# Damping of Low-Frequency Oscillations in a Power System, Based on Multiple-Model Optimal Control Strategy

Authors: Elahe Pagard; Shahrokh Shojaeian; Mohammad Mahdi Rezaei
Journal: International Transactions on Electrical Energy Systems
 (2023)

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## Abstract

This paper proposes a new control strategy based on multimodel linear optimization control (MMLOC) to damp power system low-frequency oscillations (LFO) faster. The multimodel control strategy is used to anticipate synchronous generator dynamic behavior and linear optimization control for generating a control signal uncomplicatedly. The third-order model of synchronous machine is considered for each of the models. The probability of each model’s validity is evaluated from the difference between model outputs and the same actual power system outputs via the Bayesian approach. To improve the postfault dynamics of the case study power system, which contains a static VAR compensator, a linear optimal controller (LOC) is designed. The 8th order model of the machines is used in simulations to represent the plant as accurately as possible.


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## Body

## 1. Introduction

Nowadays, the power system stability issue and the effect of low-frequency oscillations (LFO) on large and modern power systems, which include distributed generators and high voltage DC lines, have become more important than ever. In addition, small and plenty of renewable energy sources are continuously growing up, and then, the low inertia of these generators adversely affects the overall stability of the power system [1–6]. To mitigate this challenge, many investigations based on different linear and nonlinear control methods have been made in recent decades. In [3] using eigenvalue analysis have been investigated LFO in a power system. In many references, LFO in power systems has been limited to using a power system stabilizer (PSS). In power systems including flexible transmission systems (FACTs), these devices must be coordinated with PSS to better control the transient behavior of the power system. Hence, in some previous studies, PSS restructuring has been carried out using optimization algorithms to achieve better performance in system stabilization. The optimization techniques Coyote algorithm and whale algorithm are used in [7, 8], respectively. In some other studies, the design of the PSS has been modified to improve LFO such as the fractional-order PID-PSS method [9] and the lead-lag control method [10], and in other papers, the new designs for PSS have been introduced on the power system [11, 12]. Due to the nonlinear characteristics of the electronic power converter, LFO is easily created. Improvement oscillations attenuation of a virtual synchronous machine connected to a voltage source converter (VSM) has been investigated using the virtual power system stabilizer (VPSS) in [3, 13]. A study of LFO in the power system shows that the investigation of nonlinear factors that change with time, which occurs due to fault scenarios, can be analyzed the path eigenvalues [2] and can be improved [14]. The multimodel method is used to estimate machine and system dynamics in postfault conditions. This method has been investigated and simulated using the PI controller in [15]. Other samples of applying a multimodel approach to damp of LFO can be found in [16–18].Due to the nonlinear nature of modern and interconnected power systems, the methods based on nonlinear control have been discussed in many papers [19–30]. The sliding mode (SM) control in combination with fuzzy control has been used in [20, 21]. Some other papers have applied the various order SM controllers to damp LFOs [22–27]. These papers, in some cases, have proposed the fractional-order SM controllers [22, 23] and the higher-order SM controller [26, 27]. In [29], a SM control is used on an extended state observer to improve LFOs. Nonlinear control methods show better performance for LFO damping, but their implementation has certain complexities. In some papers, nonlinear control methods have been combined with linear methods to take advantage of both. The suggestion of [31] is to combine the desirable features of the PID controller and many optimal effective features of the SM controller to eliminate the chattering. In order to produce the best control signal for the excitation system, a robust SM-PID design has been performed on a single machine infinite bus (SMIB) system to increase its stability. In [32], combining PID and SM controllers have been proposed to achieve the advantages of both controllers. Also, the linear optimal control (LOC) method for damping LFOs in [33–35] is discussed. To examine the other practice of incorporating linear and nonlinear control methods, in this paper, the combination of LOC and the multiple-model control (MMC) strategy is proposed and implemented on a single machine system. The system postfault behavior is analyzed and compared with conventional SM control. Since the SM control which have been studied in other references, are very effective in controlling LFOs. Despite the fact that nonlinear control methods are effective in damping low-frequency oscillations, the structural complexity and implementation problems of these methods are considered a major drawback in the power industry. Therefore, we intend to increase the accuracy of the proposed control strategy to nonlinear control methods.


## 2. The MMC Strategy

The block diagram of MMC strategy is shown in Figure1. In this strategy, multiple identifier models are used to model dynamics behavior of synchronous generator. The models have been composed based on different set points of the synchronous generator. Each model estimates plant parameters, and then, the differences between the estimated and actual values are calculated. According to Figure 1, the MMC strategy scheme consists of three main blocks and two sub-blocks. The main parts are the plant, the model bank, and the control bank, and the subparts include the weight calculation block and the incorporation block.Figure 1
The multiple-model control strategy.


### 2.1. Studied Power System

A SMIB power system shown in Figure2 is used in this paper. The synchronous generator has been connected to the infinite bus via double transmission lines. A static var compensator (SVC) is located in the middle of one of the lines.Figure 2
Studied power system.


### 2.2. Calculation of Model Weights

The probability of each model is calculated for each synchronous generator model, according to the difference in the measured and calculated angular speed at each time. Then, based on the calculated errors and calculated probabilities in the previous step for each model, the new probabilities have been computed. A model that is less different from the plant should have a greater impact on to control of the plant. The Bayes recursive theory has been used to achieve this goal. At thekth time step, the weights of each model are computed from(1)Pi,k=exp−1/2ei,kTCfei,kPi,k−1∑j=1Nexp−1/2ei,kTCfei,kPi,k−1,ei,k=yk−yi,kc,where ei,k is the speed error of the ith model at the kth step, yk is the angular speed of the plant at the kth time step, and yci,k is the angular speed of ith model at the kth step. Cfe is the convergence coefficient of the probabilities.(2)Wi,k=Pi,k∑j=1NPj,k.After calculating the weight of each model, in each iteration, the output signals of the controllers are combined and applied to the plant. Therefore, the control signal of each model in each iteration (uj,k) is multiplied by its own weight. The multiplications are then added together to create the final control signal. According to the MMC strategy, this hybrid signal (uj) should be applied to both plant and each of the models in the model bank.(3)uk=∑i=1NWj,k⋅uj,k.


### 2.3. Model Bank

In this study, the plant is a synchronous generator connected to a power system that 8th order model has been used to describe the synchronous generator. To implement the MMC strategy, the expected response space of the system and the machine must be modeled. For this purpose, two 3rd order models for low and medium load conditions of the synchronous generator are considered. These reduced-order models are designed according to the operating point of the synchronous generator in medium load and low load conditions to be able to estimate the dynamic behavior of the synchronous generator at all operating points and in postdisturbance conditions. Each of the models in the model bank is connected to the same power system under study.


### 2.4. Controller Bank

In the control bank for each of the models in the model bank, a linear optimal controller is designed, as described in the next section. This controller generates two output signals that are applied to the synchronous generator excitation system and static var compensator (SVC) in each model.


## 2.1. Studied Power System

A SMIB power system shown in Figure2 is used in this paper. The synchronous generator has been connected to the infinite bus via double transmission lines. A static var compensator (SVC) is located in the middle of one of the lines.Figure 2
Studied power system.


## 2.2. Calculation of Model Weights

The probability of each model is calculated for each synchronous generator model, according to the difference in the measured and calculated angular speed at each time. Then, based on the calculated errors and calculated probabilities in the previous step for each model, the new probabilities have been computed. A model that is less different from the plant should have a greater impact on to control of the plant. The Bayes recursive theory has been used to achieve this goal. At thekth time step, the weights of each model are computed from(1)Pi,k=exp−1/2ei,kTCfei,kPi,k−1∑j=1Nexp−1/2ei,kTCfei,kPi,k−1,ei,k=yk−yi,kc,where ei,k is the speed error of the ith model at the kth step, yk is the angular speed of the plant at the kth time step, and yci,k is the angular speed of ith model at the kth step. Cfe is the convergence coefficient of the probabilities.(2)Wi,k=Pi,k∑j=1NPj,k.After calculating the weight of each model, in each iteration, the output signals of the controllers are combined and applied to the plant. Therefore, the control signal of each model in each iteration (uj,k) is multiplied by its own weight. The multiplications are then added together to create the final control signal. According to the MMC strategy, this hybrid signal (uj) should be applied to both plant and each of the models in the model bank.(3)uk=∑i=1NWj,k⋅uj,k.


## 2.3. Model Bank

In this study, the plant is a synchronous generator connected to a power system that 8th order model has been used to describe the synchronous generator. To implement the MMC strategy, the expected response space of the system and the machine must be modeled. For this purpose, two 3rd order models for low and medium load conditions of the synchronous generator are considered. These reduced-order models are designed according to the operating point of the synchronous generator in medium load and low load conditions to be able to estimate the dynamic behavior of the synchronous generator at all operating points and in postdisturbance conditions. Each of the models in the model bank is connected to the same power system under study.


## 2.4. Controller Bank

In the control bank for each of the models in the model bank, a linear optimal controller is designed, as described in the next section. This controller generates two output signals that are applied to the synchronous generator excitation system and static var compensator (SVC) in each model.


## 3. The Linearized Dynamic Model of the Power System under Study

The 3rd order model of synchronous generator is used as a system linear model in this paper. Also, the synchronous machine excitation system is the IEEE Type-ST1 A [36]. The parameters KA and TA are the gain and time constants related to this system, respectively.(4)dδdt=ωbωr−1,dωdt=Pm−Pe−DΔωM,(5)deq′dt=1τdo'efd−xd−xd′id−eq′,(6)defddt=1TA−efd−KAvtr−vt,where the input electrical power is calculated by(7)Pe=vdsids+vqsiqs,where M and D are the inertia constant and damping coefficient, respectively.


### 3.1. Modeling of the Synchronous Generator

The 8th order synchronous machine model [37] is used for accurate simulation:(8)dψqsdt=ωbvqs−ωrωbψds+rsxlsψmq−ψqs,dψdsdt=ωbvds−ωrωbψqs+rsxlsψmd−ψds,dψkq1dt=ωbvkq1−rkq1xlkq1ψmq−ψkq1,dψkq2dt=ωbvkq2−rkq2xlkq2ψmq−ψkq2,dψkddt=ωbvkd−rkdxlkdψmd−ψkd,dδdt=ωbωr−1,dωdt=1MPm−Pe−DΔω,where ψ is the linkage fluxes of different synchronous generator windings, kq1, kq2, and kd are the indices denoting rotor q and d axis damper windings, and fd denotes the rotor field winding. Pm and Pe are the generator’s mechanical and electrical power, respectively, and ωb is the base angular speed. To obtain control law, the first step is linearization of the studied power system. In Figure 3, one can see that(9)−Iae+vt−vBZline+vt−vCZline/2=0,(10)Ish+vC−vBZline/2+vC−vtZline/2=0,where Zline is the impedance of each line and in this paper, it is supposed to be 0.015 + j0.175 pu.Figure 3
The power system model under study.Substitutingvt = vd + jvq and Zline = R + jX in (9) and (10) and separating imaginary and real parts result in(11)vt=vB−14Zline⋅Ish+12Zline⋅Iae,vd=vB⋅cosδ−X4Ish+12R⋅id−12X.iq,vq=vB⋅sinδ−R4Ish+12X⋅id−12X.iq.Now, the obtainedvd and vq are placed in the machine’s model equations (37) and then equations id and iq are obtained from δ, e′q, and Ish and subsequently the resulting equations have been linearized:(12)vqvd=10eq′−0xd′−xq0iqid,ΔiqΔid=Dq−EqLqDd−EdLdΔδΔeq′ΔIsh,where Dq, Dd, Eq, Ed, Lq, and Ld all are functions of R, X, VB, x׳d, xq, and δ0. Next, the linearized 3rd order model of a synchronous machine is used to obtain the state-space equations:(13)pΔδ=ωbΔωr,(14)pΔωr=1MΔPm−ΔPe.Substituting linear electrical power equations in (26) results in [37](15)pΔωr=1MΔPm−K1Δδ−K2Δeq′−KpΔIsh,K1=xq−xd′id0+eq0'Dq+xq−xd′iq0Dd,K2=−xq−xd′id0+eq0'Eq−xq−xd′iq0Ed+iq0,Kp=xq−xd′id0+eq0'Lq+xq−xd′iq0Ld.Considering the excitation system of Figure4, linearization of (5) results in(16)pΔeq′=1τdo'Δefd−K4Δδ−K3Δeq′−KqΔIsh,(17)K3=1−xd−xd′Ed,K4=xd−xd′Dd,Kq=xd−xd′Ld.(18)pΔefd=1TA−Δefd+KA−Δvt+uE,(19)pΔefd=1TA−Δefd+KA−K5Δδ−K6Δeq′−KvΔIsh+uE,(20)K5=xqDqsinδ0−xd′Ddcosδ0,K6=−xqEqsinδ0+1+xd′Edcosδ0,Kv=xqLqsinδ0−xd′Ldcosδ0.Figure 4
The exciter system model.DefiningX = [Δδ Δω Δe′q Δefd] T as the state vector and u = [uEIsh] T, from (13), (15), (16), and (19), the system state-space can be expressed as follows:(21)X˙=AX+Bu,where(22)A=0ωb00−K1M0−K2M0−K4τdo′0−K3τdo′1τdo′−KAK5TA0−KAK6TA−1TA,B=000−KvM0−Kqτdo′KATA−KAKvTA.


## 3.1. Modeling of the Synchronous Generator

The 8th order synchronous machine model [37] is used for accurate simulation:(8)dψqsdt=ωbvqs−ωrωbψds+rsxlsψmq−ψqs,dψdsdt=ωbvds−ωrωbψqs+rsxlsψmd−ψds,dψkq1dt=ωbvkq1−rkq1xlkq1ψmq−ψkq1,dψkq2dt=ωbvkq2−rkq2xlkq2ψmq−ψkq2,dψkddt=ωbvkd−rkdxlkdψmd−ψkd,dδdt=ωbωr−1,dωdt=1MPm−Pe−DΔω,where ψ is the linkage fluxes of different synchronous generator windings, kq1, kq2, and kd are the indices denoting rotor q and d axis damper windings, and fd denotes the rotor field winding. Pm and Pe are the generator’s mechanical and electrical power, respectively, and ωb is the base angular speed. To obtain control law, the first step is linearization of the studied power system. In Figure 3, one can see that(9)−Iae+vt−vBZline+vt−vCZline/2=0,(10)Ish+vC−vBZline/2+vC−vtZline/2=0,where Zline is the impedance of each line and in this paper, it is supposed to be 0.015 + j0.175 pu.Figure 3
The power system model under study.Substitutingvt = vd + jvq and Zline = R + jX in (9) and (10) and separating imaginary and real parts result in(11)vt=vB−14Zline⋅Ish+12Zline⋅Iae,vd=vB⋅cosδ−X4Ish+12R⋅id−12X.iq,vq=vB⋅sinδ−R4Ish+12X⋅id−12X.iq.Now, the obtainedvd and vq are placed in the machine’s model equations (37) and then equations id and iq are obtained from δ, e′q, and Ish and subsequently the resulting equations have been linearized:(12)vqvd=10eq′−0xd′−xq0iqid,ΔiqΔid=Dq−EqLqDd−EdLdΔδΔeq′ΔIsh,where Dq, Dd, Eq, Ed, Lq, and Ld all are functions of R, X, VB, x׳d, xq, and δ0. Next, the linearized 3rd order model of a synchronous machine is used to obtain the state-space equations:(13)pΔδ=ωbΔωr,(14)pΔωr=1MΔPm−ΔPe.Substituting linear electrical power equations in (26) results in [37](15)pΔωr=1MΔPm−K1Δδ−K2Δeq′−KpΔIsh,K1=xq−xd′id0+eq0'Dq+xq−xd′iq0Dd,K2=−xq−xd′id0+eq0'Eq−xq−xd′iq0Ed+iq0,Kp=xq−xd′id0+eq0'Lq+xq−xd′iq0Ld.Considering the excitation system of Figure4, linearization of (5) results in(16)pΔeq′=1τdo'Δefd−K4Δδ−K3Δeq′−KqΔIsh,(17)K3=1−xd−xd′Ed,K4=xd−xd′Dd,Kq=xd−xd′Ld.(18)pΔefd=1TA−Δefd+KA−Δvt+uE,(19)pΔefd=1TA−Δefd+KA−K5Δδ−K6Δeq′−KvΔIsh+uE,(20)K5=xqDqsinδ0−xd′Ddcosδ0,K6=−xqEqsinδ0+1+xd′Edcosδ0,Kv=xqLqsinδ0−xd′Ldcosδ0.Figure 4
The exciter system model.DefiningX = [Δδ Δω Δe′q Δefd] T as the state vector and u = [uEIsh] T, from (13), (15), (16), and (19), the system state-space can be expressed as follows:(21)X˙=AX+Bu,where(22)A=0ωb00−K1M0−K2M0−K4τdo′0−K3τdo′1τdo′−KAK5TA0−KAK6TA−1TA,B=000−KvM0−Kqτdo′KATA−KAKvTA.


## 4. Linear Optimal Control Approach

The LOC theory described in [33–35] is restated in this section briefly. In the LOC method, by defining the energy function J as in (23), and minimizing that the optimal linear control signal u according to the feedback control law is obtained:(23)J=12∫t0tfXTQX+uTRudt,(24)u=−R−1BTKX.In (23), the weight R is the real, definite positive and symmetric matrix and the Q is the real, symmetric positive, and quasidefinite weight matrix. In this paper, they selected as follows:(25)R=1001,Q=1000000100000000.100000.The matrixK is obtained by solving the Riccati equation given in (26). Substituting the optimal K in (24), the optimal u is calculated. It generates the optimal control signal for controlling the excitation system and SVC simultaneously:(26)ATK+KA−KBR−1KT+Q=0.


## 5. Results and Discussion

The power system shown in Figure2 and the proposed MMLOC shown in Figure 1 is simulated using MATLAB®. The simulation code is built in MATLAB/m-code environment using a step size 0.1 ms. The 4th order Runge-Kutta numerical method is used to solve the 8th order synchronous generator nonlinear equations (as the real plant) and the third order of synchronous generator nonlinear equations (as the models) and the power system. The parameters of the synchronous generator can be found in Table 1. To investigate the effect of the MMLOC strategy, it is assumed that a three-phase symmetric fault has occurred in the middle of lines 1–3 in the scenario of Table 2.Table 1
The synchronous generator parameters.
ParametersValue (pu)D0.01rs0.0019xls0.19xd1.8xq1.8xlfd0.1414rfd0.000929xlkd0.08125rkdK0.01334xlkq10.8125rkq10.00178xlkq20.0939rkq20.00841Table 2
Three-phase fault planning.
StepPlanTime (sec)1Steady statet < 0.52There is a 3-phase fault in the middle of lines 1–30.5 <t < 0.73Lines 1–3 is opened on both sides0.7 <t < 1.74The fault is removed and the lines 1–3 are reclosedt > 1.7The aim of this paper is to damp LFO via applying coordinated control to the generator excitation system and SVC. Hence, the postfault dynamics of the synchronous generator have been estimated by the multimodel approach and the LOC method has been used for the production of control signals. Each model produces its own optimal linear control signal. Combining the control signals of the models together probabilistically, the final control signals are generated and applied to the plant. To evaluate the performance of the MMLOC strategy, this strategy has been compared with the SM control method and comparative results have been drawn in Figures5 and 6. To better compare the MMLOC and SMC performances, the coefficients of the SMC method have been adjusted so that the excitation voltage diagram and the terminal voltage in both methods are superimposed. The results are shown in Figures 7 and 8. Also, the generator output current has the same maximum values, which are shown in Figure 9. As in the more detailed figures drawn in Figures 7 and 9, it can be seen that the SMC method causes a lot of chattering in the voltage and current of the generator, which is a big disadvantage for the SMC control method. The final control signals contain the excitation voltage (Exfd) and the SVC susceptance (Bsvc) that are drawn in Figures 8 and 10, respectively. The control performance of rotor speed based on MMLOC, SM, and no control are shown in Table 3. It can be seen that the control performances of MMLOC and SM are much better than no control. There are the lower peak and the lower valley in MMLOC compared with those of SM strategy. The rise time is the same in both control methods because the three-phase fault has not yet been seen by the controller.Figure 5
Rotor speed of the generator with MMLOC and SMC and without control.Figure 6
Rotor angle of the generator with MMLOC and SMC and without control.Figure 7
Terminal voltage of the generator with MMLOC and SMC.Figure 8
Exciter voltage of the generator with MMLOC and SMC.Figure 9
The current generator with MMLOC and SMC.Figure 10
The control signal of SVC in MMLOC strategy.Table 3
The performance index rotor speed under three-phase fault.
Control strategyPeak (pu)Valley (pu)Rise time (s)Settling time (s)MMLOC1.0050.9950.72SMC1.0070.9940.72No control1.0070.9930.7>10The control performance of rotor angle based on MMLOC, SM, and no Control are shown in Table4. It can be seen that the control performances of MMLOC and SM are much better than no control. There is the lower peak in MMLOC compared with that of SM strategy. Similarly, the rise time is the same in both control methods because the three-phase fault has not yet been seen by the controller. Also, the settling time is almost the same in both methods and both performance index [38–42].Table 4
The performance index rotor angle under three-phase fault.
Control strategyPeak (pu)Valley (pu)Rise time (s)Settling time (s)MMLOC53.5280.832SMC62280.832No control62.614.60.96>10



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