Particle Swarm Optimization Of Sliding Mode Controller Computer Science Essay

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Energy Resources in Smart Grids. Mohammadhassan A. Sofla and Devinder Kaur, Senior Member, IEEEAbstract-This paper models a power converter of distributed energy resource in smart grid applications. LC filter is employed to meet the power quality standard of interconnection systems. A robust method is chosen to stabilize the fragile dynamics of LC filters in an environment full of uncertainties and disturbances. Particle swarm optimization is selected to design sliding surface instead of conventional liner quadratic regulator method. A multi-objective particle swarm optimization is employed to find state feedbacks gains and sliding surface gain. The objectives are to minimize the total harmonic distortion and improve transient performance. An algorithm is developed to solve this nonlinear multi-objective problem. Transient performances of the system developed under high frequency dynamics such as voltage sag and a three-phase fault are with minimum overshoot and high rise time. The total harmonic distortion of voltage source inverter is reduced to 1.2% by designed controller which is quite good according the standard of interconnection systems.

Index Terms-Current-controlled voltage source inverter, distributed energy resources, dynamic stability, particle swarm optimization, robustness, sliding mode control.

Nomenclature

Current and voltage of inverter.

Current and voltage of inverter in dq axis.

Filter inductance.

Filter Capacitance.

Inductance resistance.

Inverter input voltage.

Coupling inductance (transformer inductance).

Grid side current.

DC link voltage.

System description matrices.

System outputs matrices.

Input control signal.

Disturbance of the system.

State variables.

Proportional gain.

Integral gain.

Discredited system matrices.

Real power.

System frequency (rad/sec.) .

Integral gain.

Coefficient matrix of feedback states.

M Coefficient matrix of sliding surface.

Position of swarms.

Velocity of swarms.

Local best position.

Global best position.

Coefficients of velocity update of PSO.

Introduction

INTERCONECTION of distributed energy resources (DERs) to power grids is a major concern against outspread of them. In fact, the use of DERs in future smart grids depends on the performance of interconnection systems (ICSs). The standard performance of ICSs is addressed in IEEE standard 1547 published in 2003 [1]. Proper operation of ICS is the result of accurate reference signal tracking, disturbance rejection and smooth sinusoidal output. DC output and non-synchronized energy resources have to employ power converters to interconnect to AC power systems. Due to the power converters usage, the output voltage encompasses undesired harmonics in the output voltage. LC filters are employed in the output of the voltage source inverters (VSIs) to pick the desired output frequency (60 Hz). The important point is that in applications where the penetration of DER is high and loads are close to the DERs (like microgrids), it is necessary to use LC filters [2]. But, stability problems rise due to the dynamics of LC filters. Dynamics of the LC filter can move to instability due to transients. Fig. 1 shows a microgrid in the grid-connected mode.

Dynamics of LCL (last L due to the transformer inductance) filter are dominant dynamics of ICS. Classical control based on frequency domain analysis has been developed for the linear time-invariant (LTI) description of VSIs. Usually, an input-output description of the VSI has been developed based on dominant poles of the LCL filter [3]. Although, this method has the simple design procedure and successful implementation, but it is a model-based design which is suitable in rather simple and linear single-input and single-output

Fig. 1. Single line diagram of a distribution network with several DERs connected via VSIs.

(SISO) systems.

But, conditions to use classical control may not be met in applications of DERs in smart grids. Nonlinearity of components, multi-input and multi-output (MIMO) nature of a three-phase VSI, uncertainties in the filter and DC-link, time varying characteristics of loading and DER's response to power system transients make the modern control analysis reasonable [4]. The ability of VSI to ride through all transients in power systems should be considered in designing control approach. On the other side, VSI not only should meet steady state operation standards but also its dynamics should not influence the operation of power system. Different control approaches based on the state space model of VSIs have been developed. But, the main challenge of these models is their low robustness against parametric variations, noises and disturbances. For example, control methods such as predictive control needs precise values of LCL filter. Robust control methods such as sliding mode control (SMC) [5] and repetitive [6] controls have been developed to overcome this drawback.

A robust method such as SMC is a good candidate for this application. But, this method is a nonlinear method and its design is a difficult task. Usually, the design procedure is based on experience. In some cases, the robustness of SMS is optimized by means of linear quadratic regulator (LQR). This method uses a linear model of the system to minimize the quadratic criterion. However, biologically inspired methods are promising in solving nonlinear problems. Among them, particle swarm optimization (PSO) has better performance due to its fast, robust and easy implementation. Performances of controller based on classic control and improved SMC-PSO are presented and discussed.

In this paper a method is used to design a sliding mode controller to control current in grid-connected VSIs with LC output. This method is based on particle swarm optimization which is a fats and robust biologically inspired method.

Modeling of power converters and design procedure of a SMC controller are considered in Section III. A design procedure based on multi-objective PSO is presented in Section IV. The results and proposed scenarios to validate the effectiveness of the proposed controller are discussed in Section V. Final section concludes this paper.

Fig. 2. Single line diagram of a DER connected via power converter and LCL filter.

Modeling and Control of Inverter-based DERs

Characteristics of System

Interconnection system (ICS) of DERs connected to power systems is facing different voltage disturbances such as faults, voltage flicker, voltage sag and harmonics. These transients cause oscillations which leads LCL filters to instability. Other character of inverter-based DERs is considerable uncertainties in the plant. There are different uncertainties in subsystems such as LCL filter, operating point and DC-link voltage. Dynamics of VSIs can be modeled by dynamics of LCL filter.

Modeling and Control of Power Converters

Average model of a VSI shown in Fig. 2 in abc reference frame can be written as follows:

The order of system (1) is four. This model is transformed to synchronous dq reference frame to facilitate the control implementation. Negative sequence of the symmetrical components is also considered to improve performance of the proposed controller under unbalanced situation (Eq. (2)). Values of LCL filter parameters which are used for a 100 KVA DER are as follows:

Positive and negative synchronous reference frame of the system is represented in (2):

This system is still a four order system for each sequence. If the decoupling of d and q parameters is almost accomplished, then this MIMO system can be analyzed based by SISO systems theorem.

Let the following state space representation be for each sequence (Bold parameters are matrices or vectors):

In this modeling, the grid current id is supposed to be a disturbance to the dynamics of LCL system. The outputs need to be tracked are currents of the VSI. However, voltage sensors are also needed to calculate the output power of VSI.

The states of this system are controllable since the rank of matrix Co is perfect.

Remark 1. A system is perfect controllable if the rank of the following controllability matrix is perfect:

Discretizing (3) might make chatter problem. The following discretizing is free of chatter which was developed in [7]:

This makes this control method sensitive to uncertainties in the plant model. The sliding surface of this system can be

Fig. 3. Sliding mode control diagram.

defined as follows:

Matrix defines the desired dynamics in which the system will slide. Fig. 3 shows a SMC diagram.

The system (5) displays robust stability if the following Lyapunov function is reached [7]:

Following input for the system (5) guarantees this equation. However, comprehensive discussion on stability of this system can be found in [7]. Based on the Filipov's method the following input is to be designed:

Because of practical limitation of DC link voltage, the following equation should be used for :

where is the maximum voltage that PWM can generate, which equals to the maximum DC link voltage. Fig. 3 represents the main idea of SMC.

The reference is supposed to be zero because the reactive power of the DER is supposed to be zero. The reference current in d axis is found by the following controller.

where z is the discrete Laplace operator.

PSO to Design Sliding Mode Controller

Designing a sliding surface to achieve the desired dynamic performance is a difficult task due to the nonlinearity in the system. LQR method uses approximate linear model of the system to find the gains of the sliding surface and states [8]. Particle swarm method is a biologically inspired method which is able to solve such problems. PSO has been used in SMC design by some researchers [9]-[10]. Its results are

Fig. 4. Proposed method to design SMC for power converters with LC output.

promising to achieve desired robustness as well as an acceptable transient performance. However, in application in power converters, total harmonic distortion (power quality) has to be considered in the PSO algorithm. Thus, a multi-objective PSO is needed here. Different functions can be designed to model the objective function. In [10] a method has been developed which is used in this paper.

The objective function is proposed as follows:

in which, f and g represent transient performance and power quality functions, respectively (Fig. 4).

Function f calculates the rise time and overshoot of the output. On the other side, g calculates the THD of the output power. These functions are evaluated by a steady state and voltage sag in Simulink. In an iteration of following algorithm, the Simulink file runs and f and g are calculated and reported to the PSO m-file. Function can be a linear function of f and g or can be defined by other methods. Here, a linear function of f and g is developed.

In this algorithm, fitness function is negative of objective function because PSO tried to maximize the objective function. The PSO algorithm tries to find and to optimize SMC.

Initialize the swarm positions and velocities as follows:

where and are the vectors of the lower and upper limits of the controller gain respectively, and rand(0,1) is a random number between 0 and 1.

c1 = 2.05;

c2 = 2.05;

phi = c1 + c2;

= 2.0/abs(2.0-phi-sqrt(phi^2-4*phi));

2. Carry out time-domain simulation.

3. Objective function and fitness evaluation using (12)

4. if fitness(x) > fitness () then; ;

end

5. if fitness(x)> fitness () then; ;

end

6. Velocity update using:

7. Position update using:

9. Check for convergence

if 'convergence is met' then

output 'M and '

else go to step 2

end

END

The search space of states and sliding surface gains are based on the desired surface dynamics and robustness. Fig. 4 shows the diagram of the proposed method to design SMC current controller for grid connected VSI with LCL output.

Results and Discussion

Table I shows the parameters, values of the current controlled VSI, LCL filter and transformer. A discrete sliding mode is developed for each sequence. Then, the PSO algorithm is run with 50 iterations and 100 swarms. For a 100 kVA DER, the performance of the VSI is tested via a voltage disturbance and steady state THD. Based on the performance of controller, PSO updates the gains every iteration. Final values of proposed system are found as follows for each symmetric sequence:

Although, sliding mode is a robust controller based on the modern control theorem, but frequency response analysis of it provides some useful information such as the bandwidth of current controller. Fig. 5 shows the frequency response of the system for current which is derived in Simulink by frequency domain analyzer. As it is observable, the controller can track the reference signal in the high frequency modes.

The steady state performance of the system with the proposed controller is presented in Fig. 6.

Search space for the PSO algorithm comes from the desired dynamics for the proposed power conversion system. PSO algorithm searches for optimum values of and within that search space and runs the actual model of the system in Simulink.

THD and regulation of the proposed controller has an excellent performance. The THD is about 1.2% and the current regulation is about 0.03%.

Performance of the system under transient is also tested by

TABLE I

Specification and components of VSI

Specification

Value

Components

Value

DC link voltage

520 V

Inductor (H)

200

Grid voltage (rms)

440 V

Capacitor (F)

40

System frequency (Hz)

60

Inductor resistance ()

0,01

Coupling inductor

200

Switching frequency (kHz)

10

Fig. 5. Frequency response analysis (approximate) of inverter current.

Fig. 6. Steady state performance of a VSI by SMC-PSO controller.

Fig. 7. Transient performance of the proposed controller under voltage sag.

Fig. 8. Transient performance of the proposed controller under three phase fault.

voltage sag. The search space for M is determined by the desired robustness. The desired robustness depends on the strength of disturbances which can lead the VSI to instability. On the other side M influences THD of the output signal.

Fig. 7 shows transient performance of the system under voltage sag. As it is shown the VSI has a very small overshoot and rise time. Fig. 8 also represents performance of the power converter under a three phase fault. This performance also validates the effectiveness of the proposed controller to handle high frequency dynamics.

Conclusion

In this paper, an advance controller has been designed to control power conversion of an inverter-based distributed energy resource. A sliding mode control has been developed which shows high robustness and low THD for voltage source inverter. Particle swarm optimization which is a fast and robust biologically inspired method has been employed to design the SMC.

A PSO algorithm has been developed to solve the multi-objective optimization problem. The proposed PSO has found the optimum values of SMC in order to represent a good transient response as well as an acceptable THD.

The power conversion system is simulated in Simulink. PSO m-file calls this Simulink file in every iteration to evaluate the fitness function. The future work is to implement this system in a laboratory prototype and to analyze the performance of the controller in the islanded mode.

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