Received 25 August 2014; revised 18 September 2014; accepted 26 September 2014

ABSTRACT

The use of the supplementary controllers of a High Voltage Direct Current (HVDC) based on Voltage Source Converter (VSC) to damp low Frequency oscillations in a weakly connected system is surveyed. Also, singular value decomposition (SVD)-based approach is used to analyze and assess the controllability of the poorly damped electromechanical modes by VSC-HVDC different control channels. The problem of supplementary damping controller based VSC-HVDC system is formulated as an optimization problem according to the time domain-based objective function which is solved using quantum-behaved particle swarm optimization (QPSO). Individual designs of the HVDC controllers using QPSO method are evaluated. The effectiveness of the proposed controllers on damping low frequency oscillations is checked through eigenvalue analysis and non-linear time simulation under various disturbance conditions over a wide range of loading.

Keywords:

VSC-HVDC, Power System Stability, Quantum Particle Swarm Optimization, Supplemetary Damping Controller

1. Introduction

Large interconnected AC systems have many well-known advantages. However, larger interconnected AC systems also increase the system complexity from the operation point of view, and might adversely decrease the system reliability [1] -[3] . Steady state stability, lack of reactive power supply, voltage stability, electromechanical oscillations and transient stability are common problems that can happen in expanded power systems and transmit a large amount of power over long distance transmission lines [3] [4] . Increasing power system complexity gives rise to low frequency oscillations in the range of 0.2 - 3.0 Hz. If not well damped, these oscillations may keep growing in magnitude until loss of synchronism results [5] . In order to damp these power system oscillations and increase system oscillations stability, the installation of power system stabilizer (PSS) is both economical and effective [5] [6] . However, PSSs may adversely affect voltage profile, result in leading power factor, and may not be able to suppress oscillations resulting from severe disturbances, especially those three-phase faults which may occur at the generator terminals [6] . Flexible AC transmission systems devices, such as Static VAR Compensators (SVC), Thyristor Control Series Compensators (TCSC), Static Synchronous Compensators (ST-ATCOM), and Unified Power Flow Controller (UPFC), are one of the recent propositions to alleviate such situations by controlling the power flow along the transmission lines and improving power oscillations damping [6] [7] . Recently High Voltage Direct Current (HVDC) systems have greatly increased. They interconnect large power systems offering numerous technical and economic advantages. This interest results from practical characteristics and performance that include for example, nonsynchronous interconnection, control of power flow and modulation to increase stability limits [8] . It is well known that HVDC may improve the transient and dynamic performance of the interconnected AC/DC system due to its fast electronic control of power flow also transient stability of the AC systems in a composite AC-DC system can be improved by taking advantage of the fast controllability of HVDC converters [6] - [10] . The conventional HVDC has several limitations and undesirable characteristics including being physically large and requiring the AC network with sufficient short-circuit ratio [11] - [13] . The Voltage Source Converter based on HVDC (VSC HVDC), which uses modern semiconductors with self-commuted ability, overcomes the disadvantages of conventional HVDC and is therefore more suitable for a weak AC network or a passive network without any power sources [13] . The control of the voltage magnitude and the phase angle of the VSCs make the use of separate control for active and reactive power possible. The active power loop can be set to control either the active power or the dc-side voltage [14] [15] . A traditional lead-lag damping controller structure is preferred by the power system utilities because of the ease of on-line tuning and also lack of assurance of the stability by some adaptive or variable structure methods [7] [16] [17] . Having several local optimum parameters for a lead-lag controller, using of traditional optimization approach is not suitable for such a problem. Thus, the heuristic methods as solution for finding global optimization are developed [18] - [20] . Particle swarm optimization (PSO) is a novel population based on metaheuristic, which utilizes the swarm intelligence generated by the cooperation and competition between the particle in a swarm and has emerged as a useful tool for engineering optimization [21] . This new approach features many advantages; it is simple, flexible, and fast and can be coded in few lines. Also, its storage requirement is minimal. However, the main disadvantage is that the PSO algorithm is not guaranteed to be globally convergent. In order to overcome this drawback and improve optimization synthesis, in this paper, a quantum-behaved PSO technique is proposed for optimal tuning of HVDC based damping controller for enhancing of power systems low frequency oscillations damping. In this paper a novel approach is presented to model parallel AC/DC power system namely Phillips-Heffron model based d-q algorithm in order to study dynamical stability of system. In addition, a block diagram representation is formed to analyze the system stability characteristics. Also, singular value decomposition (SVD) is used to choose damping control signal which has most effect on damping the electromechanical (EM) mode oscillations. A very powerful tool commonly used for this purpose is Popov-Belevitch-Hautus (PBH), which can be used to evaluate the EM mode controllability of the PSS and the different VSC HVDC controllers. A single machine infinite bus (SMIB) system equipped with a PSS and a VSC HVDC controller is used in this study. The problem of damping controllers design is formulated as an optimization problem to be solved using QPSO. The aim of the optimization is to search for the optimum controller parameter settings that maximize the minimum damping ratio of the system.

2. Problem Statement

Figure 1 shows a SMIB system equipped with a VSC HVDC. As it can be seen the infinite bus is supplied by HVAC parallel connected with a VSC HVDC power transmission system. The VSC HVDC consists of two coupling transformer, two three-phase IGBT based voltage source converters (VSCs). These two converters are connected either back-to-back or joined by a DC cable, depending on the application.

The AC side of each converter is connected to the line through a coupling transformer. The first voltage

source converter behaves as a rectifier. It regulates the DC link voltage and maintains the magnitude of the voltage at the connected terminal. The second voltage source converter acts as a controlled voltage source, which controls power flow in VSC HVDC feeder. The four input control signals to the VSC HVDC are where are the amplitude modulation ratio and are phase angle of the control signals of each VSC respectively.

2.1. Power System Nonlinear Model

By applying Park’s transformation and neglecting the resistance and transients of the coupling transformers, the VSC HVDC can be modeled:

(1)

(2)

(3)

(4)

(5)

where and are the middle bus voltage, infinite bus voltage, flowed current to rectifier and inverter respectively. And are the DC link capacitance and voltage, respectively. and are the DC capacitances and voltages of rectifier and inverter respectively.

The non-linear model of the SMIB system of Figure 1 is:

(6)

(7)

(8)

(9)

where:, , , , , where and are the input and output power , respectively; and the inertia constant and damping coefficient , respectively; the synchronous speed; and the rotor angle and speed, respectively; and the generator internal, field and terminal voltages, respectively; the open circuit field time constant; and the d-axis, d-axis transient reactance, and q-axis reactance, respectively; and the exciter gain and time constant, respectively; the reference voltage.

Also, from Figure 1 we have:

(10)

(11)

(12)

where, , and are the armature current, rectifier voltage, infinite bus current and voltage respectively. From Equations (10)-(12) we can have:

(13)

(14)

And for inverter side:

(15)

(16)

2.2. Power System Linearized Model

By linearizing Equations (1)-(7) (13)-(16):

(17)

(18)

(19)

(20)

where:

(21)

(22)

(23)

(24)

Substitute Equations (21)-(23) in (17)-(20) we can obtain the state variable of the power system installed with the VSC HVDC to be (state space model):

and

(25)

where and are the linearization of the input control signals of the VSC HVDC and PSS output respectively. The linearized dynamic model of Equation (25) can be shown by Figure 2. In this figure and are defined:

It can be seen that the configuration of the Phillips-Heffron model is exactly the same as that installed with SVC, TCSC, TCPS, UPFC and STATCOM. Also from Equation (25) it can be seen that there five choice of input control signals of the VSC HVDC to superimpose on the damping function of the VSC HVDC and. Therefore, in designing the damping controller of the VSC HVDC, besides setting its parameters, the selection of the input control signal of the VSC HVDC to superimpose on the damping function of the VSC HVDC is also important.

3. PSO versus QPSO

In a PSO system [21] -[23] , multiple candidate solutions coexist and cooperate simultaneously. Each solution candidate, called a “particle”, flies in the problem space (similar to the search process for food of a bird swarm) looking for the optimal position. A particle with time adjusts its position to its own experience, while adjusting to the experience of neighboring particles. If a particle discovers a promising new solution, all the other particles will move closer to it, exploring the region more thoroughly in the process.

PSO starts [22] with a population of random solutions “particles” in a D-dimension space. The i_th particle is represented by. Each particle keeps track of its coordinates in hyperspace, which are associated with the fittest solution it has achieved so far. The value of the fitness for particle (pbest) is also stored as. The global version of the PSO keeps track of the overall best value (gbest), and its location, obtained thus far by any particle in the population [21] [22] . PSO consists of, at each step, changing the velocity of each particle toward its pbest and gbest according to following Equations:

(26)

(27)

where, = pbest and = gbest

PSO algorithm is as follow:

Step 1: Initialize an array of particles with random positions and their associated velocities to satisfy the inequality constraints.

Step 2: Check for the satisfaction of the equality constraints and modify the solution if required.

Step 3: Evaluate the fitness function of each particle.

Step 4: Compare the current value of the fitness function with the particles’ previous best value (pbest). If the current fitness value is less, then assign the current fitness value to pbest and assign the current coordinates (positions) to pbest.

Step 5: Determine the current global minimum fitness value among the current positions.

Step 6: Compare the current global minimum with the previous global minimum (gbest). If the current global minimum is better than gbest, then assign the current global minimum to gbest and assign the current coordinates (positions) to gbest.

Step 7: Change the velocities according to Equation (26).

Step 8: Move each particle to the new position according to Equation (27) and return to Step 2.

Step 9: Repeat Step 2 - 8 until a stopping criterion is satisfied or the maximum number of iterations is reached.

The main disadvantage is that the PSO algorithm is not guaranteed to be global convergent [24] . The dynamic behavior of the particle is widely divergent forming that of that the particle in the PSO systems in that the exact values of and cannot be determined simultaneously. In quantum world, the term trajectory is meaningless, because and of a particle cannot be determined simultaneously according to uncertainty principle. Therefore, if individual particles in a PSO system have quantum behavior, the PSO algorithm is bound to work in a different fashion. In the quantum model of a PSO called here QPSO, the state of a particle is depicted by wave function W(x, t) instead of position and velocity [23] [24] . Employing the Monte Carlo method, the particles move according to the following iterative equation:

(28)

where and are values generated according to a uniform probability distribution in range [24] , the parameter is called contraction expansion coefficient, which can be tuned to control the convergence speed of the particle. In the QPSO, the parameter must be set as to guarantee convergence of the particle [23] . Where Mbest called mean best position is defined as the mean of the pbest positions of all particles, i.e.:

. (29)

The procedure for implementing the QPSO is given by the following steps [23] [24] :

Step 1: Initialization of swarm positions: Initialize a population (array) of particles with random positions in the n-dimensional problem space using a uniform probability distribution function.

Step 2: Evaluation of particle’s fitness: Evaluate the fitness value of each particle.

Step 3: Comparison to pbest (personal best): Compare each particle’s fitness with the particle’s pbest. If the current value is better than pbest, then set the pbest value equal to the current value and the pbest location equal to the current location in n-dimensional space.

Step 4: Comparison to gbest (global best): Compare the fitness with the population’s overall previous best. If the current value is better than gbest, then reset gbest to the current particle’s array index and value.

Step 5: Updating of global point: Calculate the Mbest using Equation (29).

Step 6: Updating of particles’ position: Change the position of the particles according to Equation (28), where c1 and c2 are two random numbers generated using a uniform probability distribution in the range [0, 1].

Step 7: Repeating the evolutionary cycle: Loop to Step 2 until a stop criterion is met, usually a sufficiently good

3.1. PSS and VSC-HVDC Damping Controller

(30)

The VSC-HVDC damping controllers are of the structure shown in Figure 3 which can be . However, an electrical torque in-phase with the speed deviation is to be produced in order to enhance damping of the system oscillations. It includes gain block, signal-washout block and lead-lag compensator. The parameters of the damping controller are obtained using QPSO algorithm.

VSC-HVDC Controller Design Using QPSO

To obtain optimal parameters, this paper employs QPSO [24] to enhance optimization synthesis and find the global optimum value of fitness function. The objective function (which must be minimized) is defined as follows [25] :

(31)

where is the time range of simulation and is the total number of operating points for which the optimization is carried out. The design problem can be formulated as the following constrained optimization problem, where the constraints are the controller parameters bounds [22] :

(32)

Typical ranges of the optimized parameters are [0.01 - 100] for and [0.01 - 1] for, , and. The proposed approach employs QPSO algorithm to solve this optimization problem and search for an optimal or near optimal set of controller parameters.

3.2. Controllability Measurement Based on SVD

Controllability shows how the state variables describing the behavior of a system can be influenced by its inputs.

More accurately, the dynamical system or the pair (A, B) is said to be state controllable if, for any initial state any time and any final state there exist an input u(t) such that. Otherwise the system is said to be state uncontrollable.

In damping of power oscillations, it is necessary to determine controllability for specific eigenvalues (electromechanical mode). A very powerful tool commonly used for this purpose is Popov-Belevitch-Hautus (PBH) test which is described as below. It includes in evaluating the rank of matrices:

(33)

which is the kth eigenvalue of the matrix A, I is the identity matrix, is the column of B corresponding to i_th input. The mode of linear system in state space form is controllable if matrix has full row rank. The rank of matrices can be evaluated by their singular values. The singular values are defined as below:

If is a complex matrix, then there exist unitary matrices and with dimensions of and, respectively, such that:

(34)

where,

With where and are the singular values of.

The minimum singular value represents the distance of the matrix from all the matrices with a rank of [26] . This property can be used to quantify modal controllability and observability [26] . The matrix H (and J) can be written as where is a column vector corresponding to the input. The minimum singular value, of the matrix indicates the capability of the input to control the mode associated with the eigenvalue. Actually, the higher, the higher the controllability of this mode by the input considered. As such, the controllability of the EM mode can be examined with all inputs in order to identify the most effective one to control the mode. Thus, the choice of input through the PBH test is done by selecting those with the largest of the minimum singular values of matrix.

4. Simulation Results

Power system information is given in Appendix A. Constant coefficients in modeling are calculated according information which given in appendix B. In this paper, we consider (rotor speed deviation) as outputs and five inputs which are i.e. modulation index, phase angle of rectifier and inverter respectively and finally PSS input. Selecting an affective coupling between inputs-output for damping oscillation of the power system is one of the most important goals of this paper. Following section consider this topic.

4.1. Controllability Measure by Using PBH Test

SVD based on PBH is employed to measure the controllability of the electromechanical mode (EM) mode from each of the five inputs:. The minimum singular value is estimated over a wide range of operating conditions. For SVD analysis, ranges from 0.01 to 1.5 Pu and. At each loading condition, the system model is linearized, the EM mode is identified, and the SVD-based controllability and observability measure is implemented. For comparison purposes, the minimum singular value for all inputs at and Pu is shown in Figure 4, respectively. From these figures, the following can be noticed:

・ EM mode controllability via is always higher than that of any other input.

・ The capabilities of to control the EM mode is higher than that of.

・ All control signals have low EM mode controllability in low load condition except.

According to what said above, to design supplementary controller based VSC-HVDC, applying damping signal to can have most affection on oscillation mode.

4.2. Using QPSO to Obtain Parameters of Supplementary Controllers

The QPSO algorithm is used to obtain the optimal parameter settings of each of the supplementary controllers so

that the objective function is optimized. The final parameters are given in Table 1.

These supplementary controllers are used by VSC-HVDC system in different loading condition (Table 2). According results which are obtained from SVD analysis, it is obvious that best input for applying damping signal is. In this paper, for comparison purposes, a supplementary controller is designed by PSO method for this input. This controller is used in condition and nonlinear system.

Parameters of PSO based damping controller are as: Testing model consists of small changing in mechanical power which is applied at. Testing nonlinear model includes small changing in mechanical power at and three phase fault at infinite bus at time that is removed after 7 cycles. Figures 5-13 show the linear power system response in condition and. Because of results which obtained by SVD analysis and also system response for condition, in we just used from rectifier and PSS inputs for applying damping signal and are omitted. According to these figures, damping controller based on QPSO damps active power, rotor speed oscillations and load angle better than PSO-based compensator for loading condition. Also, using of can guarantee best damping results. Figures 14-19 show the nonlinear power responses. According to these figures, QPSO based damping controller applying to damps active power, terminal voltage and rotor speed oscillations better than PSO.

5. Conclusion

In this paper, SVD has been employed to evaluate the electromechanical mode controllability to PSS and the four VSC HVDC control signals. It has been shown that the electromechanical mode is most powerfully controlled via a wide range of loading conditions. Also, the quantum-behaved particle swarm optimization algorithm has been successfully applied to the robust design of VSC HVDC based damping controllers. The effectiveness of the proposed VSC HVDC controllers for improving transient stability performance of a power system are demonstrated by a weakly connected power system subjected to different severe disturbances. The

non-linear time-domain simulation results show the effectiveness of the proposed controller and their ability to provide good damping of low frequency oscillations.

References

Appendix

The test system parameters are (all in pu):

Machine and Exciter:

Transmission line and transformer reactance:

VSC HVDC:

Coefficients are:

, , ,

, , , ,

, , , ,

, , , ,

, , , ,

, , , , , ,

, , , ,

, , , ,

, , , ,

, , , ,

, , , ,

, , , ,

, , ,

, , ,

, , , ,

, , , ,

, , ,