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A PD-type Multi Input Single Output SSSC damping controller design employing hybrid improved differential evolution-pattern search approach
Accepted Manuscript Title: A PD-type multi input single output SSSC damping controller design employing hybrid improved differential evolution-pattern search approach Author: Sidhartha Panda S. Harish Kiran S.S. Dash C. Subramani PII: DOI: Reference:
S1568-4946(15)00250-1 http://dx.doi.org/doi:10.1016/j.asoc.2015.04.023 ASOC 2913
To appear in:
Applied Soft Computing
Received date: Revised date: Accepted date:
19-5-2014 6-4-2015 6-4-2015
Please cite this article as: S. Panda, S.H. Kiran, S.S. Dash, C. Subramani, A PD-type multi input single output SSSC damping controller design employing hybrid improved differential evolution-pattern search approach, Applied Soft Computing Journal (2015), http://dx.doi.org/10.1016/j.asoc.2015.04.023 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Start
A
Specify the DE parameters: Strategy, NP, Gmax, Fmax, Fmin, CRmax, Crmin
Specify mesh size, mesh expansion factor, mesh contraction factor, maximum number of function evaluations and iterations
Initialize the population
Iter. = 0
G=0 Evalute the ffitness of individuals by time domain simulation
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Yes
Is termination conditions reached? Iter. = Iter. +1
No
Iter. = Iter. +1
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Generate mutant vector and construct trial vector
Stop
Set the starting point
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Evalute trial vector
Construct Pattern vectors and create mesh points No
Is fitness of trial vector fitness of current individual ?
Replace the current individual with the trial vector
Is the poll successful?
No
Contract the mesh size
Yes
A
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G = Gmax ?
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Graphical Abstract
Yes
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Size of new population < Old population ? No
No
Expand the mesh size
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Update G, F and CR
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Yes
Yes
Evalute the mesh points
Discard the trial vector
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Highlights A novel PD type MISO controller is proposed for SSSC based damping controller. Hybrid improved DE and PS approach is proposed to optimize the controller parameters. In improved DE, control parameters F and CR are varied during optimization runs.
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Both single machine infinite bus and multi-machine power systems are considered.
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an
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Comparative results are provided to show the superiority of the proposed design approach.
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A PD-type multi input single output SSSC damping controller design employing hybrid improved differential evolution-pattern search approach *
Sidhartha Pandaa, S. Harish Kiranb, S. S. Dashc and C. Subramanid
a
Department of Electrical and Electronics Engineering, VSSUT, Burla, Odisha-768018, India b,c,d
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Department of Electrical Engineering, SRM University, Chennai-603203, India
*
Corresponding author: e-mail: [email protected], Phone +91-9438251162
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Abstract—In this paper, a Proportional Derivative (PD)-type Multi Input Single Output (MISO) damping controller is
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designed for Static Synchronous Series Compensator (SSSC) controller. Both local and remote signals with associated time delays are chosen as the input signal to the proposed MISO controller. The design problem is formulated as an
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optimization problem and a hybrid Improved Differential Evolution and Pattern Search (hIDEPS) technique is employed to optimize the controller parameters. The improvement in Differential Evolution (DE) algorithm is introduced by changing two of its most important control parameters i.e. Scaling Factor F and Crossover Constant CR with an objective
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of achieving improved performance of the algorithm. The superiority of proposed Improved DE (IDE) over original DE and hIDEPS over IDE has also been demonstrated. To show the effectiveness and robustness of the proposed design
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approach, simulation results are presented and compared with DE and Particle Swarm Optimization (PSO) optimized Single Input Single Output (SISO) SSSC based damping controllers for both Single Machine Infinite Bus (SMIB) power
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system and multi-machine power system. It is noticed that the proposed approach provides superior damping performance compared to some approaches available in literature.
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Key words—Power system stability; static synchronous series compensator (SSSC); improved differential evolution algorithm; pattern search; multi input single output controller.
1. INTRODUCTION
Active power oscillations in power transmission systems may arise in corridors between interconnected areas as a result of poor damping of the interconnection [1]. Active power oscillation limits the power transmission capacity of interconnections between areas. Power System Stabilizers (PSS) are generally employed to damp these oscillations, but PSS are not effective in some cases, particularly when inter-area oscillations of typically 0.2-0.7 Hz are present. Alternatively, Flexible AC Transmission Systems (FACTS) controllers can be employed to damp the power system oscillations [2]. Static Synchronous Series Compensator (SSSC) is one of the important members of series FACTS controller [3]. If a SSSC is installed in a power system to enhance the power transfer controllability, a supplementary 3
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damping controller could be designed for SSSC to damp the power system oscillations [4]. Despite the availability of a variety of controller, the fixed gain, lead-lag compensation type of controller structure continues to be the most popular with the electrical utilities because of the ease of on-line tuning and also lack of assurance of the stability by some adaptive or variable structure techniques [5-6]. Most of the previous works on stability
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and damping improvement by SSSC are based on Single Input Single Output (SISO) based lead lag controllers using either local signal or remote signal [7-9]. To avoid additional costs associated with communication, input signal should preferably be locally measurable. However, local control signals, although easy to get, may not contain the desired
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oscillation modes. So, compared to wide-area signals, they are not as highly controllable and observable. Owing to the
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recent advances in optical fibre communication and global positioning systems, the wide-area measurement system can realize phasor measurement synchronously and deliver it to the control centre even in real time. Hence both local and remote signals can be used reliably as control input signals. In this paper, a Multi Input Single Output (MISO) controller is
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proposed as SSSC based damping controller. While considerable work has been reported for the improvement of controller structure of a Proportional Integral Derivative (PID) controller, surprisingly, hardly any attempt has been made
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to improve the structure of a lead lag controller. The structure of a lead lag controller consists of a gain block which acts as a proportion gain and there is scope to add an additional gain term i.e. derivative term to improve the system response
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and the performance of the controller. In view of the above, a PD-type MISO controller for SSSC is proposed in the present work to damp power system oscillations.
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The problem of FACTS controller parameter tuning is a complex task. The conventional techniques that are reported in literature pertaining to the tuning of FACTS controller suffer from heavy computation burden and the search process is
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likely to be trapped in local minima as the optimal solution may not be obtained. The growth in size and complexity of electric power systems has necessitated the use of intelligent systems that combine knowledge, techniques and methodologies from various sources for the real-time control of power systems. In recent years, a lot of interest has been drawn to the applications of intelligent techniques to power system problems. Differential Evolution (DE) is a populationbased direct search algorithm for global optimization capable of handling non-differentiable, non-linear and multi-modal objective functions, with few, easily chosen, control parameters [10]. DE uses a greedy selection procedure with inherent elitist features and has fewer control parameters, which can be tuned effectively [11]. But, the success of DE in solving a particular problem significantly depends on suitable choice of control parameter values namely the Scaling Factor (F) and Crossover Constant (CR) [12]. It is advantageous to use appropriate F and CR values at different stages of evolution/search process instead of using fixed F and CR values for the entire search process [13-14]. The key to achieving high performance for any meta-heuristic algorithm is to maintain a good balance between exploitation and 4
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exploration in the search process. DE being a global optimizing method is designed to explore the search space and most likely will give an optimal/near-optimal solution. On the other hand, local optimizing methods like Pattern Search (PS) are designed to exploit the local area, but they are usually not good at exploring wide search space and hence generally not applied alone for global optimization problems [15-16]. Due to their respective strength and weakness, there is motivation
hybrid improved DE and PS (hIDEPS) for the design of a SSSC based damping controller.
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for the hybridization of DE and PS. In view of the above, an attempt has been made in this paper for the application of a
In this paper, a Multi Input Single Output (MISO) controller is proposed for SSSC to damp power system oscillations
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following a disturbance. The MISO controller consists of two PD-type lead lag controllers with both remote signal (speed
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deviation signal) and local signal (tie-line power deviation signal). The design problem of proposed controller is formulated as an optimization problem and hIDEPS technique is employed to find the optimal controller parameters. The performance of the proposed controller is evaluated in two test systems subjected to different transient disturbances. To
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show the effectiveness and robustness of the proposed approach, simulation results are presented and compared with some
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SISO based damping controllers approaches reported in literature [7-8].
2. MATHEMATICAL MODELING OF SYSTEM UNDER STUDY
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2.1 Single Machine Infinite Bus Power System with SSSC A Single Machine Infinite Bus (SMIB) power system shown in Fig. 1 is considered at the first instance to design the PD-
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type MISO damping controller for SSSC. The system consists of a synchronous generator connected to an infinite-bus through a step-up transformer and a SSSC through a double circuit transmission line. The generator is provided with
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Hydraulic Turbine and Governor (HTG) and excitation system. The HTG consists of a hydraulic turbine, a governor system, and a servomotor. The excitation system consists of a voltage regulator and DC exciter, as recommended in IEEE Recommended Practice for Excitation System Models for Power System Stability Studies [17]. In Fig. 1, T/F represents the transformer; VS and VR are the generator terminal and infinite-bus voltages respectively; V1 and V2 are the bus voltages; VDC and Vcnv are the DC voltage source and output voltage of the SSSC converter respectively; I is the line current and PL is the total real power flow in the transmission line respectively. 2.2Modeling of Machine The dynamics of the stator, field and damper windings are included in the present analysis. Two-axis reference frame (d-q frame) is used to express the stator and rotor quantities. All rotor quantities are referred to stator (represented by primed variables) as given in (1)-(8): 5
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d q R q dt
V d R S id
d q R d dt
' kq 1
V kq'
2
R
' kd
' i kd
' kq 1
R
Where, d
d dt
' i kq
' ' R kq 2 i kq
d dt
2
1
' fd
(3)
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V
i 'fd
' kd
d dt d ' kq dt
(4)
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' kd
' fd
' kq 1
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V
R
2
' L d i d L md ( i 'fd i kd )
' 'fd L' fd i 'fd L md ( i d i kd )
' kq 2
L 'kq
2
' i kq
2
L mq i q
' L q i q L mq i kq
' L 'kq 1 i kq 1 L mq i q
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In the above equations, the subscripts:
' kq 1
(6)
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' ' kd L'kd i kd L md ( i d i 'fd )
q
(5)
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' fd
(2)
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V q R S iq V
(1)
d and q stands for d -axis and q -axis quantities
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R and S stands for rotor and stator quantities
f and k stands for field and damper winding l and m stands for leakage and magnetizing inductance
The mechanical equations are given by: d 1 r ( Pe F r r P m ) dt J
(7)
d dt
(8)
r
Where, r and are angular velocity and angular position of the rotor respectively, Pe and Pm represent electrical and mechanical power respectively, J and Fr represent inertia and friction of rotor respectively. 6
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3 THE PROPOSED APPROACH 3.1 Structure of SSSC based Controller The proposed MISO controller structure consists of two PD-type lead lag controllers as shown in Fig. 2. Each lead lag structure consists of a proportional gain and a derivative gain block, a signal washout block and two-stage phase
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compensation block. Derivative mode improves stability of the system. However, when the input signal has sharp corners, the derivative term will produce unreasonable size control inputs to the plant. Also, any noise in the control input signal
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will result in large control output signals. These reasons often limit the practical applications of derivative term in the controller. The practical solution to the these problems is to put a first filter on the derivative term and tune its pole so that
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the chattering due to the noise does not occur since it attenuates high frequency noise. In view of the above a filter is used for the derivative term in the present paper.
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The phase compensation block provides the appropriate phase-lead characteristics to compensate for the phase lag between input and the output signals. The signal washout block with time constants acts as a high-pass filter to allow
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signals associated with oscillations in input signal as unchanged signals. Without the high-pass filter, steady changes in input would alter the output. The phase compensation blocks give the exact phase-lead characteristics to overcome the
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phase lag between input and the output signals. Vqref stands for the reference injected voltage as desired by the steady state power flow control loop. In the present study Vqref is assumed constant during large disturbance transient period as it
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acts quite slowly. The necessary value of compensation is obtained according to the change in the SSSC injected voltage
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Vq1 and Vq 2 which is added to Vqref .
3.2 Problem Formulation
In lead-lag structured controllers, the washout time constants are generally pre-specified [17]. In the present study, TWL = TWR = 10 s and filter coefficient N =100 are used. The proportional and derivative gains ( K DL , K DR & K PL , K PR ) and the
time constants T1L , T2 L , T3 L , T4 L & T1R , T2 R , T3 R T4 R are to be calculated. During steady state conditions Vq1 , Vq 2 and Vref are constant. During dynamic conditions the series injected voltage Vq is adjusted to damp power system oscillations
and is given by: Vq Vqref Vq1 Vq 2
(9)
An integral time absolute error of the speed deviations is taken as the objective function given by equation (10).
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t
J 10 4 | | t dt
(10)
0
Where, is the speed deviation; and t is the time range of the simulation. To calculate the objective function, a 5-cycle 3-phase fault is applied at the middle of one transmission line which is cleared by line outage for 5-cycles. Thus, the
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design problem is formulated as the following optimization problem: Minimize J
(11)
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Subject to
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K PL min K PL K PL max , K PR min K PR K PR max K IL min K IL K IL max , K IR min K IR K IR max
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T1L min T1L T1L max , T1R min T1R T1R max T2 L min T2 L T2 L max , T2 R min T2 R T2 R max min
T3 L T3 L
max
, T3 R min T3 R T3 R max
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T3 L
4 OPTIMIZATION TECHNIQUE
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4.1 Differential Evolution (DE)
(12)
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T4 L min T4 L T4 L max , T4 R min T4 R T4 R max
Differential Evolution (DE) algorithm is a simple but efficient search technique proposed by Storn and Price [10]. In DE
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algorithm, an optimization task consisting of D variables is represented by a D-dimensional vector. At the beginning, a population of NP solution vectors is initialized within the parameter bounds. Mutation, crossover and selection operators are applied to update the population. DE algorithm uses two generations; old generation and new generation of equal population size. Individuals of the current population become target vectors for the next generation. The mutation operation produces a mutant vector for each target vector, by adding the weighted difference between two randomly chosen vectors to a third vector. A trial vector is generated by the crossover operation by mixing the parameters of the mutant vector with those of the target vector. The trial vector replaces the target vector in the next generation if has a better fitness value than the target vector. The DE evolutionary operators are described below [10-11]: 4.1.1 Initialization of parameter At the beginning of a DE run i.e. at Generation G = 0, search parameters are initialized within the search space
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D D 1 constrained by the specified minimum and maximum bounds X min {x1min,G , , xmin, G } and X max { xmax,G ,, xmax,G } . The
initial value of j-th parameter in the i-th individual at first generation (G = 0) is generated by: j j j xij, 0 xmin rand (0,1) ( xmax xmin ),
j 1,2, , D
(13)
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4.1.2 Mutation Operator A parent vector X i ,G from the current generation is selected (known as target vector) and a mutant vector Vi , G is created by
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the differential mutation operation (known as donor vector). A child is produced by combining the donor with the target vector (known as trial vector). For each target vector X i , G at the generation G, its associated mutant vector
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Vi , G {vi1, G , , viD,G } can be generated through a number of mutation strategies. Several strategies can be employed in DE
optimization algorithm. The strategy in a DE algorithm is denoted by DE/x/y/z, where x represents the mutant vectors, y
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represents the number of difference vectors used in the mutation process and z represents the crossover scheme used in the crossover operation. A comparison of various strategies has been presented in [18] and the best strategy reported is
Vi , G X best , G F .( X r i ,G X r i , G ) F .( X r i ,G X r i , G ) 1
2
3
4
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DE/best/2/bin given by:
(14)
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The indices r1i , , r4i are randomly generated once for each mutant vector and are mutually exclusive integers within the range [1, NP]. X best ,G is the best individual vector within the best fitness value in the population at generation G and
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4.1.3 Crossover Operator
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mutation strategy is executed binomial crossover.
To enhance the potential diversity of the population, the crossover operation is applied to each pair of target vectors X i ,G and its corresponding mutant vector Vi ,G employed to generate a trial vector U i , G {ui1, G , , uiD, G } . The donor vector
enters the trial vector with a probability. The crossover operation can be expressed as: vij,G if uij,G j xi ,G if
(rand j CR or ( j jrand )
(rand j ,i CR ) or ( j jrand )
(15)
j 1,, D
The crossover rate CR is a user specified constant within the range and controls the fraction of parameter values copied from the mutant vector. jrand is a randomly selected integer in the range [1, D]. The condition j jrand makes certain that the trial vector differs from its corresponding target vector by at least one parameter.
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4.1.4 Selection Operator Selection operation is performed to keep the population size constant over successive generations. The objective function value of each trial vector is compared to that of its corresponding target vector in the current population. If the trial vector has better or equal objective function value than the corresponding target vector, the trial vector replaces the target vector
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and enters the population of the next generation. Otherwise, the target vector remains in the population for the next generation. The selection operation can be represented by:
(16)
i [1, , NP]
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U i , G if f (U i , G ) f ( X i , G ) X i ,G 1 otherwise X i , G
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4.2 Previous Work on DE Parameter Tuning
The performance of the DE algorithm is highly dependent on the choice of control parameter. Inappropriate selection of
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control parameters may lead to premature convergence or stagnation of the algorithm. The important control parameters that influence the performance of DE algorithm are: scaling factor F and crossover rate CR. It has been reported that the
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effectiveness, efficiency, and robustness of the DE algorithm depend on the control parameters setting and the control parameter settings may be different for different functions or the same function with different requirements [19]. Many
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empirical strategies have been reported in literature for selecting the control parameter settings. It has been reported that the searching capability and convergence speed of DE algorithm are very sensitive to the choice of control parameters F, CR and NP and suggested 0.6, 0.3-0.9 and 3D-8D for F, CR and NP respectively [20].
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Researchers have also proposed a number of techniques to change the control parameters during the run. In [21], an
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optimal value of 0.5 for CR and variable F in the range 0.4 to 1 for each generation was suggested. In [22], the factor F was linearly reduced with increasing generation count. A uniform distribution between 0.5 and 1.5 with a mean value of 1 was employed in [23] for selecting F. Normal distribution between 0.5 to 1.5 was employed in [24] for choosing CR for each individual. Self-adapted schemes were employed for the scaling factor F in [24] and crossover rate CR in [25].
4.3 Hybrid Improved DE and Pattern Search Approach It is observed from the literature survey that various conflicting conclusions have been drawn with regard to the rules for setting the control parameters. Furthermore, in most of the cases conclusions lack sufficient justification as their validity is restricted to the problems, strategies, and parameter values considered in the investigations. So it can be concluded that choosing suitable control parameter values is a problem-dependent task.
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4.3.1 Improved Differential Evolution (IDE) In majority of population based search methods, the individuals are encouraged to sample diverse zones of the search space during the early stages of the search process. During the later stages of search process, it is important to regulate the movements of trial solutions finely so that they can explore the interior of a relatively small space in which the suspected
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global optimum lies. To meet the above objective, F and CR values are reduced with time from a predetermined maximum to a predetermined minimum value. The ranges of F and CR values are suitably chosen to maintain both exploitation (with small values) and exploration (with large values) power of the algorithm throughout the evolution process. Following
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expressions is used to vary the F and CR values during generations:
CR CRmin CRG CRmax max Gmax
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G G
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F Fmin FG Fmax max Gmax
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Linear decrease (IDE-1):
Exponential decrease (IDE-2):
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FG Fmax e ( K1G Gmax )
Parabolic decrease (IDE-3):
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CRG CRmax e ( K 2 G Gmax )
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FG FG 1 Fmin .Gmax G / Gmax Fmin . e
G G max / 4
CRG CRG 1 CRmin .Gmax G / Gmax CRmin . e
(17)
(18)
(19) (20)
2
G G max / 4
(21) 2
(22)
Where FG and CRG are the F and CR values at generation G, Gmax is the maximum generation, Fmax, Fmin and CRmax, CRmin are the maximum and minimum values of F and CR respectively. The constants K1 and K2 values are so chosen that F and CR values are at their maximum limit at the first generation (G=0) and minimum values at the last generation (G=Gmax). 4.3.2 Fine Tuning by Pattern Search Pattern Search (PS) optimization technique is a derivative free evolutionary algorithm suitable to solve a variety of optimization problems that lie outside the scope of the standard optimization methods. It is simple in concept, easy to
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implement and computationally efficient. It possesses a flexible and well-balanced operator to enhance and adapt the global search and fine tune local search [26]. By examining the neighborhood of the current solution, pattern search is very effective to exploit the local regions. PS performs individual learning by exploiting small local regions effectively in relatively short periods of time. In addition, its convergence to local minima for constrained problems as well as
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unconstrained problems has been proven in [26]. The PS algorithm computes a sequence of points that may or may not approaches to the optimal point. The algorithm starts with a set of points called mesh, around the initial points. The initial points or current points are provided by the IDE technique. The mesh is created by adding the current point to a scalar
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multiple of a set of vectors called a pattern. If a point in the mesh is having better objective function value, it becomes the
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current point at the next iteration [15-16]. The Pattern search begins at the initial point X0 that is given as a starting point by the IDE algorithm. At the first iteration, with a scalar = 1 called mesh size, the pattern vectors or direction vectors are constructed as [0 1], [1 0], [-1 0] and [0 -1]. The direction vectors are added to the initial point X0 to compute the mesh
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points as X0 + [0 1], X0 + [1 0], X0 + [-1 0] and X0 + [0 -1]. The algorithm computes the objective function at the mesh points in the same order. The algorithm polls the mesh points by computing their objective function values until it finds
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one whose value is smaller than the objective function value of X0. Then the poll is said to be successful when the objective function value decreases at some mesh point and the algorithm sets this point equal to X1. After a successful
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poll, the algorithm steps to iteration 2 and multiplies the current mesh size by 2. As the mesh size is increased by multiplying by a factor i.e. 2, this is called the expansion factor. So in 2nd iteration, the mesh points are: X1 + 2 *[0 1], X1
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+ 2 * [1 0], X1 +2 * [-1 0] and X1 + 2 *[0 -1] and the process is repeated until stopping criteria is met. Now if in a particular iteration, none of the mesh points has a smaller objective function value than the value at initial/current point at
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that iteration, the poll is said to be unsuccessful and same current point is used in the next iteration. Also, at the next iteration, the algorithm multiplies the current mesh size by 0.5, a contraction factor, so that the mesh size at the next iteration is smaller and the process is repeated until stopping criteria is met.
5 RESULTS AND DISCUSSIONS
5.1 Application of hIDEPS Algorithm The model of the system under study shown in Fig. 1 is developed in MATLAB/SIMULINK environment. Simulations are executed on a Pentium 4, 3 GHz, 504 MB RAM computer, in the MATLAB 7.0.1 environment. A sensor time constant of 15 ms for local input signal and transmission delay of 50 ms [6-7, 27] in addition to the sensor delay of 15 ms for remote input signal are assumed.
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In [28], DE and its various strategies are applied for the optimal design of shell-and-tube heat exchangers and shown that DE/best/. . . strategies are better than DE/rand/. . .. from ‘more likeliness’ as well as ‘speed’ point of view. In reference [18] a detailed analysis of effect of DE strategy and controller parameters for a dynamic problem has been presented. The suggested DE strategy and parameters are: strategy 9 (DE/best/2/bin), Scaling Factor F = 0.2 and Crossover Constant CR
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= 0.6, Population size NP = 40 and Generation G = 30. In the present paper the same control parameters are used but the parameters F and CR values are decreased exponentially as per (17) - (22). Series of algorithm runs are executed to select the minimum ranges of F and CR. The maximum and minimum value of F and CR are chosen as 1.0 and 0.1 respectively.
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The optimization processes is run 10 times and the results are summarized in Table I. For better illustration of evolution
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process, the minimum, maximum, mean and standard deviations of objective function values obtained in 10 runs are also given in Table I. To show the advantage of proposed IDE approach, the corresponding values with constant F and CR and decrease of F and CR values during the optimization run i.e. IDE (linear), IDE (exponential) and IDE (parabolic) are also
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given in Table I. It is clear from Table I that the minimum objective function value is obtained by constant F and CR are 107.097, 108.737 and 101.096 with F=1 & CR=1, F=01 & CR=0.1 and F=0.2 & CR=0.6 respectively. The minimum J
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value obtained decreases to 98.751, 93.661 and 91.118 for linear, exponential and parabolic decrease of F and CR values during the optimization run. It is clear from Table I that better results are obtained when the F and CR values are
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decreased in parabolic manner during the optimization run. Further, it can be noticed from Table I that form an evolutionary point of view, proposed IDE (parabolic) outperforms other approaches in terms of minimum, average and
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maximum J values. The final values of controller parameters obtained IDE (parabolic) algorithm are taken as the initial points for PS algorithm. PS is executed with a mesh size of 1, mesh expansion factor of 2 and mesh contraction factor of
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0.5. The maximum number of generations and function evaluations and are set to 10 and 100 respectively. The objective function value further decreases to 84.731 by the proposed hIDEPS approach as given in Table I. The flow chart of proposed hIDEPS algorithm is shown in appendix C. The final controller parameters and performance comparison for different cases discussed latter are given in Table II. For comparison, the corresponding values with DE optimized SSSC based controller with local input signal [7] and PSO optimized SSSC based controller with remote input signal [8] are also given in Table II. To test the usefulness and robustness of the designed controller, various operating conditions and different disturbances are considered. The response with proposed hIDEPS optimized PD-type MISO SSSC controller is shown with solid lines, the response with DE optimized Single Input Single Output (SISO) SSSC-based damping controller with local input signal [7] is shown with dotted lines and the response with PSO optimized SISO SSSC controller with remote input signal [8] is shown with dashed lines. The following cases are considered:
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5.1.1 Case-1: Nominal Loading ( Pe 0.85 p.u., 0 51.50 ) A 5-cycle, 3-phase fault is applied in the transmission line near Bus2 at t = 1.0 sec. The fault is cleared by tripping the faulted line and the line is reclosed after 5-cycles. The speed deviation, power angle and the SSSC injected voltage responses for the above severe disturbance are shown in Figs. 3-5 respectively. It can be observed from Figs. 3-5 that
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proposed hIDEPS optimized PD-type MISO SSSC controller exhibits better damping characteristic compared to DE and PSO optimized SSSC based controller with local [7] and remote [8] input signals respectively.
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5.1.2 Case-2: Light Loading ( Pe 0.5 p.u., 0 29.5 0 )
To evaluate the performance of the designed controller to operating condition, location and type of disturbance, the
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generator loading is changed to light loading condition and one of the transmission lines is tripped for 200 ms. The load near Bus 1 is also disconnected for the same duration. The system response for above contingency is depicted in Fig. 6
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which clearly shows the robustness of designed controller to changes in operating condition and type of disturbance. Moreover, hIDEPS optimized PD-type MISO SSSC controller outperforms both DE and PSO optimized SSSC controllers
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[7-8]. 5.1.3 Case-3: Heavy Loading ( Pe 1.0 p.u., 0 60.7 0 )
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The effectiveness of the designed controller is also demonstrated at heavy loading condition under another severe disturbance. A 3-pahe fault of 5-cycle duration is applied in one of the transmission line near Bus 3 at t=1.0 s. The fault is removed by tripping the faulty line for 53 ms. The original system is restored after the fault clearance. The system
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response for the above contingency is shown in Fig. 7. It is clear from Fig. 7 that the system is unstable with both DE
Ac ce
optimized SISO controller with local input signal and PSO optimized SISO controller with remote signal. The stability of the system is maintained and power system oscillations are effectively damped with proposed hIDEPS optimized PD-type MISO controller. It is also evident from Table II that for all the cases, minimum objective function value is obtained with the proposed approach compared to some reported approaches [7, 8]
5.2 Extension to Multi-machine Power System The proposed approach of designing PD-type MISO SSSC controller is further extended to a three-machine six-bus power system shown in Fig. 8. It is similar to the power systems used in references [6-9, 29]. The system consists of three generators divided in to two subsystems and are connected via an intertie. Following a disturbance, the two subsystems swing against each other resulting in instability. To improve the stability the line is sectionalized and a SSSC is assumed on the mid-point of the tie-line. The relevant data for the system is given in appendix.
14
Page 14 of 27
The objective function in this case is expressed as: t
J 10 4 ( | L | | I | ) t dt
(23)
0
Where, ΔωL and ΔωI are the speed deviations of local and inter-area modes of oscillations respectively and t is the time
ip t
range of the simulation The same approach as explained for SMIB system is followed to optimize the SSSC-based damping controller parameters
cr
for 3-machine case. Speed deviations of generators G1 & G3 and line power deviation of nearest Bus (i.e. Bus 5) are selected as the input signal to the proposed PD-Type MISO SSSC based damping controller. For the design purpose, a
us
severe disturbance is applied to the system under study. The optimized controller parameters and performance comparison for multi-machine power system is given in Table III. The following cases are considered:
an
5.2.1 Case-1:3-Phase Fault Disturbance
A 5-cycle, 3-phase self clearing fault is applied at line near Bus 6 at t = 1 s. The system response for the above
M
contingency is shown in Figs. 9-10 which clearly show the robustness of designed controller to changes in type of disturbance. It can also be noticed that proposed hIDEPS optimized PD-type MISO SSSC controller gives better transient
5.2.2 Case-2:Line Outage Disturbance
ed
response compared to DE and PSO based SISO controllers.
One of the parallel transmission lines between Bus 6 and Bus 1 is tripped for 10 cycles and the inter-area and local modes
pt
of oscillations are shown in Figs. 11-12. It is evident from Figs. 11-12 that superior damping characterises are observed with proposed hMDEPS optimized PD-type MISO SSSC controller compared to both DE optimized SSSC based
Ac ce
controller with local input signal [7] and PSO optimized SSSC based controller with remote input signal [8]. 5.2.3 Case-3:Simultaneous 3-Phase Fault & Small Disturbance A 3-phase fault is applied at Bus 2 at t=1.0 s. The fault is cleared after 5-cycles and at the same time the load at Bus 2 is disconnected for 10 cycles. The system responses for Case-3 are shown in Figs. 13-14 from which it is clear that the proposed hIDEPS optimized PD-type MISO SSSC controller damps the modal oscillations effectively, for simultaneous 3-phase fault and small disturbances. The objective function values for the above three cases are also provided in Table III for better illustration of advantage of the proposed hIDEPS MISO approach Simulation results show that the performance of the proposed hIDEPS optimized PD-type MISO SSSC controller is superior to DE and PSO optimized SISO controllers in all the cases. It is also obvious from simulation results that the proposed approach is highly effective in damping inter-area mode of oscillations. In addition, the impact of the control on 15
Page 15 of 27
the local modes is not negative and the local modes are also damped. Even though, the studied systems are simple, the structure and parameters are realistic. The systems are ideally suitable for studies related to the stability and control of local and inter-area modes, without the overwhelming complexity of actual inter-connected power systems for stability studies [6-9, 28]. By studying the above simple systems, the basic characteristics of the controller can be assessed and
ip t
analyzed, and conclusions can be drawn to give an insight for the implementation of SSSC in a large realistic power system.
cr
6 CONCLUSION
us
In this study, a hybrid technique involving an improved Differential Evolution (IDE) and Pattern Search (PS) approach (hIDEPS) is proposed to design PD-type Multi Input Single Output (MISO) SSSC based damping controller. The design objective is to improve power system stability by minimizing the time trajectory deviations following a disturbance.
an
Modification in the original DE algorithm is introduced by decreasing the Scaling Factor (F) and Crossover Constant (CR) from 1.0 to 0.1 in linear, exponential and parabolic manners. It is observed that better results are obtained when the F and
M
CR values are decrease in a parabolic manner during the optimization run. To fine tune the algorithm, Pattern Search (PS) technique is also employed. The superiority of the proposed controller structure and optimization technique is
ed
demonstrated for both Single Machine Infinite Bus and multi-machine power systems by comparing results with some of the approaches reported in literature. It is noticed that proposed hIDEPS optimized PD-type MISO controller provides
pt
superior damping performance in comparison to both remote and local input signals based conventional lead lag type Single Input Single Output (SISO) controllers.
Ac ce
APPENDIX
All data are in per unit unless specified otherwise. A. Single-Machine Infinite Bus Power System [7, 8]: Generator: SB = 2100 MVA, H =3.7 s, VB = 13.8 kV, f = 60 Hz, RS = 2.8544 e -3, Xd =1.305, Xd’= 0.296, Xd’’= 0.252, Xq = 0.474, Xq’ = 0.243, Xq’’ = 0.18, Td =1.01 s, Td’= 0.053 s, Tqo’’= 0.1 s., Load at Bus2: 250MW, Transformer: 2100 MVA, 13.8/500 kV, 60 Hz, R1 =R2= 0.002, L1 = 0,L2=0.12, D1/Yg connection, Rm = 500, Lm = 500, Transmission line: 3-Ph, 60 Hz, Length = 300 km each, R1 = 0.02546 Ω/ km, R0= 0.3864 Ω/ km, L1= 0.9337e-3 H/km, L0 = 4.1264e-3 H/ km, C1 = 12.74e-9 F/ km, C0 = 7.751e-9 F/ km, Hydraulic Turbine and Governor: Ka = 3.33, Ta = 0.07, Gmin = 0.01, Gmax = 0.97518, Vgmin = - 0.1 pu/s, Vgmax = 0.1 pu/s, Rp = 0.05, Kp = 1.163, Ki = 0.105, Kd = 0, Td = 0.01 s, =0, Tw = 2.67 s, Excitation
16
Page 16 of 27
System: TLP = 0.02 s, Ka =200, Ta = 0.001 s, Ke =1, Te =0, Tb = 0, Tc =0, Kf = 0.001, Tf = 0.1 s, Efmin = 0, Efmax = 7, K p = 0 B. Three-Machine Power System [7, 8]: Generators: SB1 = 4200 MVA, SB2 = SB3 = 2100 MVA, H =3.7 s, VB = 13.8 kV, f = 60 Hz, RS = 2.8544 e -3, Xd =1.305, Xd’= 0.296, Xd’’= 0.252, Xq = 0.474, Xq’ = 0.243, Xq’’ = 0.18, Td = 1.01 s, Td’ = 0.053 s, Tqo’’= 0.1 s., Loads: Load2=Load3=25MW,
Load4=250MW,
Transformers:
SBT1=2100MVA,
SBT2=
ip t
Load1=7500MW+1500MVAR,
SBT2=2100MVA, 13.8/500 kV, f = 60 Hz, R1 =R2= 0.002, L1 = 0, L2=0.12, D1/Yg connection, Rm = 500, Lm = 500,
cr
Transmission lines: 3-Ph, 60 Hz, Line lengths: L1 = 175 km, L2=50 km, L3=100 km, R1 = 0.02546 Ω/ km, R0= 0.3864 Ω/ km, L1= 0.9337e-3 H/km, L0 = 4.1264e-3 H/ km, C1 = 12.74e-9 F/ km, C0 = 7.751e-9 F/ km
us
SSSC: Converter rating: Snom = 100 MVA; Vnom = 500 kV; Frequency: f = 60 Hz; Maximum rate of change of reference voltage (Vqref ) = 3 pu/s; Converter impedances: R = 0.00533, L = 0.16; DC link nominal voltage: VDC = 40 kV; DC link -6
F; Injected Voltage regulator gains: KP = 0.00375, KI = 0.1875; DC Voltage
an
equivalent capacitance CDC = 375 x 10
regulator gains: K P =0.1 x 10 -3, KI = 20 x 10 -3 ; Injected voltage magnitude limit: Vq = ± 0.2
M
Initial operating conditions: Pe1 = 3480.6 MW; Qe1 = 2577.2 MVAR, Pe2 = 1280 MW; Qe2 = 444.27 MVAR, Pe3 = 880
Ac ce
pt
ed
MW; Qe3 = 256.33 MVAR
17
Page 17 of 27
C. Flow chart of proposed hybrid modified differential evolution and pattern search approach Start
A
Specify the DE parameters: Strategy, NP, Gmax, Fmax, Fmin, CRmax, Crmin
Specify mesh size, mesh expansion factor, mesh contraction factor, maximum number of function evaluations and iterations
Initialize the population
ip t
Iter. = 0
G=0 Evalute the ffitness of individuals by time domain simulation
Yes
Iter. = Iter. +1
Generate mutant vector and construct trial vector
Stop
cr
Is termination conditions reached? No
Iter. = Iter. +1
us
Set the starting point
Evalute trial vector
No
Is fitness of trial vector fitness of current individual ?
No
Yes
Is the poll successful?
No
Contract the mesh size
pt
Yes
G = Gmax ?
A
Ac ce
No
Expand the mesh size
ed
Update G, F and CR
M
Replace the current individual with the trial vector
Size of new population < Old population ?
Evalute the mesh points
Discard the trial vector
Yes
Yes
an
Construct Pattern vectors and create mesh points
18
Page 18 of 27
REFERENCES [1] M. Noroosian, G. Andersson, Damping of power system oscillations by use of controllable components, IEEE Trans. Power Del. 9 (4) (1994) 2046–2054. [2] N. G. Hingorani, L. Gyugyi, Understanding FACTS: concepts and technology of flexible ac transmission systems, New York: IEEE Press, 2000. compensation of transmission lines, IEEE Trans. Power Del. 12 (1997) 406–417.
ip t
[3] L. Gyugyi, C. D. Schauder, K. K. Sen, Static synchronous series compensator: a solid-state approach to the series [4] H. F. Wang, Static synchronous series compensator to damp power system oscillations, Elect. Power Systems Res.
cr
54 (2000) 113–119.
[5] A.D. Del Rosso, C.A. Canizares, V.M. Dona, A study of TCSC controller design for power system stability improvement, IEEE Trans. Power Syst. 18 (2003) 1487–1496.
us
[6] S. Panda, N. K. Yegireddy, S. Mahapatra, Hybrid BFOA-PSO approach for coordinated design of PSS and SSSCbased controller considering time delays, Int. J. Elect. Power & Energy Syst. 49 (2013) 221–233. [7] S. Panda, Differential evolution algorithm for SSSC-based damping controller design considering time delay, J. of
an
the Franklin Inst. 348 (20110 1903–1926.
[8] S. Panda, N. P. Padhy, R. N. Patel, Power system stability improvement by PSO optimized SSSC-based damping controller”, Elect. Power Comp. & Systs. 36 (2008) 468–490.
M
[9] S. Panda, S. C. Swain, P. K. Rautray, R. K. Malik, G. Panda, Design and analysis of SSSC-based supplementary damping controller, Sim. Modelling Practice & Theory. 18 (2010) 1199–1213. [10] R. Stron, K. Price, Differential evolution – A simple and efficient adaptive scheme for global optimization over
ed
continuous spaces, J. Global Optimization. 11 (1995) 341–359. [11] S. Das, P. N. Suganthan, Differential Evolution: A Survey of the State-of-the-Art, IEEE Trans. Evol. Compt. 15 (2011) 4–31.
pt
[12] J. Brest, S. Greiner, B. Boskovic, M. Mernik, V. Zumer, Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems, IEEE Trans. Evol. Compt., 10 (2005) 646–657.
Ac ce
[13] A. K. Qin, V. L. Huang, P. N. Suganthan, Differential evolution algorithm with strategy adaptation for global numerical optimization, IEEE Trans. Evol. Compt. 13 (2009) 398–417. [14] S.M. Islam, S. Das, S. Ghosh, S. Roy, P. N. Suganthan, An adaptive differential evolution algorithm with novel mutation and crossover strategies for global numerical optimization, IEEE Trans. Evol. Compt. 42 (2012) 482–500. [15] Y. Bao, Z. Hu, T. Xiong, A PSO and pattern search based memetic algorithm for SVMs parameters optimization, Neurocomputing. 117 (2013) 98–106. [16] A. K. Al-Othman, N. A. Ahmed, M. E. AlSharidah, H. A. AlMekhaizim, A hybrid real coded genetic algorithm – pattern search approach for selective harmonic elimination of PWM AC/AC voltage controller, Int. J. Elect. Power & Energy Syst. 44 (2013) 123–133. [17] P. Kundur, Power System Stability and Control. New York: McGraw-Hill, 1994. [18] B. Mohanty, S. Panda, P. K. Hota, Controller parameters tuning of differential evolution algorithm and its application to load frequency control of multi-source power system, Int. J. Elect. Power & Energy Syst. 54 (2014) 77–85.
19
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[19] J. Liu, J. Lampinen, A fuzzy adaptive differential evolution algorithm, Soft Compt.—a Fusion of Found. Methodologies & Appln. 9 (2005) 448–462. [20] R. Gamperle, S.D. Muller, P. Koumoutsakos, A parameter study for differential evolution, Proc. Int. Conf. Advances in Intelligent Systems, Fuzzy Systems, Evolutionary Computation, Switzerland, 10 (2002) 293–298. [21] M. M. Ali, A. Torn, Population set-based global optimization algorithms: some modifications and numerical studies, Comput. Oper. Res. 31 (2004) 1703–1725.
ip t
[22] S. Das, A. Konar, U. K. Chakraborty, Two improved differential evolution schemes for faster global search, Proc. Int. Conf. Genetic Evolut. Comput. Conf., Washington, DC, (2005) 991–998.
[23] U. K. Chakraborty, S. Das, A. Konar, Differential evolution with local neighborhood, Proc. Congr. Evolut. Comput.,
cr
Vancouver, BC, Canada, (2006) 2042–2049.
[24] M.G.H. Omran, A. Salman, A.P. Engelbrecht, Self-adaptive differential evolution, Lecture Notes in Artificial
us
Intelligence. Berlin, Germany: Springer-Verlag, (2005) 192–199.
[25] H. A. Abbass, The self-adaptive Pareto differential evolution algorithm, Proc. Congr. Evolut. Comput., Honolulu, HI, (2002) 831–836. 583.
an
[26] E. D. Dolan, R. M. Lewis, V. Torczon, On the local convergence of pattern search, SIAM J. Optim. 14 (2003) 567– [27] S. Ray, G.K. Venayagamoorthy, E.H. Watanabe, A computational approach to optimal damping controller design for
M
a GCSC, IEEE Trans. Power Delv. 23 (2008) 1673-1681.
[28] B.V. Babua, S.A. Munawarb, Differential evolution strategies for optimal design of shell-and-tube heat exchangers, Chemical Engineering Science. 62 (2007) 3720 – 3739
ed
[29] S. Mishra, P. K. Dash, P. K. Hota, M. Tripathy, Genetically optimized neuro-fuzzy IPFC for damping modal
Ac ce
pt
oscillations of power system, IEEE Trans. Power Systs. 17 (4) (2002) 1140-1147.
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Page 20 of 27
Figures
V1
PL
Vq T/F
VR
V2
I
Tr. Line
Bus1
Bus2
Vcnv
Generator
Bus3
Infinite-bus
ip t
VS
Load
SSSC
cr
VSC
VDC
Proportional Gain
1 1 sTTD
Inpput1
Sensor & Delay
1 1 sTTD
Inpput2
K DL
+_
Filter
+
Sensor & Delay
Washout
Filter N
+
Integrator
sTWR 1 sTWR
+
1 s
K PR
sTWL 1 sTWL
1 s
_
+
Integrator
DerivativeGain K DR
+
N
1 sT1L 1 sT2 L
an
DerivativeGain
PL
M
K PL
us
Fig. 1. Single machine infinite bus power system with SSSC
Washout
1 sT3 L Vq1 1 sT4 L
2-Stage Lead-Lag Vqref
1 sT1R 1 sT2 R
1 sT3 R 1 sT4 R
Vq max
+
+
+
Vq
Output
Vq min
Vq 2
2-Stage Lead-Lag
ed
Proportional Gain
Fig. 2. Structure of proposed PD-type MISO controller for SSSC -3
pt
x 10
SISO: PL signal: DE [7] SISO: signal: PSO [8]
Ac ce
8 6
Proposed PD-type MISO: hIDEPS
4
) . u. p(
2 0
-2 -4 -6
0
1
2
3
4
5
6
Time (sec)
Fig. 3. Speed deviation response for Case-1
21
Page 21 of 27
75 SISO: PL signal: DE [7]
70
SISO: signal: PSO [8] Proposed PD-type MISO: hIDEPS
65 60
55
ip t
) g. e d(
50
40
0
1
2
3
Time (sec)
4
5
6
us
35
cr
45
Fig. 4. Power angle response for Case-1
an
0.2 0.15
M
0.1 0.05
-0.05
ed
q
-0.1 -0.15 -0.2 1
pt
V
0
2
SISO: PL signal: DE [7] SISO: signal: PSO [8] Proposed PD-type MISO: hIDEPS
3
4
5
6
7
8
Time (sec)
Ac ce
) . u. p(
Fig. 5. SSSC injected voltage for Case-1
22
Page 22 of 27
-3
4
x 10
SISO: PL signal: DE [7] 3
SISO: signal: PSO [8] Proposed PD-type MISO: hIDEPS
2
1
ip t
) . u. p(
0
cr
-1 -2
0
1
2
3
4
5
Time (sec)
6
us
-3
7
8
Fig. 6. Speed deviation response for Case-2
an
200
ed
50
0
0
pt
100
1
2
3
Ac ce
) g. e d(
SISO: signal: PSO [8] Proposed PD-type MISO: hIDEPS
M
150
SISO: PL signal: DE [7]
4
5
6
7
8
9
10
Time (sec)
Fig. 7. Power angle response for Case-3
Bus2
L2
Bus4
G2
T2
L1
Bus5
Bus6
L1
Bus1
Load2
Bus3
L3
G1
SSSC
L1
G3
L1
T1 Load1
T3
Load3
Load4
Fig. 8. Three-machine six-bus power system
23
Page 23 of 27
-3
x 10
SISO:PL signal: DE [7] SISO: signal: PSO [8] Proposed PD-type MISO: hIDEPS
1
0.5
0
-0.5
-1 2
3
4
5
Time 9sec)
6
7
us
1
cr
2 1
ip t
) . u. p(
8
9
10
Fig. 9. Inter-area mode of oscillations for Case-1 -3
an
x 10 1.5
0.5
0
-0.5
-1.5
1
pt
-1
2
3
4
5
6
7
8
9
10
Time (sec)
Ac ce
3 2
ed
) . u. p(
SISO: signal: PSO [8] Proposed PD-type MISO: IDEPS
M
1
SISO: PL signal: DE [7]
Fig. 10. Local mode of oscillations for Case-1
24
Page 24 of 27
-3
2.5
x 10
SISO: PL signal: DE [7]
2
SISO: signal: PSO [8] Proposed PD-type MISO: hIDEPS
1.5 1
) . u. p( 2 1
0
ip t
0.5
-0.5
-1.5 0
1
2
3
4
5
Time (sec)
6
7
us
-2
cr
-1
8
9
10
Fig. 11. Inter-area mode of oscillations for Case-2 -4
an
x 10 6
2
3 2
0
ed
) . u. p(
SISO: signal: PSO [8] Proposed PD-type MISO: hIDEPS
M
4
SISO: PL signal: DE [7]
-2
2
3
4
5
6
7
8
9
10
Time (sec)
Ac ce
1
pt
-4
Fig. 12. Local mode of oscillations for Case-2
25
Page 25 of 27
-3
4
x 10
SISO: P signal: DE 7] L
3
SISO: signal: PSO [8] Proposed PD-type MISO: hIDEPS
2
) . u. p(
0
ip t
2 1
-1 -2
0
1
2
3
4
5
Time (sec)
6
7
8
us
-3
cr
1
9
10
Fig. 13. Inter-area mode of oscillations for Case-3 -3
an
x 10 2 1.5
) . u. p(
ed
0 -0.5
-1.5 1
pt
-1
2
3
Ac ce
0.5 3 2
SISO:Lead lag: signal: PSO [8] Proposed PD-type MISO: hIDEPS
M
1
SISO: PL signal: DE [7]
4
5
6
7
8
9
10
Time (sec)
Fig. 14. Local mode of oscillations for Case-3
26
Page 26 of 27
Tables
St.dev.
ip t
7.552 7.636 7.172 6.727 7.577 6.203 -
us
cr
TABLE I SIMULATION RESULTS OVER 10 INDEPENDENT RUNS Objective function Min. Ave. Max. value/Technique J J J DE (F=1, CR=1) 107.097 116.424 129.299 DE (F=0.1, CR=0.1) 108.737 121.557 133.073 DE (F=0.2, CR=0.6) 101.096 115.324 126.472 IDE (Linear) 98.751 110.311 121.815 IDE (Exponential) 93.661 109.985 125.391 IDE (Parabolic) 91.118 104.881 118.234 hIDEPS 84.731 -
an
TABLE II CONTROLLER PARAMETERS AND PERFORMANCE COMPARISON FOR SINGLE MACHINE INFINITE BUS POWER SYSTEM Parameters/ hIDEPS/ DE/ PSO/ Technique/ MISO SISO [7] SISO [8] Controller PL 2.5493
45.5084
KD
0.0741
0.0714
_
_
T1
0.1193
0.8498
0.583
0.2828
T2
0.8135
0.3663
0.5843
0.3
T3
0.6394
0.5844
0.0371
0.2765
0.7916
0.9895
0.9875
0.3
56.5627 37.5398 117.0536
55.5725 16.1653 86.6105
42.6596 15.6180 51.9753
pt
J: Case 1 J: Case-2 J: Case 3
ed
T4
M
KP
0.0068
73.9296
Ac ce
TABLE III CONTROLLER PARAMETERS AND PERFORMANCE COMPARISON FOR MULTI-MACHINE POWER SYSTEM Parameters/ hIDEPS/ DE/ PSO/ Technique/ MISO SISO [7] SISO [8] PL Controller KP
2.6938
92.4532
0.0082
59.4152
KI
0.4731
0.6418
_
_
T1
0.1148
0.7432
0.1809
0.3292
T2
0.6741
0.6151
0.9107
0.3
T3
0.2838
0.6207
0.501
0.2303
T4
0.2777
0.4782
0.7793
0.3
84.203 57.794 117.053
26.208 42.887 86.611
J: Case 1 J: Case-2 J: Case 3
21.085 34.636 51.975
27
Page 27 of 27
S1568-4946(15)00250-1 http://dx.doi.org/doi:10.1016/j.asoc.2015.04.023 ASOC 2913
To appear in:
Applied Soft Computing
Received date: Revised date: Accepted date:
19-5-2014 6-4-2015 6-4-2015
Please cite this article as: S. Panda, S.H. Kiran, S.S. Dash, C. Subramani, A PD-type multi input single output SSSC damping controller design employing hybrid improved differential evolution-pattern search approach, Applied Soft Computing Journal (2015), http://dx.doi.org/10.1016/j.asoc.2015.04.023 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Start
A
Specify the DE parameters: Strategy, NP, Gmax, Fmax, Fmin, CRmax, Crmin
Specify mesh size, mesh expansion factor, mesh contraction factor, maximum number of function evaluations and iterations
Initialize the population
Iter. = 0
G=0 Evalute the ffitness of individuals by time domain simulation
ip t
Yes
Is termination conditions reached? Iter. = Iter. +1
No
Iter. = Iter. +1
cr
Generate mutant vector and construct trial vector
Stop
Set the starting point
us
Evalute trial vector
Construct Pattern vectors and create mesh points No
Is fitness of trial vector fitness of current individual ?
Replace the current individual with the trial vector
Is the poll successful?
No
Contract the mesh size
Yes
A
pt
G = Gmax ?
Ac ce
Graphical Abstract
Yes
M
Size of new population < Old population ? No
No
Expand the mesh size
ed
Update G, F and CR
an
Yes
Yes
Evalute the mesh points
Discard the trial vector
1
Page 1 of 27
Highlights A novel PD type MISO controller is proposed for SSSC based damping controller. Hybrid improved DE and PS approach is proposed to optimize the controller parameters. In improved DE, control parameters F and CR are varied during optimization runs.
ip t
Both single machine infinite bus and multi-machine power systems are considered.
Ac ce
pt
ed
M
an
us
cr
Comparative results are provided to show the superiority of the proposed design approach.
2
Page 2 of 27
A PD-type multi input single output SSSC damping controller design employing hybrid improved differential evolution-pattern search approach *
Sidhartha Pandaa, S. Harish Kiranb, S. S. Dashc and C. Subramanid
a
Department of Electrical and Electronics Engineering, VSSUT, Burla, Odisha-768018, India b,c,d
ip t
Department of Electrical Engineering, SRM University, Chennai-603203, India
*
Corresponding author: e-mail: [email protected], Phone +91-9438251162
cr
Abstract—In this paper, a Proportional Derivative (PD)-type Multi Input Single Output (MISO) damping controller is
us
designed for Static Synchronous Series Compensator (SSSC) controller. Both local and remote signals with associated time delays are chosen as the input signal to the proposed MISO controller. The design problem is formulated as an
an
optimization problem and a hybrid Improved Differential Evolution and Pattern Search (hIDEPS) technique is employed to optimize the controller parameters. The improvement in Differential Evolution (DE) algorithm is introduced by changing two of its most important control parameters i.e. Scaling Factor F and Crossover Constant CR with an objective
M
of achieving improved performance of the algorithm. The superiority of proposed Improved DE (IDE) over original DE and hIDEPS over IDE has also been demonstrated. To show the effectiveness and robustness of the proposed design
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approach, simulation results are presented and compared with DE and Particle Swarm Optimization (PSO) optimized Single Input Single Output (SISO) SSSC based damping controllers for both Single Machine Infinite Bus (SMIB) power
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system and multi-machine power system. It is noticed that the proposed approach provides superior damping performance compared to some approaches available in literature.
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Key words—Power system stability; static synchronous series compensator (SSSC); improved differential evolution algorithm; pattern search; multi input single output controller.
1. INTRODUCTION
Active power oscillations in power transmission systems may arise in corridors between interconnected areas as a result of poor damping of the interconnection [1]. Active power oscillation limits the power transmission capacity of interconnections between areas. Power System Stabilizers (PSS) are generally employed to damp these oscillations, but PSS are not effective in some cases, particularly when inter-area oscillations of typically 0.2-0.7 Hz are present. Alternatively, Flexible AC Transmission Systems (FACTS) controllers can be employed to damp the power system oscillations [2]. Static Synchronous Series Compensator (SSSC) is one of the important members of series FACTS controller [3]. If a SSSC is installed in a power system to enhance the power transfer controllability, a supplementary 3
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damping controller could be designed for SSSC to damp the power system oscillations [4]. Despite the availability of a variety of controller, the fixed gain, lead-lag compensation type of controller structure continues to be the most popular with the electrical utilities because of the ease of on-line tuning and also lack of assurance of the stability by some adaptive or variable structure techniques [5-6]. Most of the previous works on stability
ip t
and damping improvement by SSSC are based on Single Input Single Output (SISO) based lead lag controllers using either local signal or remote signal [7-9]. To avoid additional costs associated with communication, input signal should preferably be locally measurable. However, local control signals, although easy to get, may not contain the desired
cr
oscillation modes. So, compared to wide-area signals, they are not as highly controllable and observable. Owing to the
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recent advances in optical fibre communication and global positioning systems, the wide-area measurement system can realize phasor measurement synchronously and deliver it to the control centre even in real time. Hence both local and remote signals can be used reliably as control input signals. In this paper, a Multi Input Single Output (MISO) controller is
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proposed as SSSC based damping controller. While considerable work has been reported for the improvement of controller structure of a Proportional Integral Derivative (PID) controller, surprisingly, hardly any attempt has been made
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to improve the structure of a lead lag controller. The structure of a lead lag controller consists of a gain block which acts as a proportion gain and there is scope to add an additional gain term i.e. derivative term to improve the system response
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and the performance of the controller. In view of the above, a PD-type MISO controller for SSSC is proposed in the present work to damp power system oscillations.
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The problem of FACTS controller parameter tuning is a complex task. The conventional techniques that are reported in literature pertaining to the tuning of FACTS controller suffer from heavy computation burden and the search process is
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likely to be trapped in local minima as the optimal solution may not be obtained. The growth in size and complexity of electric power systems has necessitated the use of intelligent systems that combine knowledge, techniques and methodologies from various sources for the real-time control of power systems. In recent years, a lot of interest has been drawn to the applications of intelligent techniques to power system problems. Differential Evolution (DE) is a populationbased direct search algorithm for global optimization capable of handling non-differentiable, non-linear and multi-modal objective functions, with few, easily chosen, control parameters [10]. DE uses a greedy selection procedure with inherent elitist features and has fewer control parameters, which can be tuned effectively [11]. But, the success of DE in solving a particular problem significantly depends on suitable choice of control parameter values namely the Scaling Factor (F) and Crossover Constant (CR) [12]. It is advantageous to use appropriate F and CR values at different stages of evolution/search process instead of using fixed F and CR values for the entire search process [13-14]. The key to achieving high performance for any meta-heuristic algorithm is to maintain a good balance between exploitation and 4
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exploration in the search process. DE being a global optimizing method is designed to explore the search space and most likely will give an optimal/near-optimal solution. On the other hand, local optimizing methods like Pattern Search (PS) are designed to exploit the local area, but they are usually not good at exploring wide search space and hence generally not applied alone for global optimization problems [15-16]. Due to their respective strength and weakness, there is motivation
hybrid improved DE and PS (hIDEPS) for the design of a SSSC based damping controller.
ip t
for the hybridization of DE and PS. In view of the above, an attempt has been made in this paper for the application of a
In this paper, a Multi Input Single Output (MISO) controller is proposed for SSSC to damp power system oscillations
cr
following a disturbance. The MISO controller consists of two PD-type lead lag controllers with both remote signal (speed
us
deviation signal) and local signal (tie-line power deviation signal). The design problem of proposed controller is formulated as an optimization problem and hIDEPS technique is employed to find the optimal controller parameters. The performance of the proposed controller is evaluated in two test systems subjected to different transient disturbances. To
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show the effectiveness and robustness of the proposed approach, simulation results are presented and compared with some
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SISO based damping controllers approaches reported in literature [7-8].
2. MATHEMATICAL MODELING OF SYSTEM UNDER STUDY
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2.1 Single Machine Infinite Bus Power System with SSSC A Single Machine Infinite Bus (SMIB) power system shown in Fig. 1 is considered at the first instance to design the PD-
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type MISO damping controller for SSSC. The system consists of a synchronous generator connected to an infinite-bus through a step-up transformer and a SSSC through a double circuit transmission line. The generator is provided with
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Hydraulic Turbine and Governor (HTG) and excitation system. The HTG consists of a hydraulic turbine, a governor system, and a servomotor. The excitation system consists of a voltage regulator and DC exciter, as recommended in IEEE Recommended Practice for Excitation System Models for Power System Stability Studies [17]. In Fig. 1, T/F represents the transformer; VS and VR are the generator terminal and infinite-bus voltages respectively; V1 and V2 are the bus voltages; VDC and Vcnv are the DC voltage source and output voltage of the SSSC converter respectively; I is the line current and PL is the total real power flow in the transmission line respectively. 2.2Modeling of Machine The dynamics of the stator, field and damper windings are included in the present analysis. Two-axis reference frame (d-q frame) is used to express the stator and rotor quantities. All rotor quantities are referred to stator (represented by primed variables) as given in (1)-(8): 5
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d q R q dt
V d R S id
d q R d dt
' kq 1
V kq'
2
R
' kd
' i kd
' kq 1
R
Where, d
d dt
' i kq
' ' R kq 2 i kq
d dt
2
1
' fd
(3)
ip t
V
i 'fd
' kd
d dt d ' kq dt
(4)
cr
' kd
' fd
' kq 1
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V
R
2
' L d i d L md ( i 'fd i kd )
' 'fd L' fd i 'fd L md ( i d i kd )
' kq 2
L 'kq
2
' i kq
2
L mq i q
' L q i q L mq i kq
' L 'kq 1 i kq 1 L mq i q
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In the above equations, the subscripts:
' kq 1
(6)
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' ' kd L'kd i kd L md ( i d i 'fd )
q
(5)
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' fd
(2)
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V q R S iq V
(1)
d and q stands for d -axis and q -axis quantities
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R and S stands for rotor and stator quantities
f and k stands for field and damper winding l and m stands for leakage and magnetizing inductance
The mechanical equations are given by: d 1 r ( Pe F r r P m ) dt J
(7)
d dt
(8)
r
Where, r and are angular velocity and angular position of the rotor respectively, Pe and Pm represent electrical and mechanical power respectively, J and Fr represent inertia and friction of rotor respectively. 6
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3 THE PROPOSED APPROACH 3.1 Structure of SSSC based Controller The proposed MISO controller structure consists of two PD-type lead lag controllers as shown in Fig. 2. Each lead lag structure consists of a proportional gain and a derivative gain block, a signal washout block and two-stage phase
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compensation block. Derivative mode improves stability of the system. However, when the input signal has sharp corners, the derivative term will produce unreasonable size control inputs to the plant. Also, any noise in the control input signal
cr
will result in large control output signals. These reasons often limit the practical applications of derivative term in the controller. The practical solution to the these problems is to put a first filter on the derivative term and tune its pole so that
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the chattering due to the noise does not occur since it attenuates high frequency noise. In view of the above a filter is used for the derivative term in the present paper.
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The phase compensation block provides the appropriate phase-lead characteristics to compensate for the phase lag between input and the output signals. The signal washout block with time constants acts as a high-pass filter to allow
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signals associated with oscillations in input signal as unchanged signals. Without the high-pass filter, steady changes in input would alter the output. The phase compensation blocks give the exact phase-lead characteristics to overcome the
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phase lag between input and the output signals. Vqref stands for the reference injected voltage as desired by the steady state power flow control loop. In the present study Vqref is assumed constant during large disturbance transient period as it
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acts quite slowly. The necessary value of compensation is obtained according to the change in the SSSC injected voltage
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Vq1 and Vq 2 which is added to Vqref .
3.2 Problem Formulation
In lead-lag structured controllers, the washout time constants are generally pre-specified [17]. In the present study, TWL = TWR = 10 s and filter coefficient N =100 are used. The proportional and derivative gains ( K DL , K DR & K PL , K PR ) and the
time constants T1L , T2 L , T3 L , T4 L & T1R , T2 R , T3 R T4 R are to be calculated. During steady state conditions Vq1 , Vq 2 and Vref are constant. During dynamic conditions the series injected voltage Vq is adjusted to damp power system oscillations
and is given by: Vq Vqref Vq1 Vq 2
(9)
An integral time absolute error of the speed deviations is taken as the objective function given by equation (10).
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t
J 10 4 | | t dt
(10)
0
Where, is the speed deviation; and t is the time range of the simulation. To calculate the objective function, a 5-cycle 3-phase fault is applied at the middle of one transmission line which is cleared by line outage for 5-cycles. Thus, the
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design problem is formulated as the following optimization problem: Minimize J
(11)
cr
Subject to
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K PL min K PL K PL max , K PR min K PR K PR max K IL min K IL K IL max , K IR min K IR K IR max
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T1L min T1L T1L max , T1R min T1R T1R max T2 L min T2 L T2 L max , T2 R min T2 R T2 R max min
T3 L T3 L
max
, T3 R min T3 R T3 R max
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T3 L
4 OPTIMIZATION TECHNIQUE
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4.1 Differential Evolution (DE)
(12)
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T4 L min T4 L T4 L max , T4 R min T4 R T4 R max
Differential Evolution (DE) algorithm is a simple but efficient search technique proposed by Storn and Price [10]. In DE
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algorithm, an optimization task consisting of D variables is represented by a D-dimensional vector. At the beginning, a population of NP solution vectors is initialized within the parameter bounds. Mutation, crossover and selection operators are applied to update the population. DE algorithm uses two generations; old generation and new generation of equal population size. Individuals of the current population become target vectors for the next generation. The mutation operation produces a mutant vector for each target vector, by adding the weighted difference between two randomly chosen vectors to a third vector. A trial vector is generated by the crossover operation by mixing the parameters of the mutant vector with those of the target vector. The trial vector replaces the target vector in the next generation if has a better fitness value than the target vector. The DE evolutionary operators are described below [10-11]: 4.1.1 Initialization of parameter At the beginning of a DE run i.e. at Generation G = 0, search parameters are initialized within the search space
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D D 1 constrained by the specified minimum and maximum bounds X min {x1min,G , , xmin, G } and X max { xmax,G ,, xmax,G } . The
initial value of j-th parameter in the i-th individual at first generation (G = 0) is generated by: j j j xij, 0 xmin rand (0,1) ( xmax xmin ),
j 1,2, , D
(13)
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4.1.2 Mutation Operator A parent vector X i ,G from the current generation is selected (known as target vector) and a mutant vector Vi , G is created by
cr
the differential mutation operation (known as donor vector). A child is produced by combining the donor with the target vector (known as trial vector). For each target vector X i , G at the generation G, its associated mutant vector
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Vi , G {vi1, G , , viD,G } can be generated through a number of mutation strategies. Several strategies can be employed in DE
optimization algorithm. The strategy in a DE algorithm is denoted by DE/x/y/z, where x represents the mutant vectors, y
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represents the number of difference vectors used in the mutation process and z represents the crossover scheme used in the crossover operation. A comparison of various strategies has been presented in [18] and the best strategy reported is
Vi , G X best , G F .( X r i ,G X r i , G ) F .( X r i ,G X r i , G ) 1
2
3
4
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DE/best/2/bin given by:
(14)
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The indices r1i , , r4i are randomly generated once for each mutant vector and are mutually exclusive integers within the range [1, NP]. X best ,G is the best individual vector within the best fitness value in the population at generation G and
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4.1.3 Crossover Operator
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mutation strategy is executed binomial crossover.
To enhance the potential diversity of the population, the crossover operation is applied to each pair of target vectors X i ,G and its corresponding mutant vector Vi ,G employed to generate a trial vector U i , G {ui1, G , , uiD, G } . The donor vector
enters the trial vector with a probability. The crossover operation can be expressed as: vij,G if uij,G j xi ,G if
(rand j CR or ( j jrand )
(rand j ,i CR ) or ( j jrand )
(15)
j 1,, D
The crossover rate CR is a user specified constant within the range and controls the fraction of parameter values copied from the mutant vector. jrand is a randomly selected integer in the range [1, D]. The condition j jrand makes certain that the trial vector differs from its corresponding target vector by at least one parameter.
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4.1.4 Selection Operator Selection operation is performed to keep the population size constant over successive generations. The objective function value of each trial vector is compared to that of its corresponding target vector in the current population. If the trial vector has better or equal objective function value than the corresponding target vector, the trial vector replaces the target vector
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and enters the population of the next generation. Otherwise, the target vector remains in the population for the next generation. The selection operation can be represented by:
(16)
i [1, , NP]
cr
U i , G if f (U i , G ) f ( X i , G ) X i ,G 1 otherwise X i , G
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4.2 Previous Work on DE Parameter Tuning
The performance of the DE algorithm is highly dependent on the choice of control parameter. Inappropriate selection of
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control parameters may lead to premature convergence or stagnation of the algorithm. The important control parameters that influence the performance of DE algorithm are: scaling factor F and crossover rate CR. It has been reported that the
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effectiveness, efficiency, and robustness of the DE algorithm depend on the control parameters setting and the control parameter settings may be different for different functions or the same function with different requirements [19]. Many
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empirical strategies have been reported in literature for selecting the control parameter settings. It has been reported that the searching capability and convergence speed of DE algorithm are very sensitive to the choice of control parameters F, CR and NP and suggested 0.6, 0.3-0.9 and 3D-8D for F, CR and NP respectively [20].
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Researchers have also proposed a number of techniques to change the control parameters during the run. In [21], an
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optimal value of 0.5 for CR and variable F in the range 0.4 to 1 for each generation was suggested. In [22], the factor F was linearly reduced with increasing generation count. A uniform distribution between 0.5 and 1.5 with a mean value of 1 was employed in [23] for selecting F. Normal distribution between 0.5 to 1.5 was employed in [24] for choosing CR for each individual. Self-adapted schemes were employed for the scaling factor F in [24] and crossover rate CR in [25].
4.3 Hybrid Improved DE and Pattern Search Approach It is observed from the literature survey that various conflicting conclusions have been drawn with regard to the rules for setting the control parameters. Furthermore, in most of the cases conclusions lack sufficient justification as their validity is restricted to the problems, strategies, and parameter values considered in the investigations. So it can be concluded that choosing suitable control parameter values is a problem-dependent task.
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4.3.1 Improved Differential Evolution (IDE) In majority of population based search methods, the individuals are encouraged to sample diverse zones of the search space during the early stages of the search process. During the later stages of search process, it is important to regulate the movements of trial solutions finely so that they can explore the interior of a relatively small space in which the suspected
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global optimum lies. To meet the above objective, F and CR values are reduced with time from a predetermined maximum to a predetermined minimum value. The ranges of F and CR values are suitably chosen to maintain both exploitation (with small values) and exploration (with large values) power of the algorithm throughout the evolution process. Following
cr
expressions is used to vary the F and CR values during generations:
CR CRmin CRG CRmax max Gmax
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G G
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F Fmin FG Fmax max Gmax
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Linear decrease (IDE-1):
Exponential decrease (IDE-2):
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FG Fmax e ( K1G Gmax )
Parabolic decrease (IDE-3):
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CRG CRmax e ( K 2 G Gmax )
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FG FG 1 Fmin .Gmax G / Gmax Fmin . e
G G max / 4
CRG CRG 1 CRmin .Gmax G / Gmax CRmin . e
(17)
(18)
(19) (20)
2
G G max / 4
(21) 2
(22)
Where FG and CRG are the F and CR values at generation G, Gmax is the maximum generation, Fmax, Fmin and CRmax, CRmin are the maximum and minimum values of F and CR respectively. The constants K1 and K2 values are so chosen that F and CR values are at their maximum limit at the first generation (G=0) and minimum values at the last generation (G=Gmax). 4.3.2 Fine Tuning by Pattern Search Pattern Search (PS) optimization technique is a derivative free evolutionary algorithm suitable to solve a variety of optimization problems that lie outside the scope of the standard optimization methods. It is simple in concept, easy to
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implement and computationally efficient. It possesses a flexible and well-balanced operator to enhance and adapt the global search and fine tune local search [26]. By examining the neighborhood of the current solution, pattern search is very effective to exploit the local regions. PS performs individual learning by exploiting small local regions effectively in relatively short periods of time. In addition, its convergence to local minima for constrained problems as well as
ip t
unconstrained problems has been proven in [26]. The PS algorithm computes a sequence of points that may or may not approaches to the optimal point. The algorithm starts with a set of points called mesh, around the initial points. The initial points or current points are provided by the IDE technique. The mesh is created by adding the current point to a scalar
cr
multiple of a set of vectors called a pattern. If a point in the mesh is having better objective function value, it becomes the
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current point at the next iteration [15-16]. The Pattern search begins at the initial point X0 that is given as a starting point by the IDE algorithm. At the first iteration, with a scalar = 1 called mesh size, the pattern vectors or direction vectors are constructed as [0 1], [1 0], [-1 0] and [0 -1]. The direction vectors are added to the initial point X0 to compute the mesh
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points as X0 + [0 1], X0 + [1 0], X0 + [-1 0] and X0 + [0 -1]. The algorithm computes the objective function at the mesh points in the same order. The algorithm polls the mesh points by computing their objective function values until it finds
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one whose value is smaller than the objective function value of X0. Then the poll is said to be successful when the objective function value decreases at some mesh point and the algorithm sets this point equal to X1. After a successful
ed
poll, the algorithm steps to iteration 2 and multiplies the current mesh size by 2. As the mesh size is increased by multiplying by a factor i.e. 2, this is called the expansion factor. So in 2nd iteration, the mesh points are: X1 + 2 *[0 1], X1
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+ 2 * [1 0], X1 +2 * [-1 0] and X1 + 2 *[0 -1] and the process is repeated until stopping criteria is met. Now if in a particular iteration, none of the mesh points has a smaller objective function value than the value at initial/current point at
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that iteration, the poll is said to be unsuccessful and same current point is used in the next iteration. Also, at the next iteration, the algorithm multiplies the current mesh size by 0.5, a contraction factor, so that the mesh size at the next iteration is smaller and the process is repeated until stopping criteria is met.
5 RESULTS AND DISCUSSIONS
5.1 Application of hIDEPS Algorithm The model of the system under study shown in Fig. 1 is developed in MATLAB/SIMULINK environment. Simulations are executed on a Pentium 4, 3 GHz, 504 MB RAM computer, in the MATLAB 7.0.1 environment. A sensor time constant of 15 ms for local input signal and transmission delay of 50 ms [6-7, 27] in addition to the sensor delay of 15 ms for remote input signal are assumed.
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In [28], DE and its various strategies are applied for the optimal design of shell-and-tube heat exchangers and shown that DE/best/. . . strategies are better than DE/rand/. . .. from ‘more likeliness’ as well as ‘speed’ point of view. In reference [18] a detailed analysis of effect of DE strategy and controller parameters for a dynamic problem has been presented. The suggested DE strategy and parameters are: strategy 9 (DE/best/2/bin), Scaling Factor F = 0.2 and Crossover Constant CR
ip t
= 0.6, Population size NP = 40 and Generation G = 30. In the present paper the same control parameters are used but the parameters F and CR values are decreased exponentially as per (17) - (22). Series of algorithm runs are executed to select the minimum ranges of F and CR. The maximum and minimum value of F and CR are chosen as 1.0 and 0.1 respectively.
cr
The optimization processes is run 10 times and the results are summarized in Table I. For better illustration of evolution
us
process, the minimum, maximum, mean and standard deviations of objective function values obtained in 10 runs are also given in Table I. To show the advantage of proposed IDE approach, the corresponding values with constant F and CR and decrease of F and CR values during the optimization run i.e. IDE (linear), IDE (exponential) and IDE (parabolic) are also
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given in Table I. It is clear from Table I that the minimum objective function value is obtained by constant F and CR are 107.097, 108.737 and 101.096 with F=1 & CR=1, F=01 & CR=0.1 and F=0.2 & CR=0.6 respectively. The minimum J
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value obtained decreases to 98.751, 93.661 and 91.118 for linear, exponential and parabolic decrease of F and CR values during the optimization run. It is clear from Table I that better results are obtained when the F and CR values are
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decreased in parabolic manner during the optimization run. Further, it can be noticed from Table I that form an evolutionary point of view, proposed IDE (parabolic) outperforms other approaches in terms of minimum, average and
pt
maximum J values. The final values of controller parameters obtained IDE (parabolic) algorithm are taken as the initial points for PS algorithm. PS is executed with a mesh size of 1, mesh expansion factor of 2 and mesh contraction factor of
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0.5. The maximum number of generations and function evaluations and are set to 10 and 100 respectively. The objective function value further decreases to 84.731 by the proposed hIDEPS approach as given in Table I. The flow chart of proposed hIDEPS algorithm is shown in appendix C. The final controller parameters and performance comparison for different cases discussed latter are given in Table II. For comparison, the corresponding values with DE optimized SSSC based controller with local input signal [7] and PSO optimized SSSC based controller with remote input signal [8] are also given in Table II. To test the usefulness and robustness of the designed controller, various operating conditions and different disturbances are considered. The response with proposed hIDEPS optimized PD-type MISO SSSC controller is shown with solid lines, the response with DE optimized Single Input Single Output (SISO) SSSC-based damping controller with local input signal [7] is shown with dotted lines and the response with PSO optimized SISO SSSC controller with remote input signal [8] is shown with dashed lines. The following cases are considered:
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5.1.1 Case-1: Nominal Loading ( Pe 0.85 p.u., 0 51.50 ) A 5-cycle, 3-phase fault is applied in the transmission line near Bus2 at t = 1.0 sec. The fault is cleared by tripping the faulted line and the line is reclosed after 5-cycles. The speed deviation, power angle and the SSSC injected voltage responses for the above severe disturbance are shown in Figs. 3-5 respectively. It can be observed from Figs. 3-5 that
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proposed hIDEPS optimized PD-type MISO SSSC controller exhibits better damping characteristic compared to DE and PSO optimized SSSC based controller with local [7] and remote [8] input signals respectively.
cr
5.1.2 Case-2: Light Loading ( Pe 0.5 p.u., 0 29.5 0 )
To evaluate the performance of the designed controller to operating condition, location and type of disturbance, the
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generator loading is changed to light loading condition and one of the transmission lines is tripped for 200 ms. The load near Bus 1 is also disconnected for the same duration. The system response for above contingency is depicted in Fig. 6
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which clearly shows the robustness of designed controller to changes in operating condition and type of disturbance. Moreover, hIDEPS optimized PD-type MISO SSSC controller outperforms both DE and PSO optimized SSSC controllers
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[7-8]. 5.1.3 Case-3: Heavy Loading ( Pe 1.0 p.u., 0 60.7 0 )
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The effectiveness of the designed controller is also demonstrated at heavy loading condition under another severe disturbance. A 3-pahe fault of 5-cycle duration is applied in one of the transmission line near Bus 3 at t=1.0 s. The fault is removed by tripping the faulty line for 53 ms. The original system is restored after the fault clearance. The system
pt
response for the above contingency is shown in Fig. 7. It is clear from Fig. 7 that the system is unstable with both DE
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optimized SISO controller with local input signal and PSO optimized SISO controller with remote signal. The stability of the system is maintained and power system oscillations are effectively damped with proposed hIDEPS optimized PD-type MISO controller. It is also evident from Table II that for all the cases, minimum objective function value is obtained with the proposed approach compared to some reported approaches [7, 8]
5.2 Extension to Multi-machine Power System The proposed approach of designing PD-type MISO SSSC controller is further extended to a three-machine six-bus power system shown in Fig. 8. It is similar to the power systems used in references [6-9, 29]. The system consists of three generators divided in to two subsystems and are connected via an intertie. Following a disturbance, the two subsystems swing against each other resulting in instability. To improve the stability the line is sectionalized and a SSSC is assumed on the mid-point of the tie-line. The relevant data for the system is given in appendix.
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The objective function in this case is expressed as: t
J 10 4 ( | L | | I | ) t dt
(23)
0
Where, ΔωL and ΔωI are the speed deviations of local and inter-area modes of oscillations respectively and t is the time
ip t
range of the simulation The same approach as explained for SMIB system is followed to optimize the SSSC-based damping controller parameters
cr
for 3-machine case. Speed deviations of generators G1 & G3 and line power deviation of nearest Bus (i.e. Bus 5) are selected as the input signal to the proposed PD-Type MISO SSSC based damping controller. For the design purpose, a
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severe disturbance is applied to the system under study. The optimized controller parameters and performance comparison for multi-machine power system is given in Table III. The following cases are considered:
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5.2.1 Case-1:3-Phase Fault Disturbance
A 5-cycle, 3-phase self clearing fault is applied at line near Bus 6 at t = 1 s. The system response for the above
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contingency is shown in Figs. 9-10 which clearly show the robustness of designed controller to changes in type of disturbance. It can also be noticed that proposed hIDEPS optimized PD-type MISO SSSC controller gives better transient
5.2.2 Case-2:Line Outage Disturbance
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response compared to DE and PSO based SISO controllers.
One of the parallel transmission lines between Bus 6 and Bus 1 is tripped for 10 cycles and the inter-area and local modes
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of oscillations are shown in Figs. 11-12. It is evident from Figs. 11-12 that superior damping characterises are observed with proposed hMDEPS optimized PD-type MISO SSSC controller compared to both DE optimized SSSC based
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controller with local input signal [7] and PSO optimized SSSC based controller with remote input signal [8]. 5.2.3 Case-3:Simultaneous 3-Phase Fault & Small Disturbance A 3-phase fault is applied at Bus 2 at t=1.0 s. The fault is cleared after 5-cycles and at the same time the load at Bus 2 is disconnected for 10 cycles. The system responses for Case-3 are shown in Figs. 13-14 from which it is clear that the proposed hIDEPS optimized PD-type MISO SSSC controller damps the modal oscillations effectively, for simultaneous 3-phase fault and small disturbances. The objective function values for the above three cases are also provided in Table III for better illustration of advantage of the proposed hIDEPS MISO approach Simulation results show that the performance of the proposed hIDEPS optimized PD-type MISO SSSC controller is superior to DE and PSO optimized SISO controllers in all the cases. It is also obvious from simulation results that the proposed approach is highly effective in damping inter-area mode of oscillations. In addition, the impact of the control on 15
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the local modes is not negative and the local modes are also damped. Even though, the studied systems are simple, the structure and parameters are realistic. The systems are ideally suitable for studies related to the stability and control of local and inter-area modes, without the overwhelming complexity of actual inter-connected power systems for stability studies [6-9, 28]. By studying the above simple systems, the basic characteristics of the controller can be assessed and
ip t
analyzed, and conclusions can be drawn to give an insight for the implementation of SSSC in a large realistic power system.
cr
6 CONCLUSION
us
In this study, a hybrid technique involving an improved Differential Evolution (IDE) and Pattern Search (PS) approach (hIDEPS) is proposed to design PD-type Multi Input Single Output (MISO) SSSC based damping controller. The design objective is to improve power system stability by minimizing the time trajectory deviations following a disturbance.
an
Modification in the original DE algorithm is introduced by decreasing the Scaling Factor (F) and Crossover Constant (CR) from 1.0 to 0.1 in linear, exponential and parabolic manners. It is observed that better results are obtained when the F and
M
CR values are decrease in a parabolic manner during the optimization run. To fine tune the algorithm, Pattern Search (PS) technique is also employed. The superiority of the proposed controller structure and optimization technique is
ed
demonstrated for both Single Machine Infinite Bus and multi-machine power systems by comparing results with some of the approaches reported in literature. It is noticed that proposed hIDEPS optimized PD-type MISO controller provides
pt
superior damping performance in comparison to both remote and local input signals based conventional lead lag type Single Input Single Output (SISO) controllers.
Ac ce
APPENDIX
All data are in per unit unless specified otherwise. A. Single-Machine Infinite Bus Power System [7, 8]: Generator: SB = 2100 MVA, H =3.7 s, VB = 13.8 kV, f = 60 Hz, RS = 2.8544 e -3, Xd =1.305, Xd’= 0.296, Xd’’= 0.252, Xq = 0.474, Xq’ = 0.243, Xq’’ = 0.18, Td =1.01 s, Td’= 0.053 s, Tqo’’= 0.1 s., Load at Bus2: 250MW, Transformer: 2100 MVA, 13.8/500 kV, 60 Hz, R1 =R2= 0.002, L1 = 0,L2=0.12, D1/Yg connection, Rm = 500, Lm = 500, Transmission line: 3-Ph, 60 Hz, Length = 300 km each, R1 = 0.02546 Ω/ km, R0= 0.3864 Ω/ km, L1= 0.9337e-3 H/km, L0 = 4.1264e-3 H/ km, C1 = 12.74e-9 F/ km, C0 = 7.751e-9 F/ km, Hydraulic Turbine and Governor: Ka = 3.33, Ta = 0.07, Gmin = 0.01, Gmax = 0.97518, Vgmin = - 0.1 pu/s, Vgmax = 0.1 pu/s, Rp = 0.05, Kp = 1.163, Ki = 0.105, Kd = 0, Td = 0.01 s, =0, Tw = 2.67 s, Excitation
16
Page 16 of 27
System: TLP = 0.02 s, Ka =200, Ta = 0.001 s, Ke =1, Te =0, Tb = 0, Tc =0, Kf = 0.001, Tf = 0.1 s, Efmin = 0, Efmax = 7, K p = 0 B. Three-Machine Power System [7, 8]: Generators: SB1 = 4200 MVA, SB2 = SB3 = 2100 MVA, H =3.7 s, VB = 13.8 kV, f = 60 Hz, RS = 2.8544 e -3, Xd =1.305, Xd’= 0.296, Xd’’= 0.252, Xq = 0.474, Xq’ = 0.243, Xq’’ = 0.18, Td = 1.01 s, Td’ = 0.053 s, Tqo’’= 0.1 s., Loads: Load2=Load3=25MW,
Load4=250MW,
Transformers:
SBT1=2100MVA,
SBT2=
ip t
Load1=7500MW+1500MVAR,
SBT2=2100MVA, 13.8/500 kV, f = 60 Hz, R1 =R2= 0.002, L1 = 0, L2=0.12, D1/Yg connection, Rm = 500, Lm = 500,
cr
Transmission lines: 3-Ph, 60 Hz, Line lengths: L1 = 175 km, L2=50 km, L3=100 km, R1 = 0.02546 Ω/ km, R0= 0.3864 Ω/ km, L1= 0.9337e-3 H/km, L0 = 4.1264e-3 H/ km, C1 = 12.74e-9 F/ km, C0 = 7.751e-9 F/ km
us
SSSC: Converter rating: Snom = 100 MVA; Vnom = 500 kV; Frequency: f = 60 Hz; Maximum rate of change of reference voltage (Vqref ) = 3 pu/s; Converter impedances: R = 0.00533, L = 0.16; DC link nominal voltage: VDC = 40 kV; DC link -6
F; Injected Voltage regulator gains: KP = 0.00375, KI = 0.1875; DC Voltage
an
equivalent capacitance CDC = 375 x 10
regulator gains: K P =0.1 x 10 -3, KI = 20 x 10 -3 ; Injected voltage magnitude limit: Vq = ± 0.2
M
Initial operating conditions: Pe1 = 3480.6 MW; Qe1 = 2577.2 MVAR, Pe2 = 1280 MW; Qe2 = 444.27 MVAR, Pe3 = 880
Ac ce
pt
ed
MW; Qe3 = 256.33 MVAR
17
Page 17 of 27
C. Flow chart of proposed hybrid modified differential evolution and pattern search approach Start
A
Specify the DE parameters: Strategy, NP, Gmax, Fmax, Fmin, CRmax, Crmin
Specify mesh size, mesh expansion factor, mesh contraction factor, maximum number of function evaluations and iterations
Initialize the population
ip t
Iter. = 0
G=0 Evalute the ffitness of individuals by time domain simulation
Yes
Iter. = Iter. +1
Generate mutant vector and construct trial vector
Stop
cr
Is termination conditions reached? No
Iter. = Iter. +1
us
Set the starting point
Evalute trial vector
No
Is fitness of trial vector fitness of current individual ?
No
Yes
Is the poll successful?
No
Contract the mesh size
pt
Yes
G = Gmax ?
A
Ac ce
No
Expand the mesh size
ed
Update G, F and CR
M
Replace the current individual with the trial vector
Size of new population < Old population ?
Evalute the mesh points
Discard the trial vector
Yes
Yes
an
Construct Pattern vectors and create mesh points
18
Page 18 of 27
REFERENCES [1] M. Noroosian, G. Andersson, Damping of power system oscillations by use of controllable components, IEEE Trans. Power Del. 9 (4) (1994) 2046–2054. [2] N. G. Hingorani, L. Gyugyi, Understanding FACTS: concepts and technology of flexible ac transmission systems, New York: IEEE Press, 2000. compensation of transmission lines, IEEE Trans. Power Del. 12 (1997) 406–417.
ip t
[3] L. Gyugyi, C. D. Schauder, K. K. Sen, Static synchronous series compensator: a solid-state approach to the series [4] H. F. Wang, Static synchronous series compensator to damp power system oscillations, Elect. Power Systems Res.
cr
54 (2000) 113–119.
[5] A.D. Del Rosso, C.A. Canizares, V.M. Dona, A study of TCSC controller design for power system stability improvement, IEEE Trans. Power Syst. 18 (2003) 1487–1496.
us
[6] S. Panda, N. K. Yegireddy, S. Mahapatra, Hybrid BFOA-PSO approach for coordinated design of PSS and SSSCbased controller considering time delays, Int. J. Elect. Power & Energy Syst. 49 (2013) 221–233. [7] S. Panda, Differential evolution algorithm for SSSC-based damping controller design considering time delay, J. of
an
the Franklin Inst. 348 (20110 1903–1926.
[8] S. Panda, N. P. Padhy, R. N. Patel, Power system stability improvement by PSO optimized SSSC-based damping controller”, Elect. Power Comp. & Systs. 36 (2008) 468–490.
M
[9] S. Panda, S. C. Swain, P. K. Rautray, R. K. Malik, G. Panda, Design and analysis of SSSC-based supplementary damping controller, Sim. Modelling Practice & Theory. 18 (2010) 1199–1213. [10] R. Stron, K. Price, Differential evolution – A simple and efficient adaptive scheme for global optimization over
ed
continuous spaces, J. Global Optimization. 11 (1995) 341–359. [11] S. Das, P. N. Suganthan, Differential Evolution: A Survey of the State-of-the-Art, IEEE Trans. Evol. Compt. 15 (2011) 4–31.
pt
[12] J. Brest, S. Greiner, B. Boskovic, M. Mernik, V. Zumer, Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems, IEEE Trans. Evol. Compt., 10 (2005) 646–657.
Ac ce
[13] A. K. Qin, V. L. Huang, P. N. Suganthan, Differential evolution algorithm with strategy adaptation for global numerical optimization, IEEE Trans. Evol. Compt. 13 (2009) 398–417. [14] S.M. Islam, S. Das, S. Ghosh, S. Roy, P. N. Suganthan, An adaptive differential evolution algorithm with novel mutation and crossover strategies for global numerical optimization, IEEE Trans. Evol. Compt. 42 (2012) 482–500. [15] Y. Bao, Z. Hu, T. Xiong, A PSO and pattern search based memetic algorithm for SVMs parameters optimization, Neurocomputing. 117 (2013) 98–106. [16] A. K. Al-Othman, N. A. Ahmed, M. E. AlSharidah, H. A. AlMekhaizim, A hybrid real coded genetic algorithm – pattern search approach for selective harmonic elimination of PWM AC/AC voltage controller, Int. J. Elect. Power & Energy Syst. 44 (2013) 123–133. [17] P. Kundur, Power System Stability and Control. New York: McGraw-Hill, 1994. [18] B. Mohanty, S. Panda, P. K. Hota, Controller parameters tuning of differential evolution algorithm and its application to load frequency control of multi-source power system, Int. J. Elect. Power & Energy Syst. 54 (2014) 77–85.
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[19] J. Liu, J. Lampinen, A fuzzy adaptive differential evolution algorithm, Soft Compt.—a Fusion of Found. Methodologies & Appln. 9 (2005) 448–462. [20] R. Gamperle, S.D. Muller, P. Koumoutsakos, A parameter study for differential evolution, Proc. Int. Conf. Advances in Intelligent Systems, Fuzzy Systems, Evolutionary Computation, Switzerland, 10 (2002) 293–298. [21] M. M. Ali, A. Torn, Population set-based global optimization algorithms: some modifications and numerical studies, Comput. Oper. Res. 31 (2004) 1703–1725.
ip t
[22] S. Das, A. Konar, U. K. Chakraborty, Two improved differential evolution schemes for faster global search, Proc. Int. Conf. Genetic Evolut. Comput. Conf., Washington, DC, (2005) 991–998.
[23] U. K. Chakraborty, S. Das, A. Konar, Differential evolution with local neighborhood, Proc. Congr. Evolut. Comput.,
cr
Vancouver, BC, Canada, (2006) 2042–2049.
[24] M.G.H. Omran, A. Salman, A.P. Engelbrecht, Self-adaptive differential evolution, Lecture Notes in Artificial
us
Intelligence. Berlin, Germany: Springer-Verlag, (2005) 192–199.
[25] H. A. Abbass, The self-adaptive Pareto differential evolution algorithm, Proc. Congr. Evolut. Comput., Honolulu, HI, (2002) 831–836. 583.
an
[26] E. D. Dolan, R. M. Lewis, V. Torczon, On the local convergence of pattern search, SIAM J. Optim. 14 (2003) 567– [27] S. Ray, G.K. Venayagamoorthy, E.H. Watanabe, A computational approach to optimal damping controller design for
M
a GCSC, IEEE Trans. Power Delv. 23 (2008) 1673-1681.
[28] B.V. Babua, S.A. Munawarb, Differential evolution strategies for optimal design of shell-and-tube heat exchangers, Chemical Engineering Science. 62 (2007) 3720 – 3739
ed
[29] S. Mishra, P. K. Dash, P. K. Hota, M. Tripathy, Genetically optimized neuro-fuzzy IPFC for damping modal
Ac ce
pt
oscillations of power system, IEEE Trans. Power Systs. 17 (4) (2002) 1140-1147.
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Page 20 of 27
Figures
V1
PL
Vq T/F
VR
V2
I
Tr. Line
Bus1
Bus2
Vcnv
Generator
Bus3
Infinite-bus
ip t
VS
Load
SSSC
cr
VSC
VDC
Proportional Gain
1 1 sTTD
Inpput1
Sensor & Delay
1 1 sTTD
Inpput2
K DL
+_
Filter
+
Sensor & Delay
Washout
Filter N
+
Integrator
sTWR 1 sTWR
+
1 s
K PR
sTWL 1 sTWL
1 s
_
+
Integrator
DerivativeGain K DR
+
N
1 sT1L 1 sT2 L
an
DerivativeGain
PL
M
K PL
us
Fig. 1. Single machine infinite bus power system with SSSC
Washout
1 sT3 L Vq1 1 sT4 L
2-Stage Lead-Lag Vqref
1 sT1R 1 sT2 R
1 sT3 R 1 sT4 R
Vq max
+
+
+
Vq
Output
Vq min
Vq 2
2-Stage Lead-Lag
ed
Proportional Gain
Fig. 2. Structure of proposed PD-type MISO controller for SSSC -3
pt
x 10
SISO: PL signal: DE [7] SISO: signal: PSO [8]
Ac ce
8 6
Proposed PD-type MISO: hIDEPS
4
) . u. p(
2 0
-2 -4 -6
0
1
2
3
4
5
6
Time (sec)
Fig. 3. Speed deviation response for Case-1
21
Page 21 of 27
75 SISO: PL signal: DE [7]
70
SISO: signal: PSO [8] Proposed PD-type MISO: hIDEPS
65 60
55
ip t
) g. e d(
50
40
0
1
2
3
Time (sec)
4
5
6
us
35
cr
45
Fig. 4. Power angle response for Case-1
an
0.2 0.15
M
0.1 0.05
-0.05
ed
q
-0.1 -0.15 -0.2 1
pt
V
0
2
SISO: PL signal: DE [7] SISO: signal: PSO [8] Proposed PD-type MISO: hIDEPS
3
4
5
6
7
8
Time (sec)
Ac ce
) . u. p(
Fig. 5. SSSC injected voltage for Case-1
22
Page 22 of 27
-3
4
x 10
SISO: PL signal: DE [7] 3
SISO: signal: PSO [8] Proposed PD-type MISO: hIDEPS
2
1
ip t
) . u. p(
0
cr
-1 -2
0
1
2
3
4
5
Time (sec)
6
us
-3
7
8
Fig. 6. Speed deviation response for Case-2
an
200
ed
50
0
0
pt
100
1
2
3
Ac ce
) g. e d(
SISO: signal: PSO [8] Proposed PD-type MISO: hIDEPS
M
150
SISO: PL signal: DE [7]
4
5
6
7
8
9
10
Time (sec)
Fig. 7. Power angle response for Case-3
Bus2
L2
Bus4
G2
T2
L1
Bus5
Bus6
L1
Bus1
Load2
Bus3
L3
G1
SSSC
L1
G3
L1
T1 Load1
T3
Load3
Load4
Fig. 8. Three-machine six-bus power system
23
Page 23 of 27
-3
x 10
SISO:PL signal: DE [7] SISO: signal: PSO [8] Proposed PD-type MISO: hIDEPS
1
0.5
0
-0.5
-1 2
3
4
5
Time 9sec)
6
7
us
1
cr
2 1
ip t
) . u. p(
8
9
10
Fig. 9. Inter-area mode of oscillations for Case-1 -3
an
x 10 1.5
0.5
0
-0.5
-1.5
1
pt
-1
2
3
4
5
6
7
8
9
10
Time (sec)
Ac ce
3 2
ed
) . u. p(
SISO: signal: PSO [8] Proposed PD-type MISO: IDEPS
M
1
SISO: PL signal: DE [7]
Fig. 10. Local mode of oscillations for Case-1
24
Page 24 of 27
-3
2.5
x 10
SISO: PL signal: DE [7]
2
SISO: signal: PSO [8] Proposed PD-type MISO: hIDEPS
1.5 1
) . u. p( 2 1
0
ip t
0.5
-0.5
-1.5 0
1
2
3
4
5
Time (sec)
6
7
us
-2
cr
-1
8
9
10
Fig. 11. Inter-area mode of oscillations for Case-2 -4
an
x 10 6
2
3 2
0
ed
) . u. p(
SISO: signal: PSO [8] Proposed PD-type MISO: hIDEPS
M
4
SISO: PL signal: DE [7]
-2
2
3
4
5
6
7
8
9
10
Time (sec)
Ac ce
1
pt
-4
Fig. 12. Local mode of oscillations for Case-2
25
Page 25 of 27
-3
4
x 10
SISO: P signal: DE 7] L
3
SISO: signal: PSO [8] Proposed PD-type MISO: hIDEPS
2
) . u. p(
0
ip t
2 1
-1 -2
0
1
2
3
4
5
Time (sec)
6
7
8
us
-3
cr
1
9
10
Fig. 13. Inter-area mode of oscillations for Case-3 -3
an
x 10 2 1.5
) . u. p(
ed
0 -0.5
-1.5 1
pt
-1
2
3
Ac ce
0.5 3 2
SISO:Lead lag: signal: PSO [8] Proposed PD-type MISO: hIDEPS
M
1
SISO: PL signal: DE [7]
4
5
6
7
8
9
10
Time (sec)
Fig. 14. Local mode of oscillations for Case-3
26
Page 26 of 27
Tables
St.dev.
ip t
7.552 7.636 7.172 6.727 7.577 6.203 -
us
cr
TABLE I SIMULATION RESULTS OVER 10 INDEPENDENT RUNS Objective function Min. Ave. Max. value/Technique J J J DE (F=1, CR=1) 107.097 116.424 129.299 DE (F=0.1, CR=0.1) 108.737 121.557 133.073 DE (F=0.2, CR=0.6) 101.096 115.324 126.472 IDE (Linear) 98.751 110.311 121.815 IDE (Exponential) 93.661 109.985 125.391 IDE (Parabolic) 91.118 104.881 118.234 hIDEPS 84.731 -
an
TABLE II CONTROLLER PARAMETERS AND PERFORMANCE COMPARISON FOR SINGLE MACHINE INFINITE BUS POWER SYSTEM Parameters/ hIDEPS/ DE/ PSO/ Technique/ MISO SISO [7] SISO [8] Controller PL 2.5493
45.5084
KD
0.0741
0.0714
_
_
T1
0.1193
0.8498
0.583
0.2828
T2
0.8135
0.3663
0.5843
0.3
T3
0.6394
0.5844
0.0371
0.2765
0.7916
0.9895
0.9875
0.3
56.5627 37.5398 117.0536
55.5725 16.1653 86.6105
42.6596 15.6180 51.9753
pt
J: Case 1 J: Case-2 J: Case 3
ed
T4
M
KP
0.0068
73.9296
Ac ce
TABLE III CONTROLLER PARAMETERS AND PERFORMANCE COMPARISON FOR MULTI-MACHINE POWER SYSTEM Parameters/ hIDEPS/ DE/ PSO/ Technique/ MISO SISO [7] SISO [8] PL Controller KP
2.6938
92.4532
0.0082
59.4152
KI
0.4731
0.6418
_
_
T1
0.1148
0.7432
0.1809
0.3292
T2
0.6741
0.6151
0.9107
0.3
T3
0.2838
0.6207
0.501
0.2303
T4
0.2777
0.4782
0.7793
0.3
84.203 57.794 117.053
26.208 42.887 86.611
J: Case 1 J: Case-2 J: Case 3
21.085 34.636 51.975
27
Page 27 of 27