that provide supplementary feedback stabilizing signals in the excitation systems. PSSs augment the power system stability limit and extend the power-transfer capability by enhancing the system damping of low-frequency oscillations associated with the electromechanical modes [2]. Several approaches based on modern control theory have been applied to the PSS design problem including adaptive [3-5], robust [6] variable structure [7] control. The eigenvalue sensitivity analysis has been used for PSS design in many literatures under deterministic system operating conditions. To consider the effect of more system operating factors, the technique of probabilistic eigenvalue analysis was proposed and has been applied for the parameter 978-1-4244-2824-3/08/$25.00 ©2008 IEEE

design of power system damping controllers [8, 9]. In the probabilistic eigenvalue analysis the system stability is enhanced by shifting the distribution ranges of the critical eigenvalues to the left side of the complex plane. A new approach for the optimal decentralized design of PSSs with output feedback is investigated in [10]. If PSSs with complete state feedback control scheme are adopted, the requirements of estimators and centralized controls may be used for the unavailable states and control signals. However, these increase the hardware cost and reduce the reliability of the control system. Novel intelligent control design methods such as fuzzy logic controllers [11-12] and artificial neural network controllers [13] have being used as PSSs. Unlike other classical control methods fuzzy logic and neural network controllers are model-free controllers; i.e. they do not require an exact mathematical model of the controlled system. Moreover, speed and robustness are the most significant properties in comparison to other classical schemes. H ∞ optimization techniques have been also applied; however the importance and difficulties in the selection of weighting functions of H ∞ optimization have been reported. In addition, the additive and/or multiplicative uncertainty representation can not treat situations where a nominal stable system becomes unstable after being perturbed. On the other hand, the order of the based stabilizer is as high as that of the plant. This gives rise to complex structure of such stabilizers and reduces their applicability [14, 15].

Recently, heuristic methods are widely used to solve global optimization problems. The problem of the PSSs tuning is often formulated as an optimization problem with objective functions that relates PSSs parameters to an opportune performance index. Heuristic techniques such as genetic algorithms [16-19], Tabu search algorithm [20], simulated annealing [21]; evolutionary programming [22] and swarm optimization techniques [23] have been applied earlier to the PSSs design. These evolutionary algorithms are heuristic population-based search procedures that incorporate random variation and selection operators. PSO has been motivated by the behavior of organisms, such as fish schooling and bird flocking. Generally, PSO is characterized as a simple concept, easy to implement, and computationally efficient. Unlike the other heuristic techniques, PSO has a flexible and wellbalanced mechanism to enhance the global and local

Proceedings of the 12th IEEE International Multitopic Conference, December 23-24, 2008 exploration abilities. It suffices to specify the objective function and to place finite bounds on the optimized parameters. A PSO algorithm -based approach to robust PSS design, in which several operating conditions and system configurations are simultaneously considered in the design process is presented [23]. In this paper the proposed design process for PSSs with state feedback schemes is applied to a multimachine power system. Local and available states ( Δδ , Δω , ΔE q′ and ΔE fd ) are used as the inputs of each PSS. The design problem is converted to an optimization problem and PSO is employed to solve this problem. Robustness is achieved by considering several operating conditions. It is show good damping performance when the system is perturbed. The time domain simulation results under various operation conditions are given which show that the proposed method damps the low frequency oscillation in an efficient manner. II.

PROBLEM STATEMENT

A. Power system model A power system can be modeled by a set of nonlinear differential equations as: X& = f (X ,U )

(1)

In this paper, the two-axis model [2] given in Appendix is used for time–domain simulations. When a power system operates at one operating point, the system is linearized. The model of this linearized system can be represented by[10]: X& (t ) = AX (t ) + BU (t ) = AX (t ) + B [U s (t ) + U d (t ) ]

In this paper the power system stabilizers with state feedback schemes are designed for damping power-system oscillations. The following control input vector is defined: U s (t ) = KX (t )

(3)

Where K is the feedback gain matrix with appropriate dimensions. Applying (3) to (2) we have:

X& (t ) = (A + BK )X (t ) + BU d (t )

Where AC is the matrix of the closed-loop system. B. Objective Functions Often, the closed-loop modes are specified to have some degree of relative stability. In this case, the closed loop eigenvalues are constrained to lie to the left of a vertical line corresponding to a specified damping factor. The parameters of the PSS may be selected to minimize the following objective function [22]: J1 =

∑

σ i ≥σ 0

(4)

(σ 0 − σ i ) 2

(6)

Where σ i is the real part of the ith eigenvalue, and σ 0 is a chosen threshold. The value of σ 0 represents the desirable level of system damping. This level can be achieved by shifting the dominant eigenvalues to the left of s = σ 0 line in the s-plane. This also ensures some degree of relative stability. The condition σ i ≥ σ 0 is imposed on the evaluation of J 1 to consider only the unstable or poorly damped modes that mainly belong to the electromechanical ones. The relative stability is determined by the value of σ 0 . This will place the closed-loop eigenvalues in a sector in which σ i ≤ σ 0 as shown in Fig. 1.

jω

σi ≤ σo σo

(2)

Where the input vector U (t ) is composed of the control vector U s (t ) and disturbance vector U d (t ) , both m × 1 vectors; X (t ) is an n × 1 state vector; A is an n × n plant matrix of the open-loop system and B is an n × m input matrix; n and m are the number of state variables and control signals, respectively.

(5)

Ac = A + BK

σ

Figure 1. Region of eigenvalue location for J1 objective function.

To limit the maximum overshoot, the parameters of the PSS may be selected to minimize the following objective function [26]: J2 =

∑

ζ i ≥ζ 0

(ζ 0 − ζ i ) 2

(7)

Where ζ i is the damping ratio of the ith eigen value. This will place the closed-loop eigenvalues in a wedge-shape sector in which ζ i ≥ ζ 0 as shown in Fig. 2. In the case of J 2 , ζ 0 is the desired minimum damping ratio which is to be achieved. It is necessary to mention here that if only particular eigenvalues need to be relocated, then only those eigenvalues should be taken into consideration in the computation of the objective function. This is usually the case in dynamic stability where it is desired to relocate the electromechanical modes of oscillations. The design for state

Proceedings of the 12th IEEE International Multitopic Conference, December 23-24, 2008 feedback power system stabilizers can be treated as an optimization problem with two objective functions and a set of constraints. Minimize J 1 and J 2

jω

ξi ≥ ξo

σ

Figure 2. Region of eigenvalue location for J2 objective function.

C. Constraints In the design of state feedback power system stabilizers a stable system with satisfactory damping in the low frequency modes is absolutely desired and a set of realizable feedback gains is also entirely necessary. The following constraints therefore are formulated: k uv ,min ≤ k uv ≤ k uv ,max

b) PSO uses payoff (performance index or objective function) information to guide the search in the problem space. Therefore, PSO can easily deal with non-differentiable objective functions. Additionally, this property relieves PSO of assumptions and approximations, which are often required by traditional optimization methods. c) PSO uses probabilistic transition rules and not deterministic rules. Hence, PSO is a kind of stochastic optimization algorithm that can search a complicated and uncertain area. This makes PSO more flexible and robust than conventional methods. d) Unlike GA and other heuristic algorithms, PSO has the flexibility to control the balance between the global and local exploration of the search space. This unique feature of PSO overcomes the premature convergence problem and enhances the search capability. e) Unlike the traditional methods, the solution quality of the proposed approach does not rely on the initial population. Starting anywhere in the search space, the algorithm ensures the convergence to the optimal solution. PSO algorithm Step 1) Initialization: The velocity and position of all particles are randomly set to fall into the pre-specified or allowed range. B.

Step 2): Velocity updating: At each iteration, the velocities of all particles are updated according to the following policy: vi

←

r

ωv i +

c r ( pri ,best 1

1

r − pi ) +

c

2

r r r2 ( g best − p i )

r r Where Pi and vi are the position and velocity of particle ith ,

Where k uv is the v th designed feedback gain of the uth PSS k uv ,min is the lower bound of k uv

k uv ,max is the upper bound of k uv u = 1, 2, 3 and v = 1, 2, 3, 4

In this work, the values of σ0 and ζ 0 are taken as −1.5 and 0.2 , respectively. Reasonable values for the lower and upper bounds of the feedback gain parameters should be properly chosen so that these parameters ( k uv ) can be implemented for practical considerations [14]. III.

PARTICLE SWARM OPTIMIZATION

A. Overview PSO is one of the optimization techniques and a kind of evolutionary computation technique. The method has been found to be robust in solving problems featuring nonlinearity and non-differentiability, multiple optima, and high dimensionality through adaptation, which is derived from the social-psychological theory. The features of the PSO are as follows: a) PSO is a population-based search algorithm. This property ensures PSO to be less susceptible to getting trapped on local minima.

respectively;

r r Pi ,best and g best

are the positions with the best

objective value found so far by particle ith and the entire population, respectively; w is the parameter controlling the dynamics of flying; r1 and r2 are random variables in the range [0,1]; c1 and c2 are weighting factors. Step 3) Position updating: Between successive iterations, the positions of all particles are updated according to the following rule: r r pi ← pi + v i r r Step 4) Memory updating: Update Pi ,best and g best when the corresponding conditions are met: r r pi ,best ← pi r r g best ← pi

if if

r r f ( pi ,best ) > f ( pi ) r r f ( g best ) > f ( pi )

r Where f ( x ) is the objective function to be minimized. Step 5) Termination criteria examination: The algorithm repeats Step 2 to Step 4 until certain stopping rules are r satisfied. Once terminated, the algorithm outputs the g best and r f ( g best ) as its solution [24]. The configuration of the PSO is as follows: Population: 50 Maximum iteration number: 100

Proceedings of the 12th IEEE International Multitopic Conference, December 23-24, 2008 In this scheme, each of the generators is fitted with a partial state feedback controller so that only locally available states are fed back at each generator. This implies that the state feedback matrix K of the overall system is blocked diagonal. This is schematically shown in Fig. 4. [24], where the sub matrices K 11 , K 22 and K 33 are the feedback gains of each of the three generators. The locally measured states X are fed back at the AVR reference input of each machine after multiplication by suitable feedback gains. The state feedback gains can be optimized by using PSO as described previously.

Initial inertia weight: 0.9-0.4 IV. THE OPTIMAL DESIGN FOR STATE FEEDBACK PSSS In this study an optimal design for the PSSs with state feedback schemes is presented. A multimachine power system [2] shown in Fig. 3 is taken as the test system. Each machine is described by a third-order nonlinear model and is equipped with a first order excitation system. For enhancing system stability each machine is equipped with a PSS. The detailed machine constants and the transmission network data are given in [2].

Vref 1

States of Gen.1

Vref 2

Gen.2

Vref 3

Gen.3

Load C

G2

G3

2

7

8

3

9

5

6 Load B

Load A 4

⎡ K11 ⎢ 0 ⎢ ⎢⎣ 0

1

0 K 22 0

0 ⎤ 0 ⎥⎥ K 33 ⎥⎦

G1 Figure 4. Schematic of the optimal state feedback controller Figure 3. Three-machine nine-bus power system.

When the power system operates at one operating condition the system is linearized. The resulting linearized system can be described by the following state equation (2) where:

The optimization process was carried out at the operating point specified as base case. To assess the effectiveness and robustness of the proposed method over a wide range of loading conditions, two different cases designated as case 1 and case 2 are considered. The generator and system loading levels at these cases are given in Table I.

X = [Δω1 , Δδ1 , ΔE q′1 , ΔE fd 1 , Δω2 , Δδ 2 , ΔE q′ 2 , ΔE fd 2 , Δω3 , Δδ 3 , ΔE q′ 3 , ΔE fd 3 ]T U s = [U s 1 ,U s 2 ,U s 3 ]T TABLE I.

LOADING CONDITIONS(IN PU)

Nominal

Heavy

Light

Gen

P

Q

P

Q

P

Q

G1

0.72

0.27

2.21

1.09

0.36

0.16

G2

1.63

0.07

1.92

0.56

0.80

-0.11

G3

0.85

-0.11

1.28

0.36

0.45

-0.20

1.25 0.90 1.0

0.5 0.30 0.35

2.0 1.80 1.50

0.80 0.60 0.60

0.65 0.45 0.50

0.55 0.35 0.25

Load A B C

V.

RESULTS AND DISCUSSION

In this paper, the power system shown in Fig.3 was used as the test system to design the state feedback PSSs with PSO. The proposed PSS is connected to all machines in the test system. For the each proposed PSS, the optimal setting of four parameters is determined by the PSO, i.e. 12 parameters to be optimized.

The convergence rate of the single objective functions J1 and J 2 are shown in Fig. 5. The final values of the optimized parameters with both single objective functions J1 and J 2 are given in Table II.

Proceedings of the 12th IEEE International Multitopic Conference, December 23-24, 2008

(a)

(b) Figure 5. Variations of objective functions.

TABLE II.

Gen

OPTIMAL PSSS PARAMETERS

Objective Function J1

Objective Function J 2

G1

k1 126.1673

k2 0.4876

k3 -4.2251

k4 0.0003

k1 137.8471

k2 0.8339

k3 -4.0944

k4 -0.0081

G2

136.0270

1.0063

-5.000

-0.1008

99.5888

1.0187

-5.000

-0.0700

G3

60.4620

0.8649

-3.0216

0.0074

87.7840

0.9537

-3.6022

-0.0001

TABLE III.

Without PSSs

PSSs with

J1 PSSs with

J2

EIGEN VALUES AND DAMPING RATIOS OF ELECTROMECHANICAL MODES WITH AND WITHOUT PSSS Nominal 2.0928± i4.1890, -0.4469 1.9378 ± i 5.6488, -0.3245 -0.05374± i9.3626, 0.057 -10.4225 ± i5.0172, 0.9156 -1.9423 ± i23.0484, 0.0840 -4.8568 ± i15.7463, 0.2947 -1.5241 ± i10.4004, 0.1426 -1.5205 ± i 4.7851, 0.3028 -3.1641 ± i20.1603, 0.2383 -5.5697 ± i14.4756, 0.2457 -1.9746 ± i9.0094, 0.2343 -0.9214 ± i5.7039, 0.2248

Heavy

Light

2.3625 ± i9.2299, -0.248 2.9810 ± i4.7320, -0.533 -0.1317 ± i4.5872, 0.0287 -10.9437 ± i6.1334, 0.8723 -2.0644 ± i23.2183, 0.0886 -4.3076 ± i17.1955, 0.2430 -1.5386 ± i11.3571, 0.1342 -1.9414 ± i 6.9643, 0.2685 -7.1286 ± i18.0032, 0.3682 -3.8102 ± i17.0189, 0.2169 -2.2260 ± i10.7606, 0.2026 -1.7995± i7.6161, 0.230

1.7279± i7.8521, -0.215 2.5451± i5.6182, -0.4126 -0.3929± i2.3426, 0.1654 -10.5856 ± i6.9261, 0.4442 -3.5954 ± i22.4929, 0.1578 -1.8763 ± i20.5084, 0.0911 -1.6368 ± i10.9186, 0.1483 -2.6650 ± i 4.1054, 0.5445 -8.3518 ± i17.3628, 0.4335 -3.9573 ± i15.5801,0.2462 -2.1430 ± i10.1353, 0.2069 -1.7208 ±i5.6866, 0.2896

A. Eigenvalue Analysis The system electromechanical modes, for the base case and the two operating conditions without and with the PSSs tuned using J1 and J 2 are listed in Table III. It is clear that these modes are poorly damped and some of them are unstable. It is obvious that the electromechanical–mode eigenvalues have been shifted to the left in s-plane and the system damping with the proposed method greatly improved and enhanced. B. Response to small disturbances To evaluate the performance of the proposed method a disturbance of 0.2 pu as input torque is applied to the all machines after 0.5 sec .The study is performed at three different loads and operating conditions. The results are shown in Figs. 6-8.

Fig. 6.(a)-(c) show the speed deviations of G1 , G 2 and G 3 ,respectively, under nominal condition. For cases under heavy loading condition, the simulation results are shown in Fig.7. (a)-(c), respectively. It can be concluded that the proposed PSS achieves robust performance and damp the oscillations very well over a wide range of operating conditions. The simulation results are shown in Fig.8.(a)-(c), respectively, demonstrate the speed deviations of generators under light loading conditions. It is clear that the proposed PSS provide good damping characteristics to low-frequency oscillations and greatly enhance the dynamic stability of power system.

Proceedings of the 12th IEEE International Multitopic Conference, December 23-24, 2008 x 10

8

-3

-3

2 J1 J2

J1 J2

1.5 Speed deviation (machine 3)

Speed deviation (machine1)

6

x 10

4 2 0 -2 -4

1 0.5 0 -0.5 -1 -1.5

-6 0

0.5

1

1.5

2

2.5 Time(sec)

3

3.5

4

4.5

5

-2 0

0.5

1

1.5

2

(a) 3

x 10

2.5 Time(sec)

3

3.5

4

4.5

5

(c)

-3

Figure 7. The dynamic responses under heavy loading condition.

J1 J2 Speed deviation (machine 2)

2 -3

8

1

-1

-2

-3 0

0.5

1

1.5

2

2.5 Time(sec)

3

3.5

4

4.5

5

Speed deviation (machine 1)

J1 J2

0

(b) 4

x 10

6

4

2

0

-3

-2 0

J1 J2

3

0.5

1

1.5

2

0 -1

-3 0

0.5

1

1.5

2

2.5 Time(sec)

3

3.5

4

4.5

5

(c)

4

4.5

x 10

J1 J2

1 0 -1 -2

-4 0

0.5

1

1.5

2

2.5 Time(sec)

3

3.5

4

4.5

5

(b)

-3

x 10

J1 J2

4

-3

2

x 10

J1 J2

3

1 Speed deviation (machine 3)

2 1 0 -1 -2 -3 -4 0

0.5

1

1.5

2

2.5 Time(sec)

3

3.5

4

4.5

0

-1

-2

-3

5

-4 0

(a)

0.5

1

1.5

-3

4

5

-3

Figure 6. The dynamic responses under nominal loading condition.

Speed deviation (machine 1)

3.5

2

-2

x 10

2

2.5 Time(sec)

3

3.5

4

4.5

5

(c)

J1 J2

3 Speed deviation (machine2)

3

(a) 3

1

5

2.5 Time(sec)

-3

2

Speed deviation (machine 2)

Speed deviation (machine 3)

x 10

Figure 8. The dynamic responses under light loading condition.

2

VI.

1 0 -1 -2 -3 0

0.5

1

1.5

2

2.5 Time(sec)

(b)

3

3.5

4

4.5

5

CONCLUSION

An optimal design for state feedback power system stabilizers has been presented in this paper. The design problem of the proposed PSSs is converted into an optimization problem. Only the local and available state variables: Δω , Δδ , ΔE q′ and ΔE fd are taken as the input

Proceedings of the 12th IEEE International Multitopic Conference, December 23-24, 2008 signals of each proposed PSSs, so that the implementation of the designed stabilizers becomes more feasible. The state feedback gains were obtained by using PSO algorithm and the optimal design of PSSs was successfully accomplished. The stabilizers are tuned to simultaneously shift the lightly damped electromechanical modes of all plants to a prescribed zone in the s-plane. Eigenvalue analysis give the satisfactory damping on system modes, especially the low-frequency modes, for systems with the proposed PSSs. Time-domain simulations show that the oscillations of synchronous machines can be quickly and effectively damped for power systems. The simulation results show the proposed PSSs work effectively over a wide range of loading conditions.

APPENDIX

Machine's model

δ&i = ωb (ωi − 1) ω& i =

1 ( Pmi − Pei − D i (ωi − 1) ) Mi

1 E&qi′ = E fdi − ( x di − x di′ ) i di − E qi′ ′ T doi

(

(

(

)

1 E& ′fdi = K Ai v refi − vi + u i − E fdi T Ai

)

)

1 ( K A i (v refi − v i + u i ) − E fdi ) E& fdi = T Ai

(

)

T ei = E qi′ i qi − x qi − x di′ i di i qi

δ

rotor angle

ω rotor speed

Pm mechanical input power

Pe electrical output power

E q′ internal voltage behind x d′

E fd equivalent excitation voltage

T e eclectic torque

T d′ 0 time constant of excitation circuit

K A regulator gain

T A regulator time constant

v ref reference voltage

v terminal voltage REFERENCES

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