# Wide-Area Damping Controller Design Based on the Frequency Domain Self-Excitation Method

Authors: Lei Gao; Yanchao Ma; Yonghui Nie; Shiyun Xu; Xiaojie Chu; Long Peng
Journal: Journal of Control Science and Engineering
 (2018)

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## Abstract

In this paper, a wide-area DC damping controller design method is proposed to damp the weak interarea oscillation modes. First, the low-order power system control model identification method is proposed based on the frequency domain self-excitation method to obtain the dominant modes and other related information. In what follows, the identified model is transformed into a low-order state space equation. By comparing the geometric metrics of different schemes, the various input signals and installation locations of the controller are selected. Furthermore, the damping controller parameter calculation is realized based on the self-excitation frequency domain identification method, which does not depend on the detailed model of the system. The design process is simple and it is easy to apply in the practical engineering of large-scale complex power grids. The applicability and effectiveness of the proposed controller design method are demonstrated through simulations of a two-area four-generator power system.


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## Body

## 1. Introduction

The AC/DC interconnected power grid can transmit large capacity of power from the remote power basis to the load centers through communication networks. It plays an extremely important role in the energy allocation in China. Since the AC and DC transmission systems are generally coupled with each other, however, disturbances that occur in both the AC and the DC sides may trigger weakly damped low frequency oscillations. If the oscillations are of the interarea type, a large number of generators will participate in and the interconnected power grid will confront severe operational risk [1, 2].However, due to the deficiency in observability, traditional power system stabilizers (PSS) which use local signals as inputs may fail to damp out the interarea oscillation and maintain system stability. To overcome the shortcoming of traditional PSS, wide-area damping controllers with remote signals obtained from wide-area measurement system (WAMS) have become a better choice in suppressing low frequency oscillation [3–5]. Since DC systems have fast response characteristics, it can effectively improve the stability of the system once faults have occurred in the AC system, and the DC damping control [6–9] that can suppress the interarea low frequency oscillation of AC/DC hybrid system has attracted the attention of researchers.During the wide-area damping controller design, it is important to choose feedback control signal and installation location. The residue method [10, 11] is widely used for feedback signal selection, but it can only compare the same type of signals. For different types of signals, such as generator speed, line transmission power, and bus voltage, the amplitude of the above-mentioned residual value will not be accurate due to the ratio problem. In [12], a relative gain array (RGA) method is proposed. Although this method reduces the mutual influence among multiple controllers, it has the same disadvantages as the residual method. Other methods, such as the Hankel singular value (HSV) [13], the singular value decomposition (SVD) [14], and the minimum singular values (MSV) [15], have also been proposed to calculate the corresponding observability and controllability indices, but it becomes much more complicated to calculate for practical large-scale power systems. On the other hand, most of the existing damping control design methods require the precise system model through linearizing it around an equilibrium point. This may lead to the modeling error due to information incompleteness.In this paper, the low frequency oscillation of an AC/DC hybrid system is damped by installing a wide-area damping controller in the DC system. Based on the above considerations, the improved geometric lateral index [11, 16] based on the frequency domain excitation identification theory is proposed to solve the wide-area DC supplementary damping control signal selection in AC/DC hybrid systems. This method can effectively solve the problem that different types of input signals have different amplitude ratios and can avoid solving the huge state matrix as well. Moreover, the method based on the self-excitation frequency domain identification is further utilized to configure the DC damping controller, which does not depend on the detailed model of the system. The design process is simple, and it is easy to use in the practical engineering of the complex large power grid. At the end of this study, the AC/DC hybrid system is taken as an example to demonstrate the effectiveness of the proposed controller design method.


## 2. Problem Formulation

### 2.1. DC Damping Controller

The DC damping control is a kind of small signal modulation, and the basic principle is to add a supplementary damping controller in the DC main control loop. The controller is usually installed at the rectifier side. In this regard, the power of the DC transmission line is adjusted by tuning the extinction angleα to damp the system oscillation.The control block diagram is shown in Figure1. By adding the DC additional control output If and the reference current input Idcr to the current control of the inverter can improve the damping of the system. The dynamic equation of the linearized regulator is given as(1)α˙=KrTrIdcr-Idc+If-αTrI˙dc=1Tdc-Idc+Idcr+IfThe supplementary wide-area damping controller transfer function is in the following form:(2)Hs=K1+sT1·sTW1+sTW·1+sT21+sT3mwhere Tdc, Tr, T1, T2, T3, TW are the time constants, Kr and K are the controller proportional coefficient, Idc is the DC current, and u is the feedback signal, which can be chosen as the generator speed deviation, the bus phase difference or any bus voltage, power, and so on.Figure 1
Rectifier current regulator.In this study, the selected feedback signal is processed by the damping controller to the input of the current regulator in the DC main control system, based on the principle of DC damping control. The extinction angleα is maintained within the normal range (αmin<α<αmax). In order to select the best alternative signal, the indicators defined by the improved geometric index can be calculated. In this regard, the designed damping controller can damp system oscillation.


### 2.2. System Model Analysis

By linearizing around the stable equilibrium point, the AC/DC hybrid power system equation can be given as(3)Δx˙=AΔx+BΔuΔy=CΔxwhere A is an n×n state matrix, B is an n×p input matrix, and C is a q×n output matrix.For power system withn generators, only part of the oscillation mode of the generator rotor is concerned, and the number of oscillating modes in which any one generator is significantly involved is generally 2~3. Based on this consideration, the state equation of the ith generator can be set as(4)Δδ˙Δω˙=0I-M-1Ks-M-1Ds·ΔδΔωwhere K(s) is the synchronization torque coefficient, D(s) is the damping torque coefficient, and M is the diagonal matrix composed by the corresponding elements of the generator.It can be seen from Figure2 that if there is no external disturbance Tx, the differential equation of the corresponding equivalent system can be derived as(5)MsΔδ¨+DsΔδ˙+KsΔδ=0Assume that the eigenvalue of (5) is λi=σi±jωi, then(6)MΔδ¨+DsΔδ˙+KsΔδλi≝Tx→0Figure 2
Equivalent system.In order to calculate the electromechanical mode associated with system oscillation, the self-excited method [17] is utilized by deriving the eigenvalue of (6) through the Newton iteration method. However, the application of the traditional self-excited method is not only sensitive to the initial value but also easily leads to the loss of roots. In this regard, this paper proposes a frequency domain self-excited identification method, which is applied to choose controller inputs. This method avoids the problems of traditional self-excited method, which is described in detail in the following section.


## 2.1. DC Damping Controller

The DC damping control is a kind of small signal modulation, and the basic principle is to add a supplementary damping controller in the DC main control loop. The controller is usually installed at the rectifier side. In this regard, the power of the DC transmission line is adjusted by tuning the extinction angleα to damp the system oscillation.The control block diagram is shown in Figure1. By adding the DC additional control output If and the reference current input Idcr to the current control of the inverter can improve the damping of the system. The dynamic equation of the linearized regulator is given as(1)α˙=KrTrIdcr-Idc+If-αTrI˙dc=1Tdc-Idc+Idcr+IfThe supplementary wide-area damping controller transfer function is in the following form:(2)Hs=K1+sT1·sTW1+sTW·1+sT21+sT3mwhere Tdc, Tr, T1, T2, T3, TW are the time constants, Kr and K are the controller proportional coefficient, Idc is the DC current, and u is the feedback signal, which can be chosen as the generator speed deviation, the bus phase difference or any bus voltage, power, and so on.Figure 1
Rectifier current regulator.In this study, the selected feedback signal is processed by the damping controller to the input of the current regulator in the DC main control system, based on the principle of DC damping control. The extinction angleα is maintained within the normal range (αmin<α<αmax). In order to select the best alternative signal, the indicators defined by the improved geometric index can be calculated. In this regard, the designed damping controller can damp system oscillation.


## 2.2. System Model Analysis

By linearizing around the stable equilibrium point, the AC/DC hybrid power system equation can be given as(3)Δx˙=AΔx+BΔuΔy=CΔxwhere A is an n×n state matrix, B is an n×p input matrix, and C is a q×n output matrix.For power system withn generators, only part of the oscillation mode of the generator rotor is concerned, and the number of oscillating modes in which any one generator is significantly involved is generally 2~3. Based on this consideration, the state equation of the ith generator can be set as(4)Δδ˙Δω˙=0I-M-1Ks-M-1Ds·ΔδΔωwhere K(s) is the synchronization torque coefficient, D(s) is the damping torque coefficient, and M is the diagonal matrix composed by the corresponding elements of the generator.It can be seen from Figure2 that if there is no external disturbance Tx, the differential equation of the corresponding equivalent system can be derived as(5)MsΔδ¨+DsΔδ˙+KsΔδ=0Assume that the eigenvalue of (5) is λi=σi±jωi, then(6)MΔδ¨+DsΔδ˙+KsΔδλi≝Tx→0Figure 2
Equivalent system.In order to calculate the electromechanical mode associated with system oscillation, the self-excited method [17] is utilized by deriving the eigenvalue of (6) through the Newton iteration method. However, the application of the traditional self-excited method is not only sensitive to the initial value but also easily leads to the loss of roots. In this regard, this paper proposes a frequency domain self-excited identification method, which is applied to choose controller inputs. This method avoids the problems of traditional self-excited method, which is described in detail in the following section.


## 3. Wide-Area DC Damping Controller Design

Generally speaking, the steps in designing a wide-area controller include wide-area signal selection, system reduction, controller design, and validation of the design by numerical simulations. Each of the above steps is briefly described in the following subsections.


### 3.1. Observable and Controllable Quantitative Index

In order to choose the feedback signal and the installation location of the wide-area damping controller, it is necessary to establish a unified index for the calculation of different types of signals or control points. The quantization index of the observability and controllability of the system based on the improving geometric lateral index can solve the defects of the different types of signals, which is described as follows.For system (3), by denoting a new state variable Δx=Wz, the oscillation characteristics of the system can be decoupled and the system with a diagonal state matrix can be derived as(7)z˙=Λz+B′ΔuΔy=C′z+DΔu,where W is the modal matrix of A, Λ=diagλi, B′=W-1B=NTB, C′=CW, Wi is the right eigenvector corresponding to λi, and Ni is the left eigenvector corresponding to λi.The observability index of the ith modal of the system is given as(8)goji=cos⁡αWi,cj=cjWiWicjwhile the controllability index is defined by(9)gcki=cos⁡θNi,bk=bkTNiNibkwhere · and ·, respectively, represent the modulo and Euclidean norm, cj is the jth line of the matrix C, bk is the kth column of the matrix B, α(Wi,cj) is the angle between the output phasor cj and the right eigenvector Wi, and θ(Ni,bk) is the angle between the output phasor bk and the left eigenvector Ni.Thus, the comprehensive index corresponding to the ith modal is given by(10)gcoik,j=gojigcki=cos⁡αWi,cjcos⁡θNi,bkQuantitative analysis and comparison of the observability and controllability of different types of alternative signals can be realized by the calculation of the above-mentioned indices. However, in the practical power system, it is difficult to obtain the accurate full-order system model. Based on this consideration, this study carries out power system model identification according to the frequency domain self-excited method, and then the low-order system can be utilized to approximate the original system, which greatly reduces the calculation complicatedness and improves the calculation effectiveness.


### 3.2. Control Loop Selection

The feedback signals and installation locations selection are the basis for controller design of wide-area damping controller. In this subsection, a new method to select the feedback signal of wide-area damping controller is proposed by combining the basic principle of self-excitation and the method of frequency domain identification.Based on the basic principle of the self-excited method, it can be derived from (6) that(11)Tx+dTxdsΔλi=0By analyzing (11), the following necessary conditions are obtained:(12)Tx→0dTxds→0However, since the traditional self-excitation method iteratively solves the eigenvalue of (6) by Newton's method, it may lead to the incompleteness of electromechanical modes. Therefore, the method proposed in this paper utilizes the frequency domain method to identify the transfer function of the system and then solve the relevant mode information to avoid the above-mentioned problems.By carrying out the Laplace transform of (6) and (12), the system transfer function can be derived as(13)Gs=ΔδsΔTxs=sMss2+Dss+KsIf a more accurate model of the generator is considered, such as taking the excitation system model into account, (9) still holds, but the order will increase. In this case, Gs can be expressed as follows:(14)Gs=ΔδisΔTxis=Ks-zj1s-zj2⋯s-zjis-p1s-p2⋯s-pk=Rj1s-p1+Rj2s-p2+⋯Rjks-pkIt can be analyzed from (6) and (12) that if the frequency of Tx is equal to the oscillation frequency of the system, only the torque with the amplitude close to zero can stimulate the large response of the oscillation frequency. Assume that Tx can increase the oscillation frequency to λω, if different generators are selected as the “excitation point”, then Gλiω can be expressed as(15)Gλiω=ΔδiλiωΔTxiλiω=Rj1λiω-p1+Rj2λiω-p2+⋯Rjkλiω-pkAlthough the “excitation point” varies, the system interarea oscillation mode keeps the same, where a pair of conjugate poles represents an oscillation mode. Letλi=σi±jωi, then it can be used to calculate the system oscillation frequency and damping ratio.Once the oscillation mode is determined, the index calculation of the candidate signal should be carried out, when it is necessary to transform the system into the state space form andGs can be written as(16)Gs=YsUs=bmsm+bm-1sm-1+⋯+b0sn+an-1sn-1+⋯+a0In this regard, the state space expression of the system can be derived as(17)Δx˙=EΔx+FΔuΔy=GΔx+HΔuwhere E=0In-1A0An-1 is the equivalent matrix of matrix A in (3), which contains the main mode of the original system. F=0⋯1T and I are the corresponding control matrix and substantial matrix, respectively. G=Bn-Anbn, H=bn. Equation (17) is the state space model derived by the frequency domain self-exited method.The state space equation identified by the frequency domain excitation method preserves the main mode of operation of the system. Therefore, (17) can be used to substitute the original system to calculate the quantization index of the observability and controllability of the system.The new method of selecting feedback signal of wide-area damping controller based on the self-excited frequency domain identification method proposed in this paper can be used in engineering application of large-scale power system without obtaining the full-state model of the system, and the calculation and operation are convenient, which is suitable for the engineering application of large power system.


### 3.3. Wide-Area Coordinated Controller Design

Once the state space equation is identified by the self-excited frequency domain identification, by carrying out the Laplace transform on (13), the transfer function between the input variable ΔU1 and the output variable ΔY1 can be obtained as(18)Gs=ΔU1sΔY1s=GsI-E-1F+HSuppose dominant oscillation modes for (18) are λ1,2=δ±jω, and the desired damping ratio after adding damping controller is ξ. Substituting s=jωd=j1-ξ2ω into (18) derives(19)Gjωd=ΔU1sΔY1ss=jωd=G∠ϕwhere G and ϕ are the amplitude and phase of G(s) at s=jωd, respectively.The lead lag link parameters of the wide-area DC damping controller (2) can be calculated by(20)ϕx=180-ϕα=1+sin⁡ϕx1-sin⁡ϕxT1=αωT2=T1αIn order to prevent the damping controller from outputting a constant deviation voltage, the design principle is that, in the frequency range where damping controllers provide damping, no obvious phase displacement should be introduced in the isolating links. The typical value is chosen as 1~10s, and in this study, it is set to be 5s.The advantage of the above wide-area DC damping controller parameter design based on the self-excited method is that it does not rely on the detailed model of the system, the design process is simple, the derived parameters are stable, and it is applicable to utilize in the practical complex large-scale power grid.


### 3.4. Wide-Area Damping Control Design Process

The basic steps of the selection of the feedback signal and the installation location of the wide-area damping controller based on the self-excited frequency domain identification method are as follows:Step 1.
Identify the low-order system based on the frequency domain self-excited method, and calculate the corresponding interarea oscillation modesλ1,2=δ±jω and the oscillation frequency ω.Step 2.
Select different feedback signals and installation locations of then groups of controllers, and then the system is able to be operated under the steady state with the open loop operation of the controller. The disturbance test for groupi (0<i<n) is carried out, and for the disturbance ui(t) that does not destroy the stability of the original system, record the output response data yi(t).Step 3.
Extract the variation dataΔui(t) and Δyi(t) of ui(t) and yi(t) in the common cycle. Carry out the discrete Fourier transform, respectively, and obtain the varying phasors Δui(jω) and Δyi(jω) corresponding to different frequency values, where i=1,2,…,n,0≤jω≤2π.Step 4.
The self-excited method is applied to identify the phasor dataΔui(jω) and Δyi(jω), and the corresponding transfer function Gi(s) is fitted. Transform the transfer function into the state space form and obtain the corresponding system matrices Ai, Bi, Ci, Di.Step 5.
Apply the geometric measure index method to calculate each index, which is used as the reference for the input signal of the comparison controller. Based on the above method, that best feedback signal and installation location are determined.Step 6.
Based on the state space equation and the dominant modes, identify the information such as the phaseΦ and the oscillation mode ω that are required by controller design. Calculate the controller time constants Ti,i=1,2,…,4 by (20).Step 7.
Adjust the controller gainKi to verify the damping effect, and the operation parameters of the wide-area DC damping controller are finally determined.


## 3.1. Observable and Controllable Quantitative Index

In order to choose the feedback signal and the installation location of the wide-area damping controller, it is necessary to establish a unified index for the calculation of different types of signals or control points. The quantization index of the observability and controllability of the system based on the improving geometric lateral index can solve the defects of the different types of signals, which is described as follows.For system (3), by denoting a new state variable Δx=Wz, the oscillation characteristics of the system can be decoupled and the system with a diagonal state matrix can be derived as(7)z˙=Λz+B′ΔuΔy=C′z+DΔu,where W is the modal matrix of A, Λ=diagλi, B′=W-1B=NTB, C′=CW, Wi is the right eigenvector corresponding to λi, and Ni is the left eigenvector corresponding to λi.The observability index of the ith modal of the system is given as(8)goji=cos⁡αWi,cj=cjWiWicjwhile the controllability index is defined by(9)gcki=cos⁡θNi,bk=bkTNiNibkwhere · and ·, respectively, represent the modulo and Euclidean norm, cj is the jth line of the matrix C, bk is the kth column of the matrix B, α(Wi,cj) is the angle between the output phasor cj and the right eigenvector Wi, and θ(Ni,bk) is the angle between the output phasor bk and the left eigenvector Ni.Thus, the comprehensive index corresponding to the ith modal is given by(10)gcoik,j=gojigcki=cos⁡αWi,cjcos⁡θNi,bkQuantitative analysis and comparison of the observability and controllability of different types of alternative signals can be realized by the calculation of the above-mentioned indices. However, in the practical power system, it is difficult to obtain the accurate full-order system model. Based on this consideration, this study carries out power system model identification according to the frequency domain self-excited method, and then the low-order system can be utilized to approximate the original system, which greatly reduces the calculation complicatedness and improves the calculation effectiveness.


## 3.2. Control Loop Selection

The feedback signals and installation locations selection are the basis for controller design of wide-area damping controller. In this subsection, a new method to select the feedback signal of wide-area damping controller is proposed by combining the basic principle of self-excitation and the method of frequency domain identification.Based on the basic principle of the self-excited method, it can be derived from (6) that(11)Tx+dTxdsΔλi=0By analyzing (11), the following necessary conditions are obtained:(12)Tx→0dTxds→0However, since the traditional self-excitation method iteratively solves the eigenvalue of (6) by Newton's method, it may lead to the incompleteness of electromechanical modes. Therefore, the method proposed in this paper utilizes the frequency domain method to identify the transfer function of the system and then solve the relevant mode information to avoid the above-mentioned problems.By carrying out the Laplace transform of (6) and (12), the system transfer function can be derived as(13)Gs=ΔδsΔTxs=sMss2+Dss+KsIf a more accurate model of the generator is considered, such as taking the excitation system model into account, (9) still holds, but the order will increase. In this case, Gs can be expressed as follows:(14)Gs=ΔδisΔTxis=Ks-zj1s-zj2⋯s-zjis-p1s-p2⋯s-pk=Rj1s-p1+Rj2s-p2+⋯Rjks-pkIt can be analyzed from (6) and (12) that if the frequency of Tx is equal to the oscillation frequency of the system, only the torque with the amplitude close to zero can stimulate the large response of the oscillation frequency. Assume that Tx can increase the oscillation frequency to λω, if different generators are selected as the “excitation point”, then Gλiω can be expressed as(15)Gλiω=ΔδiλiωΔTxiλiω=Rj1λiω-p1+Rj2λiω-p2+⋯Rjkλiω-pkAlthough the “excitation point” varies, the system interarea oscillation mode keeps the same, where a pair of conjugate poles represents an oscillation mode. Letλi=σi±jωi, then it can be used to calculate the system oscillation frequency and damping ratio.Once the oscillation mode is determined, the index calculation of the candidate signal should be carried out, when it is necessary to transform the system into the state space form andGs can be written as(16)Gs=YsUs=bmsm+bm-1sm-1+⋯+b0sn+an-1sn-1+⋯+a0In this regard, the state space expression of the system can be derived as(17)Δx˙=EΔx+FΔuΔy=GΔx+HΔuwhere E=0In-1A0An-1 is the equivalent matrix of matrix A in (3), which contains the main mode of the original system. F=0⋯1T and I are the corresponding control matrix and substantial matrix, respectively. G=Bn-Anbn, H=bn. Equation (17) is the state space model derived by the frequency domain self-exited method.The state space equation identified by the frequency domain excitation method preserves the main mode of operation of the system. Therefore, (17) can be used to substitute the original system to calculate the quantization index of the observability and controllability of the system.The new method of selecting feedback signal of wide-area damping controller based on the self-excited frequency domain identification method proposed in this paper can be used in engineering application of large-scale power system without obtaining the full-state model of the system, and the calculation and operation are convenient, which is suitable for the engineering application of large power system.


## 3.3. Wide-Area Coordinated Controller Design

Once the state space equation is identified by the self-excited frequency domain identification, by carrying out the Laplace transform on (13), the transfer function between the input variable ΔU1 and the output variable ΔY1 can be obtained as(18)Gs=ΔU1sΔY1s=GsI-E-1F+HSuppose dominant oscillation modes for (18) are λ1,2=δ±jω, and the desired damping ratio after adding damping controller is ξ. Substituting s=jωd=j1-ξ2ω into (18) derives(19)Gjωd=ΔU1sΔY1ss=jωd=G∠ϕwhere G and ϕ are the amplitude and phase of G(s) at s=jωd, respectively.The lead lag link parameters of the wide-area DC damping controller (2) can be calculated by(20)ϕx=180-ϕα=1+sin⁡ϕx1-sin⁡ϕxT1=αωT2=T1αIn order to prevent the damping controller from outputting a constant deviation voltage, the design principle is that, in the frequency range where damping controllers provide damping, no obvious phase displacement should be introduced in the isolating links. The typical value is chosen as 1~10s, and in this study, it is set to be 5s.The advantage of the above wide-area DC damping controller parameter design based on the self-excited method is that it does not rely on the detailed model of the system, the design process is simple, the derived parameters are stable, and it is applicable to utilize in the practical complex large-scale power grid.


## 3.4. Wide-Area Damping Control Design Process

The basic steps of the selection of the feedback signal and the installation location of the wide-area damping controller based on the self-excited frequency domain identification method are as follows:Step 1.
Identify the low-order system based on the frequency domain self-excited method, and calculate the corresponding interarea oscillation modesλ1,2=δ±jω and the oscillation frequency ω.Step 2.
Select different feedback signals and installation locations of then groups of controllers, and then the system is able to be operated under the steady state with the open loop operation of the controller. The disturbance test for groupi (0<i<n) is carried out, and for the disturbance ui(t) that does not destroy the stability of the original system, record the output response data yi(t).Step 3.
Extract the variation dataΔui(t) and Δyi(t) of ui(t) and yi(t) in the common cycle. Carry out the discrete Fourier transform, respectively, and obtain the varying phasors Δui(jω) and Δyi(jω) corresponding to different frequency values, where i=1,2,…,n,0≤jω≤2π.Step 4.
The self-excited method is applied to identify the phasor dataΔui(jω) and Δyi(jω), and the corresponding transfer function Gi(s) is fitted. Transform the transfer function into the state space form and obtain the corresponding system matrices Ai, Bi, Ci, Di.Step 5.
Apply the geometric measure index method to calculate each index, which is used as the reference for the input signal of the comparison controller. Based on the above method, that best feedback signal and installation location are determined.Step 6.
Based on the state space equation and the dominant modes, identify the information such as the phaseΦ and the oscillation mode ω that are required by controller design. Calculate the controller time constants Ti,i=1,2,…,4 by (20).Step 7.
Adjust the controller gainKi to verify the damping effect, and the operation parameters of the wide-area DC damping controller are finally determined.


## 4. Case Study

Taking the two-area four-generator AC/DC hybrid system shown in Figure3 as an example, the detailed model is built with MATLAB and verified by simulation. The specific parameters are taken from literature [18]. The wide-area damping controller is installed in the rectifier of the DC part. The optional input signals of the wide-area damping controller are the rotor speed difference of generator 1 (G1) and generator 3 (G3), the active power of the AC tie-line, the AC line voltage, AC line current amplitude, and difference of voltage phase between two sides of an AC contact line. By modifying the model parameters, the powers of transmission line between two regions are chosen as P = 150MW, P = 200MW, and P = 300MW, and thus three kinds of operation modes are derived (the positive direction of power flow is defined as from Area 1 to Area 2).Figure 3
Two-area four-generator AC/DC hybrid system.According to the procedure described in the previous section, the interarea oscillation modes are analyzed for the transmission power P = 150MW, P = 200MW, and P = 300MW, respectively, which are shown in Table1.Table 1
Interarea oscillation frequency and damping ratio without control.
Operation mode eigenvalue frequency Damping ratio P=150MW -0.1462+3.2683i 0.5202 0.0447 P=200MW -0.1301+3.8321i 0.6099 0.0339 P=300MW -0.2942+3.5092i 0.5585 0.0835By carrying out the wide-area damping controller design steps proposed in Section3.4, the frequency domain self-excited method and the improved geometric lateral index are used to calculate the comprehensive and residue indices for different schemes. The results are shown in Tables 2–4.Table 2
System indices for P=150MW.
Alternative input signal Residue index geometric laterality index frequency Damping ratio Compensation phase the rotor speed difference of G1 and G3 0.2711 0.9409 0.5536 0.0870 93.7  the active power of the AC line 2.0689e-06 0.7241 0.5700 0.0533 147.4  AC line current amplitude 0.1376 0.8042 0.556 0.0773 16.5  difference of voltage phase between two sides of AC line 1.6235e-06 0.7171 0.567 0.0395 54.6Table 3
System indices for P=200MW.
Alternative input signal Residue index geometric laterality index frequency Damping ratio Compensation phase the rotor speed difference of G1 and G3 0.3568 0.8951 0.6587 0.0473 97.2  the active power of the AC line 1.2302e-06i 0.7501 0.6331 0.0035 115.5  AC line current amplitude 0.0117 0.7276 0.6245 0.0172 31.4  difference of voltage phase between two sides of AC line 1.4371e-08 0.7330 0.5891 0.0063 29.8Table 4
System indices for P=300MW.
Alternative input signal Residue index geometric laterality index frequency Damping ratio Compensation phase the rotor speed difference of G1 and G3 0.2967 0.9554 0.5572 0.0566 92.6  the active power of the AC line 6.3309e-06 0.8236 0.5401 0.0670 115.6  AC line current amplitude 0.1087 0.7183 0.5431 0.0070 55.1  difference of voltage phase between two sides of AC line 6.0653e-08 0.7496 0.5587 0.0636 165.8As shown in Tables2–4, the best candidate signals among the three operating modes are the rotor speed difference between G1 and G3 by comparing the comprehensive and residue indices.In order to verify the validity of the analysis results, the wide-area damping control effect of different feedback signals is simulated by choosing the speed difference between G1 and G3 as the wide-area damping controller feedback signal, which is installed in the DC side of the main control loop. A three-phase short circuit occurred on the AC transmission line at t=10.4s, which is cleared at 10.9s. By carrying out the phase compensation method, parameters of the wide-area controller (2) can be calculated as(21)m=3,K=30,T1=0.02,TW=10,α=0.3218,T2=0.5035,T3=0.1620where the operating mode corresponding to P=300MW is selected as the compensation phase of the controller parameter calculation, and let θ=92.6°.From Figures4–6, it is observed that there are obvious differences in the control effect under the location of installation being selected. Considering the residue method, for the rotational speed differences of G1 and G3 as well as the amplitude of the AC link current (Figure 4), the AC tie-line powerPand the voltage phase difference of the tie-line (Figure 5), these two types of signals with different amplitudes are displayed as the larger the number of indicators, the better the control effect. However, since the magnitudes of the two types of signal indicators are significantly different, it is impossible to compare them at the same time. In this case, the improving geometric lateral index has a greater advantage, as it can compare different types of signals under the same order of magnitude. As shown in Figure 6, as the analysis results in Tables 2–4, the larger the index value the better the control effect, which verifies the effectiveness of the method in this paper.Figure 4
Comparison of power oscillation.Figure 5
Comparison of power oscillation.Figure 6
Active power oscillation of the tie-line.For the system under wide-area damping control, the operating modes of the interarea oscillation frequency and damping ratio are calculated based on the frequency domain self-excited method as shown in Table5.Table 5
Interarea oscillation frequency and damping ratio for system under control.
Operation mode eigenvalue frequency Damping ratio P=150MW -0.6208+3.5014i 0.5573 0.1746 P=200MW -0.4446+3.9169i 0.6234 0.1128 P=300MW -0.4082+3.6602i 0.5825 0.1108It can be seen from Table5 that, for system under control, the damping for each operating mode has been significantly increased, indicating that the system damping is improved under the designed controller. Simulation results of the AC tie-line transmission power with and without the designed damping controller are depicted in Figure 7. The above simulation results have verified the effectiveness and applicability of the proposed wide-area damping controller.Figure 7
Transmission power comparison of the AC tie-line.


## 5. Conclusion

In this study, a wide-area damping controller design method for AC/DC hybrid power system is proposed. The input signal of the wide-area damping controller is selected by the improved geometric laterality index, which can be chosen as different types of signals to increase the range of the optional signal to improve the controller design effect. The system model can be identified through the frequency domain self-excitation method, based on which the phase compensation method is further utilized to design the DC damping controller parameters. The proposed damping controller avoids the complex eigenvalue calculation in large-scale power systems and is not affected by the increase of system order. Moreover, the detailed system modeling is avoided, which may lead to inaccuracy and errors. The design process is simple, which is suitable for practical engineering applications.



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*Source: 2738362-2018-09-13.xml*