Nonlinear Optimal STATCOM Controller for Power System Based on

Nonlinear Optimal STATCOM Controller for Power System Based on Hamiltonian Formalism 1stAsim Halder Dept. of Applied Electronics & Instrumentation Engineering Haldia Institute of Technology Haldia, India [email protected]

2nd Debasish Mondal Dept. of Electrical Engineering RCC Institute of Information Technology Kolkata, India [email protected]

Abstract— This article presents the design of a nonlinear optimal FACTS controller based on Hamiltonian function method. The most popular FACTS device, Static Synchronous Compensator (STATCOM) has been selected for this purpose. A suitable Hamiltonian function has been formulated to optimize the desired cost function. The performance of the proposed nonlinear controller has been compared with a conventional linear model STATCOM controller based on the investigation of transient stability of a power system. The parameters of the conventional controller are estimated through a heuristic optimization method, Particle Swarm Optimization (PSO). Study on simulation results reveals that the proposed optimal nonlinear controller is more effective and works substantially better than the conventional controller even during contingencies like three-phase-to-earth fault scenario. Keywords— Flexible Alternating Current Transmission Systems (FACTS), Hamiltonian Function, Nonlinear Optimal Control, Particle Swarm Optimization (PSO), Static Synchronous Compensator (STATCOM), Transient Stability.

I. INTRODUCTION Power system networks are characterized by the multidimensional nonlinear dynamics and the system normally operates in multiple operating conditions. These nonlinearities become more prominent during any fault or any severe disturbance which may push the system towards instability and eventually catastrophic failure. Traditionally, conventional controllers like Power system Stabilizers (PSSs) are employed to overcome these problems [1]-[2]. The design and implementation of linear optimal Power system Stabilizers (PSSs) has also been reported in [3]. Flexible Alternating Current Transmission System (FACTS) devices [4] are third generation power system controllers which are being widely used at present to enhance stability in power transmission networks. Several heuristic approaches like, Genetic Algorithm (GA), Artificial Neural Networks (ANN), Particle Swarm Optimization (PSO) etc. are also being employed for optimal tuning and coordinated design of power system controllers comprising PSS and FACTS devices [5]-[7]. However, conventional controllers have slow response and they are also less effective in multiple operating situations. Thus, implementation of high-speed electronic power control device is required. In this issue, several attempts have been made by the researchers for many years to design and deploy non-linear power system controller in power systems. A non-linear optimization technique has been addressed in [8]-[9] for coordinated tuning of PSS and FACTS controllers for enhancement of multi-machine stability. In [10] a non-linear control strategy for FACTS device has been reported for enhancement of stability of a power system, where performance of the non-linear

3rd Nitai Pal Dept. of Electrical Engineering Indian Institute of Technology (Indian School of Mines) Dhanbad, India [email protected]

controller is also compared with a conventional controller. Among all the available FACTS devices, Static Synchronous Compensators (STATCOM) is the most versatile and popular one. Installing a STATCOM in a grid not only enhances its stability but also increases power transfer capability and improves power quality. A non-linear control scheme for the coordinated design of STATCOM and excitation control has been presented in [11] to enhance the transient stability of a power system. The theory of Zero-Dynamic design approach has been employed in [12] where it has been established that the non-linear control law for Thyristor Controlled Series Capacitors (TCSCs) is very effective for analysis of transient stability of a multimachine power system. In [13] Hamiltonian function method has been used to investigate robust adaptive control for a Superconducting Magnetic Energy Storage (SMES) type FACTS device in power systems stability improvement. A non-linear stabilizing controller for Unified Power Flow Controller (UPFC) is proposed in [14] where the discretetime Hamilton-Jacobi-Bellman (HJB) optimal control algorithm has been employed. This article presents the design of a non-linear optimal FACTS controller for STATCOM. The optimal control law has been formulated through suitable choice of a Hamiltonian function. The superiority and effectiveness of this approach has been proved through comparison with a conventional PSO based STATCOM controller. To the best of authors knowledge this work has not been inquired details in exiting literature. The rest part of the article is arranged as follows; the design of non-linear control law for STATCOM based on Hamiltonian function has been described in section II. The PSO based optimal parameter of conventional STATCOM controller has been estimated in section III. The execution of simulation and its results are presented in section IV. II. DESIGN OF NONLINEAR STATCOM CONTROLLER A. Power System with STATCOM A single-line diagram of a Single Machine Infinite Bus (SMIB) power system model with SATACOM controller is presented in Fig. 1 [15]. A STATCOM is a controllable reactive-power source FACTS device which can exchange desired reactive-power with the utility bus by processing the voltage and current waveforms in a voltage-source converter (VSC). Here, the STATCOM bus is placed in the middle of the transmission line and is connected to the utility bus through magnetic coupling. The non-linear state equations of the power system with STATCOM can be described as

 (t ) = ω

ωs 2H

 D(ω(t ) − ω s ) E q′ V∞V sc sin δ   PM −  −   ωs G1  

(1) 1

978-1-5386-9316-2/18/$31.00 ©2018 IEEE

δ(t ) = ω (t ) − ω s

(2)

The generator power (Pe) and the STATCOM bus voltage (Vsc) are related by the following equations [11] E q′ V∞V sc sin δ Pe = (3) G1 where

G1 = (xe2 Eq′ ) 2 + ((xe1 + xd′ )V∞ ) 2 + 2(xe1 + xd′ )xe2 Eq′ V∞ cosδ 0 In (1), (2) and (3) all the symbols have their usual significances. PM is the constant mechanical power (pu), D is the damping coefficient and H is the inertia constant. The term G1 in (3) appears as the equivalent reactance of the power system network. The value of G1 can be computed easily for the given network parameters and initial conditions of the SMIB system given in Appendix (A.1).

D(ω (t ) − ω 0 ) E q′ V∞V sc sin δ (t )   PM −  −   H  G1 ω0  f 2 = (ω (t ) − ω0 ) The objective is to minimize the above performance index (6) for the given plant (1) & (2) with respect to the control signal V sc (t) . The computation steps of finding the

where, f 1 =

ω0 

* nonlinear optimal control law for V sc (t) are described as follows;

Step1: Configuration of Hamiltonian function The Hamiltonian function (Hm) for this problem can be defined following [16] as Hm = Hm(ω(t ), δ (t ),V sc (t ), λ1 (t ), λ2 (t )) (7) where λ1 and λ2 are the Lagrangian multipliers. Thus the Hamiltonian function with Lagrangian multipliers can be expressed as,

(

)

H m = V V sc (t ) + λT (t ) f ( X (t ), u (t ), t ) =

ω 1 2 V sc (t ) + λ1 (t ) 0  H 2 

(8)

 D (ω(t ) − ω0 ) E q′ V ∞V sc sin δ (t )    PM −  −   ω0 G1   + λ2 (t )(ω(t ) − ω0 )

(9) Step2: Find

* Vsc

(t ) from ∂H m = 0 ∂V sc

∂Hm = 0 results in ∂V sc E′ V sin δ (t ) * (10) (t ) − λ1* (t ) ω0 q ∞ =0 Vsc H G1 from which the nonlinear optimal control law for the STATCOM can be obtained as E ′ V sin δ(t ) * (t ) = λ*1(t ) ω0 q ∞ (11) Vsc H G1

Solving for

Fig. 1. SMIB system with STATCOM

B. Formulation of Hamiltonian Function In Equation (1) and (2) δ (t ) and ω (t ) are the state variables of the system and Vsc is the control signal for the STATCOM. If the control variable ‘u(t)’ is chosen as, u (t ) = V sc (t ) , then (1)-(2) can be written in a standard plant model as X = f ( X (t ), u (t ), t ) with X (t ) = X {ω δ } . 0

0

0

0

Thus

ω ω ( t )   0  δ ( t )  =  H    

 D (ω (t ) − ω 0 ) E q′ V ∞ V sc sin δ (t )    PM −   (4) −   ω0 G1    (ω (t ) − ω 0 )

To derive the nonlinear optimal control law u(t) = V sc (t) for the STATCOM, the performance index has been chosen as; tf

J=



V ( X (t ), u (t ), t )dt

(5)

t0

With V ( X (t ), u (t ), t ) = V sc (t ) , the performance index can further be written as; 1 J= 2

tf

 Vsc (t )dt 2

(6)

Step3: Using results of Step2 in Step1 find the optimal H *m

(

)

H * m ω * (t ), δ * (t ), λ1* (t ), λ*2 (t ) = 2 2 2 * 2 1 *2 ω 0 E q′ V∞ sin δ (t ) *  ω 0 + λ1  λ1 (t ) 2  H H2 G12

−

E q′ V∞ sin δ * (t ) G1

λ1* (t )

(

+ λ*2 (t ) ω * (t ) − ω 0

=

 D(ω (t ) − ω 0 )  PM −  ω0 

* ω 0 E q′ V∞ sin δ (t ) 

H

G1

)

 

(12)

2 2 2 * 2 1 *2 ω 0 E q′ V∞ sin δ (t ) * ω 0 PM λ1 (t ) + λ1 (t ) H 2 H2 G12

− λ1* (t )

(

(

D ω * (t ) − ω 0

ω0

λ*2 (t ) ω * (t ) − ω 0

)

)− λ

*2 1

(t )

2 2 2 * ω 02 E q′ V∞ sin δ (t )

H2

G12

+ (13)

t0

Here, f ( X (t ), u (t ), t ) = [ f1 f 2 ]T

2

(

ω P D ω * (t ) − ω 0 = λ1* (t ) 0 M − λ1* (t ) ω0 H

)

+

2 2 2 * 2 1 *2 ω 0 E q′ V∞ sin δ (t ) * − λ1 (t ) + λ 2 (t ) ω * (t ) − ω 0 2 2 2 H G1

(

)

(14)

Step4: Computation of the Lagrangian multipliers The Lagrangian multipliers λ1 and λ2 can be solved from the following state and co-state equations of the problem;

(

 ∂Hm  ω P D ω* (t ) − ω0  = 0 M − ω (t ) = + H ω0  ∂λ1 * − λ1*

(t )

)

H

(15)

(

(16)

 ∂Hm  D  = λ1* (t ) λ1* (t ) = − − λ*2 (t ) ω0  ∂ω(t ) * 2 2 ω02 E q′ V∞ 2

 ∂Hm  1 2  = λ1* λ*2 (t ) = − H  ∂δ (t ) * 2

sin 2δ (t ) *

G12

(18)

* in (11), the optimal control law for the expression of V sc STATCOM can be obtained as

ω E ′ V sin δ (t ) * (t ) =  χ . 0 q ∞ V sc G1  Ωt  H D χ for χ := where λ*1 (t ) = ω0 Ωt

1

∆ −

∆

2

+

PI controller

(17)

Solving for λ1* from the above equations and using it in the

(19) and

* 2 2 2 1 ω0 E q′ V ∞ sin 2δ (t ) . The solution for λ1* from 2 H2 G12 (17) & (18) is given in Appendix (A.2).

Ω := −

This nonlinear control law (19) will be used in the section IV for simulation of dynamic response of the study system. III. DESIGN OF LINEAR MODEL STATCOM CONTROLLER A. Structure of Linear STATCOM Controller A linear model of STATCOM voltage controller is shown in the block-diagram (Fig. 2) [17]. An auxiliary control signal generator speed ( Δω ) is added along with the voltage signal to cater all the oscillatory modes of the system. The block-diagram presented in Fig. 2 can be expressed by the following state-space equations;

ΔX s2 = −

∆ +

)

Feedback stabilizer

1+

∆

G12

 ∂Hm   = ω* (t ) − ω0 δ(t ) = +  ∂λ2 *

K 1 1 ΔX s 2 + ω Δω − ΔV meas Tm Tm Tm

 K  K K K ΔX s3 =  − P + K I ΔX s2 + P ω Δω − P ΔVmeas T Tm T m  m   T  K 1 1 ΔVsc = − ΔV sc + ΔX s3 + 1  − P + K I ΔX s2 T2 T2 T2  Tm 

(20)

(21)

(22)

Again the equations (1)-(2) of the SMIB system can be linearized as KV ω s K ω Dω s Δω = − 1 s Δδ − − Δω − sc ΔV sc (23) 2H 2H 2H Δδ = Δω (24) Auxiliary control signal

2 2 2 * ω02 E q′ V∞ sin δ (t ) 2

T1 K P K ω TK Δ ω− 1 P Δ V meas T2 Tm T2 Tm

3

1+ 1+

1

∆

2

Lead-lag compensator

Voltage controller Fig. 2. Linear model of STATCOM voltage controller

∂Pe ∂P and KVsc = e . The expression for K1 ∂ Vsc ∂δ and KVsc can be derived easily from (3). Therefore, (20)-(22) and (23)-(24) together describe the linear state-space model of a SMIB system with STATCOM controller. This combined model can be expressed in standard state-variable form as follows; where K 1 =

ΔX = [ Astatcom ]ΔX + B statcom ΔU

(25)

where ΔX = [ Δω Δδ ΔXs2 ΔXs3 ΔVsc ] and ΔU = ΔVmeas . The system matrix and hence the swing modes of this system can be estimated from the eigenvalues analysis program of this combined model. Fig. 2 depicts that there are four designable unknown parameters of the STATCOM controller; the PI controller gains (KP, KI), lead-lag time constants (T1, T2). The parameters of the feedback stabilizer (Kω and Tm) are kept constant. PSO has been employed in the following section to get the tuned values of KP, KI, T1 and T2 of the STATCOM controller through optimization of a desired objective function. B. Optimization Problem The Particle Swarm Optimization PSO [18] is a bioinspired heuristic optimization technique based on movement and intelligence of biological swarms. The optimization technique in PSO begins with a population of random solutions in a search space and finds global optima (best particle) by updating generations. The best particle in a swarm is denoted by the gbest (global best), while its best position that has been visited by the current particle is termed as pbest (local best). The PSO achieves optimal solution by continuously updating the velocity of each particle towards its global best and local best location. The velocity and position of each particle in a swarm are estimated by the following equations;

3

v i (n) = α ∗ v i (n − 1) + a1 ∗ r1 ∗ ( gbest i − s i (n − 1))

(26)

a 2 ∗ r2 ∗ ( pbest i − s (n − 1))

si (n) = si (n − 1) + vi (n)

(27)

Here, vi (n) denotes the velocity of the ith particle, generally set to 10-20 % of the dynamic range of the variables. Scalar coefficients a1, a2 ∈ (0, 2) pull each particle in the direction of global best and the local best position respectively. r1 and r2 are two random variables. r1 and r2 are generally kept in the ranges ∈ (0, 1). α is the inertia weight of values ∈ (0, 1) . In this work the PSO runs to solve the optimization problem via minimization of the following objective function, named as Critical Swing Index (CSI), which is given by:

CSI = η = (1 − β i ) (28) where, β i denotes damping ratio of the i-th critical swing mode of the study system. Equation (28) indicates that the maximization of the damping ratio (β) results in minimization of the Critical Swing Index, η. It is evident that changes in the parameters of the controller, affect the damping ratio as well as the objective function. Thus, the optimization problem can be defined as; Minimize η [As in (28)] Subject to: K min ≤ K ≤ K max ; K min ≤ K ≤ K max ; I

I

I

P

P

P

T1min ≤ T1 ≤ T1max ; T2 min ≤ T2 ≤ T2 max

Step 2: Create initial population consisting swarm for the STATCOM parameters: KP, KI, K1 and T2. Step 3: Execute user defined MATLAB program for computation of the system matrix, eigenvalue and the corresponding damping ratio for the critical swing mode of the proposed test system. Step 4: Evaluate objective function (CSI) for the each ‘swarm’ in a current population. Step 5: Determine and store the best value (gbest) of the swarm which minimizes the objective function. Step 6: Check whether the generation/epoch exceeds maximum limit. Step 7: If epoch/generation < maximum limit, update population for next epoch and repeat from step 3. Step 8: If epoch/generation > maximum limit, stop program and produce output. The PSO searches the optimal set of the tuning parameters KP, KI, T1 and T2 of the STATCOM controller which maximize the damping ratio (β) through minimizations of the objective function (η). The set of the controller parameters; KP, KI, T1 and T2 evaluated through PSO are enlisted in Table I. The convergence profile of the objective function for the best solution with swarm size 10 and number of epoch limit 150 has been shown in Fig. 3. It has been found that the convergence is guaranteed near the epoch 50 and remains unaltered up to the maximum epoch limit and the possible attainable minimum value of the Critical Swing Index (η) is 0.4744. TABLE I. STATCOM CONTROLLER PARAMETERS

C. Implementation of PSO

PSO has been implemented here applying ‘PSO toolbox’ in MATLAB. The ‘PSO toolbox’ comprises of a main program ‘pso_Trelea_vectorized.m’ and some sub-programs & sub-routines. The optimization process has been executed by the main program through appropriate algorithm with the help of sub-programs and sub-routines available in the PSO toolbox. To evaluate the objective function (η), the main program of PSO follows user defined eigenvalue calculation program which computes eigenvalues as well as damping ratio of the system from the system matrix Astatcom given in (25).

Parameters of the controller KI

Typical range (Min, Max) 0.1, 5.0

PSO output value 2.0

KP

1.0, 10.0

3.33

T1

0.10, 1.50

1.50

T2

0.01, 0.25

0.21

Critical Swing Index (η)

0.4744

In this PSO based problem ‘swarm’ has been structured with four tuning parameters of the STATCOM controller. The swarm structure of the problem has been presented through following matrix equation;

Swarm = [K P K I T1 T2 ]

(29)

The PSO starts with an initial random population of each swarm. The maximum and minimum value of each element in a swarm has been constrained within a typical range (Table I). The objective function ‘η’ corresponding to each swarm in a population has been evaluated by the eigenvalue analysis programme of the test system. The computational steps for implementation of PSO are illustrated as follows: Step 1: Set basic parameters for PSO; population size, epoch/generation limit, dimension of input variables, PSO type etc.

Fig. 3. Minimization rate of the objective function

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IV. PERFORMANCE ANALYSIS The performance of the proposed non-linear controller for the STATCOM has been simulated here for a typical contingency like three-phase to-earth fault near generator bus of the test system (Fig. 1). Assumed that the fault is occurred at t = 1.0 sec and it is sustained only for a duration of 1.0 sec. It has been found that the rotor angle and the generator speed both are increases towards instability. The Hamiltonian based optimal non-linear controller and the conventional PSO based optimal controllers are incorporated separately for the study system. It is evident from the equation (19) that δ and ω are the dynamic input variables to the controller and Vsc is simulated control signal. All investigations are made in MATLAB for a simulation time 7 sec. It has been observed that the post fault system is being settled within 3-4 sec with application of the proposed nonlinear controller (Fig. 4 & Fig. 5). The simulation results further indicate that the dynamic oscillations, settling time and the steady state performance of the generator rotor angle (δ) and the generator speed (ω) all are being improved to a good degree with both the proposed non-linear and the conventional STATCOM controller however, the effectiveness of the non-linear controller is found noticeably high compared to the conventional one.

Thus, through these investigations it is possible to interpret that the power system with the non-linear controller can offer more effective transients stability solution in a post-fault system as contrast to the conventional one. The proposed strategy can also be implemented for the design of other nonlinear FACTS controller. APPENDIX A A.1. Typical Initial Condition Parameters of the SMIB System H = 2.01sec; D = 0.1; PM = 0.7864 pu, Rs = 0.0 pu ; Re = 0.02 pu; Td = 5.90 sec; ωs = 314 rad/sec; Xd = 1.70 pu; X'd = 0.245 pu; Xe = 0.7 pu; Xq = 1.64 pu; Vinf = 1.00 ∠ 0o pu; Vt = 1.72 ∠ 19.31o pu; δ =2.27 rad/sec; E'q=1.9646; G1=0.6521. A.2. Solution for λ1* from (17) & (18)

Given that the equation (17) and (18) are D λ1* (t ) = λ1* (t ) − λ*2 (t ) ω0 * 2 2 2 1 2 ω 0 E q′ V ∞ sin 2δ (t ) λ*2 (t ) = λ1* 2 H2 G12 Taking time derivative of (i) * (t ) = λ* (t ) D − λ* (t ) λ 1 1 2 ω0

(i) (ii)

(iii)

Replacing the expression for λ*2 in (iii) from (ii) results in 2 E ′ 2V 2 sin 2δ * (t ) 2 * (t ) = λ* (t ) D − 1 λ* ω 0 q ∞ λ 1 1 1 ω0 2 H2 G12

(iv)

2 2 * 2 D 1 ω0 E q′ V ∞ sin 2δ (t ) and Ω := − Let χ := ω0 2 H2 G12 Now the equation (iv) becomes 2 λ1* (t ) − { χλ1* (t ) + Ωλ* 1 (t)} = 0

Fig.4. Generator speed response for all cases

(v)

The equation (v) can be represented as a typical example of a differential equation with a saddle-node bifurcation. Let 2 r = χλ* (t ) + Ωλ* 1 (t) is the bifurcation parameter. At r = 0 1

there is exactly one fixed equilibrium point which is called a saddle-node bifurcation point. Thus, equating for r = 0 in (v) there can have two probable solutions;

2 λ1* (t ) = 0 and χλ1* (t ) + Ωλ* 1 (t) = 0 (vi) For λ1* (t ) = 0 the solution for λ1 is λ1* (t ) = C1t + C 2 which is an unbounded and infeasible to get saddle-node equilibrium point. C1 and C2 are two integration constants. 2 Again, solving for χλ1* (t ) + Ωλ* 1 (t) = 0 gives χ (vii) λ*1 (t ) = Ωt This is a converging and feasible solution for λ1 which can give at least one saddle-node equilibrium point. The expression of λ1 from (vii) has been utilized in (11) to get Hamiltonian based non-linear controller law. Fig.5. Rotor angle response for all cases

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