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The usage of open communication infrastructure for transmitting the control signals in the Load Frequency Control (LFC) scheme of power system introduces time delays. These time delays may degrade the dynamic performance of the power system. This paper proposes a robust method to design a controller for multi-area LFC schemes considering communication delays. In existing literature, the controller values of LFC are designed using time domain approach which is less accurate than the proposed method. In proposed method, t he controller values are determined by moving the rightmosteigenvalues of the system to the left half plane in a quasi-continuous way for a preset upper bound of time delay. Then the robustness of the proposed controller is assessed by estimating the maximumtolerable value of time delay for maintaining system stability. Simulation studies are carried out for multi-area LFC scheme equipped with the proposed controller using Matlab/simulink. From the results, it has been concluded that the proposed controller guarantees the tolerance for all time delays smaller than the preset upper bound and provides a bigger delay margin than the existing controllers.

For many years, the Load Frequency Control (LFC) plays a major role in power system operation and control. The main objective of LFC is to minimize the frequency variations when there is any change in load [

At present, there is a rapid momentum in the advancement of research to deal LFC with communication delays. X. Yu et al. [_{2}/H_{∞} control technique for three area interconnected power system with communication delay. L. Jiang et al. [_{∞} controller for LFC of two area system. Chuanke Zhang et al. [

J. Chen et al. [

This paper proposes a new method to design a robust controller for the multi area LFC scheme considering communication delays in order to maintain the frequency and tie-line power between the areas. The controller is designed to guarantee the stability of power system for any delays smaller than the preset upper bound. The paper is organized as follows. In section 2, the multi area LFC structure is modelled considering time delays and it is represented in state space form. In section 3, the detailed description of a robust method to design the controller for multi area LFC affected by communication delays is presented. In section 4, the efficiency of the proposed controller is evaluated by computing the maximum tolerable value of time delay theoretically using Frequency Sweeping Test. In section 5, simulation is performed to prove the efficiency of the designed controller against delays for multi area LFC.

This section illustrates the dynamic model of multi area LFC scheme with time delay. This is obtained by including an exponential term e^{−sτ} in the secondary control loop of the conventional LFC model [^{th} control area of multi-area LFC scheme where i = 1, 2…N. The exponential term denotes the time delay. The turbine, governor and generator are modelled by a first order transfer function [

The notations used for i^{th} control area are listed below.

τ transport delay.

T_{gi} Governor time constant.

T_{ti} Turbine time constant.

M_{i} Moment of inertia of generators.

D_{i} Damping co-efficient of generator.

R_{i} Speed droop.

ACE_{i} Area Control Error.

Δf_{i} Deviation in frequency.

ΔPt_{ie} Tie-line power flow.

ΔPt_{i} Turbine power output.

ΔPg_{i} Governor output.

β_{i} Frequency bias factor.

ΔP_{di} Total demands in area i.

T_{ij} Synchronizing coefficient between area i and area j.

The two-area LFC scheme with time delay can be expressed in state space form as

where x(t) is state vector and the state variables are

The ACE signal of area i is expressed as

The ACE signal is used as the input to load frequency controller, which is designed as

The closed loop model of two area LFC system can be obtained by modifying Equation (1) using state output feedback method and expressed as

where

K_{1} and K_{2} denote the gain values of PI controller. The controller values are determined using Continuous Pole Placement method which is described in detail in the next section.

The idea behind the proposed stabilization method is to move the unstable eigenvalues to the left half plane in a quasi-continuous way by applying small changes to the controller gain, in the meanwhile monitoring the other eigenvalues with a large real part. The proposed stabilization method is based on the Theorem given in Appendix 1.

The algorithm for the proposed method is as follows:

Step 1. Initialize the number of rightmost eigenvalues m = 1.

Step 2. Compute the rightmost eigenvalues for a particular preset upper bound of delay.

Step 3. Find the sensitivity of the m rightmost eigenvalues with respect to the changes in the controller gain K.

Step 4. Using the sensitivities computed in step 3, shift the m rightmost eigenvalues in the direction towards the left half plane by applying small changes to the controller gain K.

Step 5. Meanwhile monitor the uncontrolled eigenvalues. Stop when the stability is reached or go to step 2.

The detailed explanation about the different steps involved in the algorithm is presented in the following sections.

In 1999, Engelborghs and Roose proposed a method which computes the rightmost eigenvalues of the characteristic equation. In this method, a discretization of the time integration operator of the linearized system is obtained. The eigenvalues of the linearized system are exponential transforms of the roots of the characteristic equation. Then, the selected eigenvalues of the resulting large matrix are computed. A step length heuristic is applied to ensure that all eigenvalues are approximated accurately by discretization and the accuracy is improved by employing Newton iteration on the characteristic equation taking the approximate eigenvalues as starting values. This method is implemented in the Matlab package DDE-BIFTOOL. This package is a collection of matlab routines used to find the right most eigenvalues of the system.

The characteristics equation of the two area LFC system Equation (5) can be written as

where λ_{i} is a solution of the characteristic equation and n(v_{i}) is a normalizing condition. Differentiating the Equation (7) and Equation (8) w.r.t. a component k_{j} of K,

From Equation (9), _{j} is j^{th} unity vector.

It is assumed that m eigenvalues

From Equation (10) the small changes in gain

_{m}). Thus with the availability of

for few Newton iterations on Equation (7) are required when

sup

where

movement of the real parts of the rightmost eigenvalues of two area LFC scheme for the preset upper bound delay 4s,

Here the rightmost unstable eigenvalues whose real part is positive are shifted to negative real axis. The proposed algorithm converges to an optimum value at iteration on 142. At iteration 142 all the rightmost eigenvalues are moved to LHP. The final value of controller gain is K_{1} = [−0.0233, −0.0146]^{T}, K_{2} = [0.0127, −0.0335]^{T}. The spectrum of eigenvalues for final value of the controller gain is depicted in

The proposed controller guarantees the stability for the delays smaller than the preset upper bound. The robustness of the controller is validated by finding an index called delay margin. The delay margin of the system is defined as the maximum tolerable value of time delay after which the system goes unstable. The delay margin of the system is computed using Frequency Sweeping Test which is discussed in the next section.

Frequency sweeping Test [

Let _{d} is k. The delay margin can be defined as

where

That is, the system remains stable for all _{d}_{.} The explanation for the above theorem is presented below

The generalized eigenvalues of the matrix pencil _{ }is calculated for various frequencies.

The generalized eigenvalues of the matrix pencil

For time delay

At

The algorithm of Frequency sweeping Test for computing the delay margin is given below:

1) Obtain the maximum of real parts of all eigenvalues of the matrix A + A_{d}. If it is less than zero proceed.

2) Find the rank of matrix A_{d}. Consider rank(A_{d}) = k. then the number of crossover points from right half plane to left half plane is k.

3) Choose the frequency range and step size of frequency range.

4) For different frequencies

5) Determine the angle and frequency at which the absolute value of each eigenvalue variation reaches one. Otherwise go to step 3 and vary the frequency range.

6) Calculate the delay margin

where

Simulation studies have been carried out for multi- area LFC equipped with PI controller assuming the load change of 0.1 p.u in Area 1. The system parameters of each control area are listed in

The upper bounds of time delay in each area are preset as same value while designing the controllers. First the controller values are determined by using continuous pole placement method and then for the designed controller values, delay margin is theoretically calculated using FST. The theoretical results are presented in

The theoretical value of delay margin is calculated as 34.612 s. That is, the controller designed using the proposed method for the preset upper bound of delay 8 s not only retains stability for time delays up to 8s, it can also ensure stability till 34.612 s (delay margin). To validate the theoretical results, simulation is performed using MATLAB/SIMULINK by increasing the delay step by step from zero until the LFC system becomes unstable. The simulation results are shown in

Similarly, the controller values of three-area LFC are determined using the proposed method for preset upper bound of time delay 10 s. The results are compared with the controller gain reported in [

The delay margin of the three-area LFC system with PI controller (designed using proposed method) is theoretically computed as 21.115 s whereas the method reported in (9) can maintain the stability only up to 14 s. This

T_{g} | T_{t} | R | D | Β | M | T_{o} | |
---|---|---|---|---|---|---|---|

Area 1 | 0.1 | 0.3 | 0.05 | 1 | 21 | 10 | 0.1986 |

Area 2 | 0.17 | 0.4 | 0.05 | 1.5 | 21.5 | 12 | 0.2148 |

Area 3 | 0.2 | 0.35 | 0.05 | 2 | 21.8 | 12 | 0.183 |

Area | Controller parameters | Delay margin(s) | ||
---|---|---|---|---|

K_{P} | K_{I} | Theoretical | From simulation | |

1 | −0.0113 | −0.0446 | 34.612 | 34.6 |

2 | 0.0125 | −0.0405 |

clearly shows that the proposed controller is highly robust compared to the existing method.

To validate the theoretical delay margin values, simulation is performed for three area LFC scheme keeping the preset upper bound of time delay as 10 s and the results are shown in

Area | Controller gain and Delay margin | |||||||
---|---|---|---|---|---|---|---|---|

Existing method [ | Proposed method | |||||||

K_{P} | K_{I} | K_{D} | Delay margin (s) | K_{P} | K_{I} | Delay margin(s) | ||

Theoretical | From simulation | |||||||

1 | 0.0669 | −0.0615 | −0.0311 | 14 | 0.0509 | −0.0702 | 21.115 | 21.1 |

2 | 0.0305 | −0.0885 | −0.0325 | 0.0627 | −0.0635 | |||

3 | 0.0704 | −0.0688 | −0.0302 | 0.0425 | −0.0455 |

In this paper, a robust controller based on continuous pole placement method is designed for multi area LFC scheme affected by communication delays. The controller values are determined by shifting the rightmost eigenvalues to left half plane in a quasi continuous way for any particular preset upper bound of time delay. The proposed controller is highly robust in sustaining the stability of the system even for delays greater than the preset upper bound of time delay. Case studies have been carried out for two area and three area LFC schemes. The efficiency of the controller is validated by finding the value of time delay margin theoretically using Frequency Sweeping test. The theoretical value of delay margin is verified using simulation studies. Simulation result shows that the proposed controller gives larger stability margin.

T. Jesintha Mary,P. Rangarajan, (2016) Design of Robust Controller for LFC of Interconnected Power System Considering Communication Delays. Circuits and Systems,07,794-804. doi: 10.4236/cs.2016.76068

Lemma.

Let f(λ) and the sequence {f_{n}(λ)}_{n≥}_{1} be analytic functions on an (open) domain D ⊆ C. Suppose that {f_{n}(λ)}_{n≥}_{1} converges uniformly to f(λ) on the disc

_{n}(λ) has exactly k zeros _{n→∞}λ_{n,j} = λ_{0};

With this lemma, continuity properties of the spectrum with respect to the feedback gain K can easily be deduced.

Theorem. For the system