# Optimal Decentralized LQR Control to Enhance Multi-Area LFC System Stability

Authors: Ashraf M. Abdelhamid; Ahmed A. M. El-Gaafary
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Many studies have been made in the field of load frequency control (LFC) through the last few decades because of its importance to healthy power system. It is important to maintain frequency deviation at zero level after a load perturbation. In decentralized control, the multi-area power system is decomposed into many single input single output (SISO) subsystems and a local controller is designed for each subsystem. The controlled subsystems may have slow poles; these undesired poles may drive the aggregated overall system into the instability region. Thus, it is required to relocate these poles to much more stable places to avoid their effect upon the overall system stability. This study aims to design a new load frequency controller based on the powerful optimal linear quadratic regulator (LQR) technique. This technique can be applied over subsystem level to shift each subsystem undesired poles one by one into a prespecified stable location which in turn shift the overall system slow poles and reduce the effect of the interaction between the interconnected subsystems among each other. This procedure must be applied many times as the number of undesired poles (pairs) until all the desired poles are achieved. The main objective is considered to get a robust design when the system is affected by a physical disturbance and ±40% model uncertainties. LQR can be applied again over the aggregated system to enhance the stability degree. Simulation results are used to evaluate the effectiveness of the proposed method and compared to other research results.


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## Body

## 1. Introduction

LFC is one of the most important problems in electric power system design/operation and is becoming much more significant nowadays especially with the increasing dimensions and complexity of the interconnected power systems. The decentralized control strategies are widely used recently because of the ability of using the traditional local control methodologies [1]. In the decentralized control, the large-scale power system is divided into many separated subsystem models and so the complex control problem can be decomposed into individual subproblems that have some interactions among each other. Currently the classical control techniques can be easily applied over subsystem level [2]. Each subsystem contains a local controller which can be of different type and has access to local sensors only [2]. These all independent controllers together represent a decentralized controller, and the failure of one local controller does not necessarily lead to overall system instability.Large-scale power systems usually consist of many control areas, connected by tie-lines to exchange the power demands among them [5]. Loads at any station can appear randomly and changeable so that the system frequency and tie-line power will deviate from their nominal values [6]. There has been continuing interest in designing load frequency controllers with better performance to maintain the frequency and to keep tie-line power flows within desired values by using various decentralized control methods during the last two decades. In [7] the author has designed local independent H∞ controllers to decentralized two-area power system. He has used a condition based on the structured singular values (SSVs). Also, in [8] local independent LFCs have been designed for interconnected two-area power systems to realize satisfactory performances based on the SSV μ methods. Sequential decentralized LFCs for three area power systems based on μ synthesis has been designed, and system uncertainties and practical constraints have been considered [9]. In [10], the same methodology has been applied to stabilize four area power systems. The paper [3] has mixed H2 and H∞ control methods and has considered the connections between each area and the rest of the system as new disturbance. In [4], Particle Swarm Optimization (PSO) has been used in the AC-DC interconnected LFC system. Local controller to each area has been designed to stabilize the overall system and reduce settling time and frequency oscillation.A change in real power demand at any point of a network is reflected throughout the system by a change in frequency. Therefore, system frequency provides a good indicator to system generation and load imbalance [11, 12].Linear quadratic optimal systems based on predetermined eigenvalues are an important manner. Alternative methods are developed to construct the weighting matrixQ that enables the user to shift undesired system eigenvalues (unstable or slow) into prespecified (stable or faster) ones. The process can be repeated many times until all the desired pole locations are placed as desired. The paper [13] has developed a method to find the weighting matrix element dependent on the desired performance and the prescribed eigenvalues. The authors in [14] have introduced a methodology to achieve the prescribed eigenvalue locations using LQR to improve the undesired eigenvalues (slow or undampened) to faster and dampened ones. Closed loop poles and system performance have been combined in [15] as an objective function to find the optimal weighting matrices for LQR using intelligent optimization techniques. Also, [16] has presented an algorithm to determine the weighting matrices of the LQR according to the desired poles placement.


### 1.1. Research Objectives

The main idea of this work is to propose an optimal controller based on LQR approach and enable the system to beat specified uncertainties caused by variations of inertia constant (Mi) and tie-line synchronizing coefficient between areasiandj (Tij), parameters of all control areas, and load disturbances as follows:(i)
Maintaining system stability and performance for the overall power system and each control area throughout±40% uncertainty of Mi and Tij(ii)
Minimizing the effectiveness of step load disturbance on output signals to zero(iii)
Recognizing a minimum overshoot and undershoot values(iv)
Minimizing settling time on the frequency deviation signal and obtaining zero-steady state error for individual control area


### 1.2. Design Methodology

(i)
The first step is to divide the complex Multi-Input Multi-Output (MIMO) problem into simple SISO subproblems. (ii)
LQR technique is used to design a local controller for each decoupled subsystem to realize design objectives. (iii)
Slower eigenvalues are shifted more inside the stability region within prespecified values. (iv)
The controlled system is composed again, and the overall stability is determined. (v)
LQR is applied again over the aggregated overall system level to enhance system stability. (vi)
The proposed control methodology is testified and compared to other research results.


## 1.1. Research Objectives

The main idea of this work is to propose an optimal controller based on LQR approach and enable the system to beat specified uncertainties caused by variations of inertia constant (Mi) and tie-line synchronizing coefficient between areasiandj (Tij), parameters of all control areas, and load disturbances as follows:(i)
Maintaining system stability and performance for the overall power system and each control area throughout±40% uncertainty of Mi and Tij(ii)
Minimizing the effectiveness of step load disturbance on output signals to zero(iii)
Recognizing a minimum overshoot and undershoot values(iv)
Minimizing settling time on the frequency deviation signal and obtaining zero-steady state error for individual control area


## 1.2. Design Methodology

(i)
The first step is to divide the complex Multi-Input Multi-Output (MIMO) problem into simple SISO subproblems. (ii)
LQR technique is used to design a local controller for each decoupled subsystem to realize design objectives. (iii)
Slower eigenvalues are shifted more inside the stability region within prespecified values. (iv)
The controlled system is composed again, and the overall stability is determined. (v)
LQR is applied again over the aggregated overall system level to enhance system stability. (vi)
The proposed control methodology is testified and compared to other research results.


## 2. Model Description

In this study, the conventional model is used for each control area of a multi-area power system which is widely used in many studies [3, 4, 10]. The same system parameters are used in [10] for the three areas and in [3] for the four area. Referring to the simplified traditional-based LFC model shown in Figure 1 for control area 1, the state space realization of areai (from 4-control area power system) is given by(1)x˙i=Axi+Bui+widiyi=CixiThe state vector xi, control input ui, disturbance input wi, and the measured output yi can be defined as(2)xi=ΔfiΔPtiΔPgiΔPtie-i,ui=ΔPci,wi=ΔPdi,yi=βiΔfi+ΔPtie-iThen, the nominal state space model for a 4-area power system can be described by (3) as(3)xt=Δf1ΔPt1ΔPg1ΔPtie-1⋯Δf4ΔPt4ΔPg4ΔPtie-4Tut=u1u2u3u4Tdt=ΔPd1ΔPd2ΔPd3ΔPd4Tyt=y1y2y3y4TwhereΔfi is frequency deviation of areai,ΔPgi is governor valve position of generator in areai,ΔPci is governor load set point in areai,ΔPti is turbine power in areai,ΔPtie-i is change in net tie-line power flow in areai,ΔPdi is disturbance in areai,Mi is equivalent inertia constant for areai,Di is equivalent damping coefficient for areai,Tgi is governor time constant for areai,Tti is turbine time constant for areai,Tij is tie-line synchronizing coefficient between areasiandj,Ri is drooping characteristic for areai,Figure 1
Block diagram of control area-1 [3, 4].in which(4)
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0and(5)B1=00000000-1Tg10000000,B2=000000000-1Tg2000000,B3=0000000000-1Tg300000,B4=00000000000-1Tg40000and(6)W1=-1M1000000000000000,W2=0-1M200000000000000W3=00-1M30000000000000,W4=000-1M4000000000000The total real power imported to areai equals the sum of all inflowing line powers ΔPtie-ij from adjoining areas, i.e.,(7)Ptie-i=∑jPtie-ijThe real power in per unit transmitted across a lossless line of reactance Xij is(8)Ptie-ij=ViVjXijPrisin⁡θi-θj(9)Vi=ViejθiwherePri is the rated power of areai,Vi and θi are the amplitude and angle of the terminal voltage in areai.A simplified 4-area power system is represented in Figure2 with system nominal values as illustrated in Table 1 [3, 11]. These values make the 4-area open loop power system stable, but the open loop system performance is affected by changes in the equivalent inertia constants Mi and synchronizing coefficients Tij (Table 2) more than changes of other parameters; these parameter values may lead the system into the instability region.Table 1
Nominal power system parameter values.
D i p.u.MW/Hz M i p.u.MW T t i Sec T g i Sec R i Hz/p.u.MW Area1 0.0083 0.166 0.3 0.08 2.4 Area2 0.0088 0.222 0.33 0.072 2.7 Area3 0.0080 0.16 0.35 0.07 2.5 Area4 0.0088 0.13 0.375 0.085 2 T i j p.u.MW/Hz T 12 = T 13 = T 14 = T 21 = T 23 = T 31 = T 32 = T 41 = 0.545Table 2
Parameter variation limits lead to system instability.
T i j T 12 ≥ 0.645,  T13,T14≥0.605,  T32≥0.615,  T21,T23≤0.475, and T31≤0.445.  M i M 1 ≤ 0.056, M2,M3,andM4≤ its nominal valuesFigure 2
Simplified4-area power system.


## 3. LQR Technique

LQR is an optimal design technique which guarantees robustness, while pole placement technique gives the desired performance but does not guarantee robustness. In this procedure, a design of linear quadratic system with prescribed poles has been followed and applied to shift slow decoupled subsystem eigenvalues to defense the effect of the interconnections between subsystems that leads to much more stable overall system poles and enhances the overall system stability. This procedure was developed by Solheim [13].The optimal gain is determined to minimize the cost functionJ as follows: (10)J=12∫0∞xTQx+uTRudtwhere R and Q are the weighting matrices of the input u(t) and states x(t), respectively [17], and the control gain k is given by(11)k=R-1BTPwhere P is the unique, symmetric, positive definite solution to steady state Algebraic Riccati Equation (ARE)(12)PA+ATP-PBR-1BTP+Q=0To construct the weighting matrix Q to shift the complex conjugate poles to complex places, (13) is used with the constraint in (14) for its internal elements q~(13)q~11=2σ2-α2-2ω2-β2h11+h22(14)σ2+ω22=α2+β22+q~11h11+h22α2+β2+q~112h11h22-h122whereh are the elements of the solution of the matrix (H=M-1BP-1BTM-T) and M is an eigenvector matrix corresponding to the system matrix A,α±jβ are the old complex eigenvalues and σ±jω are the desired complex eigenvalues.If it is required to shift complex conjugate poles to real places, (15) is used with the constraint in (16)(15)q~11=σ12+σ22-2α2-β2h11+h22(16)σ12σ22=α2+β22+q~11h11+h22α2+β2+q~112h11h22-h122while if it is required to shift the real eigenvalues to real places, (17) is used(17)q~jj=Sj2-λj2hjjwhere Sj is the desired real eigenvalue and λj is the old real one.


## 4. Controller Design

The decentralized input control methodology presented in [18] is used in order to decompose the system into four decoupled subsystems (Ai) with interconnected subsystems (Aij).(18)x˙i=Aixi+∑j=1j≠isAijxj+biui,i=1,2,…,sThis system can be stabilized by a multilevel control. Each decoupled subsystem can be stabilized separately by using local controller, which can provide a desired degree of stability, while global controllers are applied to enhance the overall system stability through reducing the effect of the interconnections among subsystems [18–20], with control effort as(19)ui=uilt+uigtwhere uilt is the local controller and chosen as(20)uilt=-kiTtxiWith a constant vector ki∈Rni, uigt is the global controller and chosen as(21)uigt=-∑j=1skijTxjwhere kij∈Rnj are constant vectors.To start stabilization process, (18) must be represented in canonical form(22)x˙ic=Aicxic+∑j=1j≠isAijcxjc+bicui,i=1,2,…,swith appropriate transformations Qicand Qjc, where(23)Aic=Qic-1AiQicAijc=Qic-1AijQjcBic=Qic-1BiCic=CiQicFic=Qic-1FiSubstituting (20) and (21) into (22) with the appropriate transformations (23) yields(24)x˙ic=Aic-bickicTxic+∑j=1j≠isAijc-bickijcTxjc,i=1,2,…,sA linear nonsingular transformation matrix Ti is applied to (24) to diagonalize it as follows:(25)Aijd=Ti-1AijcTjbid=Ti-1bickijdT=kijcTTj(26)xi~˙d=Λixi~d+∑j=1j≠isAijd-bidkijdTxj~d,i=1,2,…,swhere kijd can be calculated from (27) to reduce the effect of the interconnected subsystems.(27)kijd=bidTbid-1bidTAijdTThe Overall Controller Gain.Through the process of stabilizing the MIMO system, the feedback control vector can be calculated using ki and kij which computed for subsystem level. To obtain the control function ut, substitute (20) and (21) into (19), which yields(28)uit=-kicTxic-∑j=1j≠iskijdTTj-1xjc(29)uit=-kicT+∑j=1j≠iskijdTTj-1xcwhere(30)kc=kicT+∑j=1j≠iskijdTTj-1and kc can be expressed in detail as(31)kc=k1cTk12dTT2-1⋯k1sdTTs-1k21dTT1-1k2cT⋯k2sdTTs-1⋮⋮⋮ks1dTT1-1ks2dTT2-1kscT


### 4.1. Scenario I: Decentralized Input Controller Design with LQR

Weighting function values for each area can be selected using trial and error after many trials to their best values of(32)Q1=100000100001000001,R1=0.1,Q2=1000001000010000010,R2=0.01Q3=1000000.100001000010,R3=0.1,Q4=1000000.100001000010,R4=0.1Step responses to 0.01 p.u input and also to 0.01p.u disturbance are illustrated in Figure 3. To examine the robustness of the design, Figure 4 shows step response of the controlled system with LQR to -40% uncertainties, (a) to 0.01 p.u input and (b) to 0.01 p.u disturbance.Figure 3
Step response of the controlled system for scenario I, to (a) 0.01 p.u. input and (b) 0.01 disturbance input.
(a) (b)Figure 4
Step response of the 4 areas in scenario I with-40% uncertainties of Mi, and Tij to (a) 0.01 p.u input and (b) 0.01 p.u disturbance.
(a) (b)


### 4.2. Scenario II: LQR Design with Prespecified Pole Locations

It is important to shift each area slow eigenvalue more inside the stability region to minimize the effect of their values and the effect of the interconnected subsystems on the overall system stability. Weighting functionsQ are calculated to realize this objective as discussed in Section 3.This subsection shows designing a controller for subsystem-1 and so the other subsystems will follow the same procedures.For Subsystem-1.Transform subsystem into controllable canonical form. For subsystem eigenvalues of -13.256-0.3534±j3.999-1.92, it is needed to start moving the slowest complex conjugate pair (-0.3534±j3.999) more into the stability region to a new location of (-6±j2.4).Then, diagonalize the system with an appropriate transformation to the form:(33)A1cd=102∗-13.2560000-1.920000-0.35400-4-0.35First the weighting matrix Q~i can be calculated as the form:(34)Q~i=q~110000q~220000q~330000q~44,q~33=q~44From (13) and (14), q~33 is calculated to be 118.6256. Move the second slow area-1 pole (-1.92) to the new location of (-20) using (17) with the value of q~22=50763.88. The new area-1 eigenvalues become -13.255-20-6±j2.402i, with final Q~1 value of(35)Q~1=1000050763.880000118.62560000118.6256Table 3 summarizes original and achieved pole locations for each area with the calculated weighting matrix Q.Table 3
Calculated weighting matrices required to shift area poles into desired new locations.
Nominal pole locations Shifted pole locations Weighting matrix Q Area2 - 14.315 -0.447±j2.84 -1.75 - 14.315 -4±j2 -20 Q ~ 2 = 1 0 0 0 0 1773.63 0 0 0 0 11.146 0 0 0 0 11.146 Area3 - 14.844 -0.4164±j3.49 -1.516 - 14.844 -4.1±j2.95 -20 Q ~ 3 = 1 0 0 0 0 133906.1 0 0 0 0 76.385 0 0 0 0 76.385 Area4 - 12.69 -0.4986±j3.55 -0.8084 - 12.69 -4±j3 -20 Q ~ 4 = 1 0 0 0 0 8.337 ∗ 10 6 0 0 0 0 51.966 0 0 0 0 51.966Local controllers are determined to be equal to(36)ka=10660430765329.4kb=5497282053325kc=7300310056026kd=6726285053627.8Global controllers are calculated to reduce the effect of the interaction between the interconnected subsystems to be(37)K12=-0.268-0.523-3.26-4.28K13=-0.268-0.523-3.26-4.28K14=-0.36-0.654-1.794-5.36K21=-0.353-0.804-2.447-2.33K23=-0.36-0.6514-1.78-5.33K31=-0.325-0.74-2.25-2.145K32=-0.245-0.478-2.98-3.916K41=-0.825-1.88-5.72-5.45K24=K34=K42=K43=0And the overall gain matrix can be calculated as in (31) to be(38)K=-15.88-13.3-2.35-10.330.5650000.562000-14.1-10.81-1.825-13.960.5170000.5170001.31500000000.5650000.5650000.5620000000-11.36-11.87-1.81-12.5700000000-11.165-14.65-2.36-34.11System step responses to 0.01 p.u input and 0.01 p.u disturbance are obtained and illustrated in Figure 5Figure 5
Step response of the controlled system with LQR applied over subsystem level for (a) 0.01 p.u. input and (b) 0.01 disturbance input.
(a) (b)To check the robustness of this design, it is required to find the controlled system step responses for±40% uncertainties with and without disturbances. Figures 6 and 7 illustrate these responses.Figure 6
Step response of the four areas with -40% uncertainties ofMi, and Tij.(a) for 0.01 p.u input and (b) for 0.01 p.u disturbance.
(a) (b)Figure 7
Step response of the four areas with +40% uncertainties ofMi, and Tij (a) for 0.01 p.u input and (b) for 0.01 p.u disturbance.
(a) (b)All the controlled system specifications are calculated from Figures5, 6, and 7 and summarized in Table 4.Table 4
Controlled system specifications for scenario II.
Controlled System Controlled System With -40% Uncertainties Controlled System With +40% Uncertainties 0.01 p.u Input 0.01 p.u Disturbance 0.01 p.u Input 0.01 p.u Disturbance 0.01 p.u Input 0.01 p.u Disturbance M p p 0 0 0.000126 0.00301 0 0 t p p __ __ 0.97 sec 0.88 sec __ __ M n p 0.000714 0.01586 0.001147 0.0243 0.00052 0.01184 t n p 0.37 sec 0.28 sec 0.36 sec 0.24 sec 0.39 sec 0.3 sec t s 1.61 sec 1.52 sec 1.89 sec 1.7 sec 1.82 sec 1.71 sec


## 4.1. Scenario I: Decentralized Input Controller Design with LQR

Weighting function values for each area can be selected using trial and error after many trials to their best values of(32)Q1=100000100001000001,R1=0.1,Q2=1000001000010000010,R2=0.01Q3=1000000.100001000010,R3=0.1,Q4=1000000.100001000010,R4=0.1Step responses to 0.01 p.u input and also to 0.01p.u disturbance are illustrated in Figure 3. To examine the robustness of the design, Figure 4 shows step response of the controlled system with LQR to -40% uncertainties, (a) to 0.01 p.u input and (b) to 0.01 p.u disturbance.Figure 3
Step response of the controlled system for scenario I, to (a) 0.01 p.u. input and (b) 0.01 disturbance input.
(a) (b)Figure 4
Step response of the 4 areas in scenario I with-40% uncertainties of Mi, and Tij to (a) 0.01 p.u input and (b) 0.01 p.u disturbance.
(a) (b)


## 4.2. Scenario II: LQR Design with Prespecified Pole Locations

It is important to shift each area slow eigenvalue more inside the stability region to minimize the effect of their values and the effect of the interconnected subsystems on the overall system stability. Weighting functionsQ are calculated to realize this objective as discussed in Section 3.This subsection shows designing a controller for subsystem-1 and so the other subsystems will follow the same procedures.For Subsystem-1.Transform subsystem into controllable canonical form. For subsystem eigenvalues of -13.256-0.3534±j3.999-1.92, it is needed to start moving the slowest complex conjugate pair (-0.3534±j3.999) more into the stability region to a new location of (-6±j2.4).Then, diagonalize the system with an appropriate transformation to the form:(33)A1cd=102∗-13.2560000-1.920000-0.35400-4-0.35First the weighting matrix Q~i can be calculated as the form:(34)Q~i=q~110000q~220000q~330000q~44,q~33=q~44From (13) and (14), q~33 is calculated to be 118.6256. Move the second slow area-1 pole (-1.92) to the new location of (-20) using (17) with the value of q~22=50763.88. The new area-1 eigenvalues become -13.255-20-6±j2.402i, with final Q~1 value of(35)Q~1=1000050763.880000118.62560000118.6256Table 3 summarizes original and achieved pole locations for each area with the calculated weighting matrix Q.Table 3
Calculated weighting matrices required to shift area poles into desired new locations.
Nominal pole locations Shifted pole locations Weighting matrix Q Area2 - 14.315 -0.447±j2.84 -1.75 - 14.315 -4±j2 -20 Q ~ 2 = 1 0 0 0 0 1773.63 0 0 0 0 11.146 0 0 0 0 11.146 Area3 - 14.844 -0.4164±j3.49 -1.516 - 14.844 -4.1±j2.95 -20 Q ~ 3 = 1 0 0 0 0 133906.1 0 0 0 0 76.385 0 0 0 0 76.385 Area4 - 12.69 -0.4986±j3.55 -0.8084 - 12.69 -4±j3 -20 Q ~ 4 = 1 0 0 0 0 8.337 ∗ 10 6 0 0 0 0 51.966 0 0 0 0 51.966Local controllers are determined to be equal to(36)ka=10660430765329.4kb=5497282053325kc=7300310056026kd=6726285053627.8Global controllers are calculated to reduce the effect of the interaction between the interconnected subsystems to be(37)K12=-0.268-0.523-3.26-4.28K13=-0.268-0.523-3.26-4.28K14=-0.36-0.654-1.794-5.36K21=-0.353-0.804-2.447-2.33K23=-0.36-0.6514-1.78-5.33K31=-0.325-0.74-2.25-2.145K32=-0.245-0.478-2.98-3.916K41=-0.825-1.88-5.72-5.45K24=K34=K42=K43=0And the overall gain matrix can be calculated as in (31) to be(38)K=-15.88-13.3-2.35-10.330.5650000.562000-14.1-10.81-1.825-13.960.5170000.5170001.31500000000.5650000.5650000.5620000000-11.36-11.87-1.81-12.5700000000-11.165-14.65-2.36-34.11System step responses to 0.01 p.u input and 0.01 p.u disturbance are obtained and illustrated in Figure 5Figure 5
Step response of the controlled system with LQR applied over subsystem level for (a) 0.01 p.u. input and (b) 0.01 disturbance input.
(a) (b)To check the robustness of this design, it is required to find the controlled system step responses for±40% uncertainties with and without disturbances. Figures 6 and 7 illustrate these responses.Figure 6
Step response of the four areas with -40% uncertainties ofMi, and Tij.(a) for 0.01 p.u input and (b) for 0.01 p.u disturbance.
(a) (b)Figure 7
Step response of the four areas with +40% uncertainties ofMi, and Tij (a) for 0.01 p.u input and (b) for 0.01 p.u disturbance.
(a) (b)All the controlled system specifications are calculated from Figures5, 6, and 7 and summarized in Table 4.Table 4
Controlled system specifications for scenario II.
Controlled System Controlled System With -40% Uncertainties Controlled System With +40% Uncertainties 0.01 p.u Input 0.01 p.u Disturbance 0.01 p.u Input 0.01 p.u Disturbance 0.01 p.u Input 0.01 p.u Disturbance M p p 0 0 0.000126 0.00301 0 0 t p p __ __ 0.97 sec 0.88 sec __ __ M n p 0.000714 0.01586 0.001147 0.0243 0.00052 0.01184 t n p 0.37 sec 0.28 sec 0.36 sec 0.24 sec 0.39 sec 0.3 sec t s 1.61 sec 1.52 sec 1.89 sec 1.7 sec 1.82 sec 1.71 sec


## 5. Result Discussions

For Scenario I.Figure 3 shows that all areas are stable with more oscillations and have both over and under shoot values. The overall system eignvalues can also be calculated and illustrated in Table 5. Figure 4 shows that area-2 becomes unstable. Time responses for 40% increasing uncertainties ensure that area-2 becomes unstable; in addition, the responses for other areas become more oscillatory. Therefore, this design is not a strong design.Table 5
Nominal and controlled overall system eigenvalues.
Design Scenario Nominal System I II Two Level Control - 14.844 - 34.55 - 21.966 - 34.36 - 14.314 - 17.7 - 20.53 - 21.42 ± j 1.508 - 13.26 - 15.1 - 18.2 ± j 3.078 - 21.5 - 12.69 - 13.56 - 18 ± 1.62 - 20.108 - 0.238 ± j 4.456 - 0.606 ± j 4.335 - 11.46 - 16.58 - 0.336 ± j 3.696 - 0.955 ± j 3.258 - 10.71 ± j 6.13 - 15.586 ± j 6.03 - 0.57 ± j 3.2897 - 1.215 ± j 2.9 - 6.47 - 12.7 - 1.133 ± j 2.236 - 1.782 ± j 0.925 - 3.45 - 5.61 ± j 6.15 - 2.082 - 1.0435 ± j 0.321 - 3.346 ± j 4.38 - 4.63 ± j 4.69 - 1.877 - 1.86 - 3.45 ± j 5.577 - 4 - 0.9124 - 17 ∗ 10 - 15 - 1 ∗ 10 - 15 - 7.03 - 1 ∗ 10 - 15   - 4.39 ∗ 10 - 7For Scenario II.System step responses for nominal values (Figure 5) and for ±40% uncertainties with the presence of load disturbance (Figures 6 and 7) and the numerical data in Table 4 ensure that the system is strong and stable with satisfied specifications. Overall system eigenvalues are listed in Table 5.The design using LQR with a prespecified eigenvalues is a strong design with fast response and zero frequency deviation.Second Controller Level.Table 5 ensures that because of shifting the slowest eigenvalues of the individual areas, most of the overall system eigenvalues are shifted more inside the stability region except one slow pole still as it is with local controllers. When global controllers are designed, there is no more important change over the resulting overall system eigenvalues and to enhance these locations apply LQR technique again but over the aggregated system level.Use the overall weighting matrixQ of the form(39)Q=Q1,1⋯0⋮⋱⋮0⋯Q16,16After many trials, weighting matrix elements can be selected at Q(3,3)=1000,Q(4,4)=1551,Q(7,7)=1500,Q(8,8)=1000, and the rest of the diagonal values are equal to 1, with R=1.The additional required overall gain is(40)KlevelII=0.043-0.1626.55-3.54-0.375-0.866-7.2-11.043-0.272-0.804-3.12-2.460.13-0.03626.834.0550.160.5041.45.430.1150.3051.448.520.0550.1460.1112.830.0760.211.934.81-0.495-1.043-2.425-13.27-0.468-1.45-1.37-12.963-0.26-0.55-1.427-7.2430.03570.154-0.4852.4320.3260.3-0.69710.560.10780.436-1.1226.4460.10230.26-0.0855.850.1770.218-0.4835.05For the computed overall system eignvalues in Table 5, it is noted that all of them are moved into more stable places. A much more shifting to overall system eigenvalues can be obtained if the slower eigenvalues of each subsystem are moved farther inside the stability region but with much more control effort.A better performance is obtained by the proposed method compared to that obtained by [3]. Smaller settling time, smaller overshoot, and undershoot values with the same zero frequency deviation for time responses to 0.01 p.u load disturbance in the presence of ±40% change in the same uncertain parameters. Also, this design gets a better performance for frequency deviation to load disturbance than that in [4, 21].


## 6. Conclusion

In this paper an optimal controller based LQR is proposed to design a decentralized load frequency controller to an interconnected 4-area power system and compared to that designed by other researchers. This method succeeds to realize the design objectives. It moves the undesired subsystem eigenvalues into more stable prespecified ones which in turn shifts the overall system pole locations more inside the stability region and enhances system internal stability.The simulation results demonstrate the effectiveness of the presented method to provide a robust frequency regulator when the system is affected by ±40% uncertainties. A zero-steady state frequency deviation is obtained for each area with accepted settling time, overshoot, and undershoot values; also the worst-case performance is accepted. The internal stability is enhanced when LQR is applied again over the overall controlled system level.



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*Source: 7232915-2018-08-12.xml*