# Suppression to Angular Oscillation among Synchronous Generators by Optimizing Parameters and Set-Points of Synchronous Condenser and High-Voltage DC

Authors: Shenghu Li; Diwen Tao; Lulu Li; Nan Qi; Junwei Gong
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

After losing one or two poles of the high-voltage DC (HVDC) line, power systems may experience angular oscillations among the synchronous generators (SGs), possibly followed by tripping of the SGs and blackouts of the loads. To suppress the oscillations, the difficulty lies in the optimization methods and the countermeasures. For the former, parameter optimization based on the eigen-analysis for a given scenario is not suitable for angular oscillation after a large disturbance lasting for several seconds or longer. For the latter, adjusting the active power of the SGs is effective but responds slowly. Adjusting the excitation system of the synchronous condensers (SCs) is seldom studied since the relation of the power angle with the reactive power is not clearly described. Adjusting the control parameters of the remaining HVDC has limited effect and is not enough. The set-points for the controller of the HVDC are adjusted to suppress the oscillation but are to be restored after the oscillation, which is different from optimization of the control parameters. In this paper, coordinated optimization of the control parameters and the set-points of the SC and the HVDC is proposed to suppress the angular oscillations among the SGs. At first, the relation of the reactive power of the SC with the power angles is validated to provide the basis for optimizing the control parameters and the set-points of the SC. Since the analytical expression between the power angle and the control parameters or the set-points related to the reactive power is not explicit and the optimization periods of different parameters are different, an improved analytical model of the trajectory sensitivity (TS) is proposed. Based on the indication indices, the control parameters and the set-points of the constant current controller and the excitation system to be optimized are decided. The gradients of the objective function are extended to derive a coordinated multiparameter optimization model to suppress the angular oscillation among the SGs. The numerical results validate the accuracy of the improved TS by comparing it with the perturbation method. The time-domain analysis in different scenarios shows that the angular oscillations in the AC/HVDC system are effectively suppressed with the proposed optimization model.


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## Body

## 1. Introduction

The angular oscillation among the synchronous generators (SGs) after large disturbances endangers the security of the power systems. To suppress the oscillations, one may introduce the supplementary controllers, e.g., the power system stabilizer (PSS) of the SGs or the power oscillation damper of the doubly-fed induction generators (DFIGs) [1, 2], or adjust the output of the DFIGs or the energy storage [3, 4]. In [5], the parameters of the PSS and the static var compensator are optimized in coordination with the particle swarm optimization (PSO) to suppress the oscillations. In [6], the tuning scheme of the PSS is proposed to suppress local and interarea mode oscillations with the weight function of the angular velocity deviation and the damping time. With the high-voltage DC (HVDC) widely applied [7], the oscillation in the AC/HVDC power system is more complex and needs to be suppressed [8, 9]. One may adjust the output of the SGs or the doubly-fed induction generators (DFIGs), curtail the loads, introduce the supplementary control, or optimize the parameters of the controllers. The latter is more economical and may be applied to the HVDC [10, 11] and the AC equipment [12, 13]. The energy storage system is applied to suppress the oscillation based on the model predictive control [14]. The impact of the control parameters of the DFIG on the oscillation is verified by the eigen-analysis [15, 16].As seen in [17], increasing the reactive power of the synchronous generator (SC) may suppress the oscillation, but the existing studies focus on optimization to the active power and ignore the effect of the reactive power on the oscillation. In [18], the improved active/reactive power control of DFIG is proposed to damp the interarea low-frequency oscillation. In [19], the optimization model of the DC power and the capacitors is proposed by simplifying the nonlinear control model into the quadratic programming model. The suppression schemes are independent, without coordinating the HVDC and other equipment. Tuning parameters is often for the control parameters, while little attention is given to the set-points. In [20], power system stability is investigated by adjusting the set-points of the HVDC in the small-signal stability analysis. The set-points are optimized to enhance the dynamic response during and after the fault with automatic adjustment [21, 22].To suppress the angular oscillation in the AC/HVDC power system, some difficulties are to be solved.(1)
How to coordinate the parameters of both the HVDC and the AC equipment. The manual adjustment is not practical for multiple parameters, hence coordinated optimization is required. But many optimization models use the heuristic methods, which are easy to implement but are time-consuming and may not achieve global optimization. An analytical optimization model is needed.(2)
How to quantify the relation between the power angles of the SGs and the parameters and decide which parameters should be optimized. The eigen-sensitivity model [23, 24] based on the linearized system model at a given operating point is not suitable for large disturbances, followed by a dynamic process lasting for several seconds. The trajectory sensitivity (TS) quantifies the relationship between the dynamic response and faults or large disturbances [25, 26]. In [27], the optimization to the PSS is proposed with the trajectory sensitivity and the conjugate gradient method. In [28], the control parameters of the DFIG are optimized with the PSO and the trajectory sensitivity. It may be derived by the perturbation method or the analytical method [29, 30]. The former is simpler but requires more calculations. The latter is computationally more efficient for adjusting multiple parameters. It is difficult to derive the analytical expression of the TS from the parameters related to the active and reactive powers since the latter may be shown in the power flow equations instead of the dynamic modeling [31].(3)
How to optimization to the control parameters and the set-points together. The control parameters, such as PI parameters and time constants, affect the dynamic process but have no impact during the steady state. They are often adjusted before the transient process. The set-points of the voltage or current will change both power flow and the postfault dynamic response [32]. For example, temporarily changing the reference of the constant current control (CCC) of the nonfault pole helps to suppress oscillation [33]. Adjusting the set-points of voltage and power helps to delay the critical clearing time and maintain system stability [34]. However, it is difficult to derive the TS-based optimization model for the control parameters and the set-points, because they have different periods of optimization.To suppress the angular oscillation among the SGs, this paper proposes a coordinated optimization model for the HVDC and the SC based on the trajectory sensitivity. Since the relation between the reactive power and the power angle is not explicit and the optimization periods of different parameters are different, an analytical model of the TS is improved. Considering the impact on the power angle, the indication indices are calculated based on the mean TS and the control parameters, and the set-points of the CCC of the HVDC and the excitation system of the SC to be optimized are determined. The gradients of the objective function are extended to derive a coordinated multiparameter optimization model. The numerical results validate the accuracy and the suppression effect of the proposed models.The prime contributions of this work are summarized as follows:(1)
A new coordinated optimization to the HVDC and the SC is proposed to suppress the angular oscillations.(2)
An improved analytical expression of the trajectory sensitivity is newly derived and can be used for the optimization of multiple parameters with different periods of optimization.(3)
The applicability of the proposed optimization for different fault scenarios is verified by introducing the index of the importance degree of the node.(4)
The coordinated optimization can reduce the adjustment range of set-points for HVDC, which is safer for the HVDC than the independent optimization without the SC.The following of this paper is arranged as follows. In Section2, the relation of the reactive power of the SC with the power angles is validated, which is the basis for optimizing the control parameters and the set-points of the SC. In Section 3, an improved TS model including the relation between the control parameters and the set-points of the SC and the power angles of the SGs and allowing different optimization periods for the parameters is proposed. In Section 4, an optimization model based on coordinating the control parameters and the set points of the HVDC and the SC is proposed to suppress the oscillations in the AC/HVDC power system. In Section 5, the case studies and analysis are given to validate the accuracy of the improved trajectory sensitivity and the suppression effect with the optimization model. In Section 6, some conclusions are given.


## 2. Impact of the Reactive Power on the Power Angle Stability

### 2.1. Effect of the Reactive Power on the Voltage and the Active Power

The power flow of the AC/HVDC system is solved with (1) iteratively, where P = [PAC, PAC-DC], Q = [QAC, QAC-DC], θ = [θAC, θAC-DC], V = [VAC, VAC-DC], P is the active power, Q is the reactive power, V is the voltage magnitude, and θ is the voltage angle. H, N, M, and L are the sub-Jacobian matrixes. Δ denotes the increment.(1)ΔPΔQ=HNMLΔθΔV.By eliminating the first row, the sensitivity of the voltage with respect to the reactive power is obtained in the following equation. Taking the same increment in the reactive power, the relative voltage sensitivitySRV is given by (3).(2)∂V∂Q=L-MH−1N−1,(3)SRVi,j=ΔViΔVj=∂Vi/∂Qi∂Vj/∂Qj.After a fault in the system, the voltage change of the short-circuit bus ΔVFau, is used as the reference to calculate the voltage of the other buses based on the SRVi,Fau.(4)ΔVi=ΔVFau×SRVi,Fau.The transmission power of the HVDC is reduced and the reactive power of the AC buses is changed after the DC blocking. Considering the change of the bus voltage and the power, (5) is derived from (1). The bus voltage and the active power jointly affect the reactive power. When the reactive power support is not enough, the voltage may be deteriorated and the power flow may be limited.(5)ΔQ=MH−1ΔP+L−MH−1NΔV.


### 2.2. Effect of the Reactive Power on the Power Angle

To analyze the effect of the reactive power output of the SC (QSC) on the power angle, (1) is extended by taking the node of SC as the PQ node to obtain the sensitivities of the voltage magnitude and angle of generator bus (VG and θG) to QSC, which are used as an intermediate quantity to indirectly obtain the sensitivity of the power angle to QSC. As is known, the power angle can be defined by the following equation:(6)δ=arctanVGy+XqIGx+RsIGyVGy+RsIGx−XqIGy,where I is the current, R is the resistance, and X is the reactance. The subscript G denotes SG, x and y denote axis x and axis y, q denotes quadrature axis, and s denotes the stator.Considering thatQSC is not included in (6), the derivative method for the compound function is used to derive the sensitivities of VGx, VGy, IGx, and IGyto QSC. VGx and VGy are decided by VG and θG, so the sensitivities of the former to QSC may be derived from the sensitivities of the latter to QSC in (7). The sensitivities of the bus injection current, IGx and IGy, to QSC are given in (8), where Y is the admittance matrix.(7)∂VGx∂QSC=cosθG∂VG∂QSC+VG∂cosθG∂QSC,∂VGy∂QSC=sinθG∂VG∂QSC+VG∂sinθG∂QSC,(8)∂IGx∂QSC∂IGy∂QSC=∂VGx∂QSC−∂VGy∂QSC∂VGy∂QSC∂VGx∂QSCReYImY+∂ReY∂QSC−∂ImY∂QSC∂ImY∂QSC∂ReY∂QSCVGxVGy.With the partial derivative of (6) to QSC, the sensitivity of the power angle to QSC is obtained in the following equation:(9)∂δ∂QSC=∂δ∂VGx∂VGx∂QSC+∂δ∂VGy∂VGy∂QSC+∂δ∂IGx∂IGx∂QSC+∂δ∂IGy∂IGy∂QSC.By substituting (1), (5), (7), and (8) into (9), it is found that the active and reactive power interacts with the bus voltage, which together affects the power angle. The relation of the reactive power of the SC and the power angle is validated and the coordinated optimization to the HVDC and the SC may be introduced to improve the power angular oscillations.


## 2.1. Effect of the Reactive Power on the Voltage and the Active Power

The power flow of the AC/HVDC system is solved with (1) iteratively, where P = [PAC, PAC-DC], Q = [QAC, QAC-DC], θ = [θAC, θAC-DC], V = [VAC, VAC-DC], P is the active power, Q is the reactive power, V is the voltage magnitude, and θ is the voltage angle. H, N, M, and L are the sub-Jacobian matrixes. Δ denotes the increment.(1)ΔPΔQ=HNMLΔθΔV.By eliminating the first row, the sensitivity of the voltage with respect to the reactive power is obtained in the following equation. Taking the same increment in the reactive power, the relative voltage sensitivitySRV is given by (3).(2)∂V∂Q=L-MH−1N−1,(3)SRVi,j=ΔViΔVj=∂Vi/∂Qi∂Vj/∂Qj.After a fault in the system, the voltage change of the short-circuit bus ΔVFau, is used as the reference to calculate the voltage of the other buses based on the SRVi,Fau.(4)ΔVi=ΔVFau×SRVi,Fau.The transmission power of the HVDC is reduced and the reactive power of the AC buses is changed after the DC blocking. Considering the change of the bus voltage and the power, (5) is derived from (1). The bus voltage and the active power jointly affect the reactive power. When the reactive power support is not enough, the voltage may be deteriorated and the power flow may be limited.(5)ΔQ=MH−1ΔP+L−MH−1NΔV.


## 2.2. Effect of the Reactive Power on the Power Angle

To analyze the effect of the reactive power output of the SC (QSC) on the power angle, (1) is extended by taking the node of SC as the PQ node to obtain the sensitivities of the voltage magnitude and angle of generator bus (VG and θG) to QSC, which are used as an intermediate quantity to indirectly obtain the sensitivity of the power angle to QSC. As is known, the power angle can be defined by the following equation:(6)δ=arctanVGy+XqIGx+RsIGyVGy+RsIGx−XqIGy,where I is the current, R is the resistance, and X is the reactance. The subscript G denotes SG, x and y denote axis x and axis y, q denotes quadrature axis, and s denotes the stator.Considering thatQSC is not included in (6), the derivative method for the compound function is used to derive the sensitivities of VGx, VGy, IGx, and IGyto QSC. VGx and VGy are decided by VG and θG, so the sensitivities of the former to QSC may be derived from the sensitivities of the latter to QSC in (7). The sensitivities of the bus injection current, IGx and IGy, to QSC are given in (8), where Y is the admittance matrix.(7)∂VGx∂QSC=cosθG∂VG∂QSC+VG∂cosθG∂QSC,∂VGy∂QSC=sinθG∂VG∂QSC+VG∂sinθG∂QSC,(8)∂IGx∂QSC∂IGy∂QSC=∂VGx∂QSC−∂VGy∂QSC∂VGy∂QSC∂VGx∂QSCReYImY+∂ReY∂QSC−∂ImY∂QSC∂ImY∂QSC∂ReY∂QSCVGxVGy.With the partial derivative of (6) to QSC, the sensitivity of the power angle to QSC is obtained in the following equation:(9)∂δ∂QSC=∂δ∂VGx∂VGx∂QSC+∂δ∂VGy∂VGy∂QSC+∂δ∂IGx∂IGx∂QSC+∂δ∂IGy∂IGy∂QSC.By substituting (1), (5), (7), and (8) into (9), it is found that the active and reactive power interacts with the bus voltage, which together affects the power angle. The relation of the reactive power of the SC and the power angle is validated and the coordinated optimization to the HVDC and the SC may be introduced to improve the power angular oscillations.


## 3. The Improved Trajectory Sensitivity considering Different Periods of Optimization

### 3.1. Dynamic Modeling of the AC/HVDC Power System

In this paper, the SGs adopt the 12th order model, including the electromagnetic equation (the 3rd order), the motion equation (the 2nd order), the governor (the 3rd order), and the excitation system (the 4th order). The model of the SC is based on [35], and it includes the electromagnetic equation (the 3rd order), the motion equation (the 2nd order), and the excitation system (the 4th order), and the excitation system (the 4th order). Figure 1 shows the model of the SC, where KA and TA are the gain and the time constant of the main regulator, KF and TF are the gain and the time constant of the rate feedback, Vref is the exciter reference voltage, Vf0 is the initial value of the output voltage of the exciter, and VM, VA, VE, and VF are the voltage outputs of the measurement, the main regulator, the exciter, and the rate feedback, respectively. The model of the HVDC is described in [36], where the CCC is applied to the rectifier and the constant extinction angle (CEA) is applied to the inverter [37]. Figure 2 shows the model of the HVDC, where kpr and kir are PI parameters of the CCC, IDC,ref, and αref are the references for the direct current and the firing-delayed angle.Figure 1
Model of SC.Figure 2
HVDC transmission system.


### 3.2. Improved Analytical Expression of TS in the AC/HVDC Power System

To solve the problem that the trajectory sensitivities of system variables are often found with respect to control parameters instead of set-points without considering different periods of optimization, the improved TS is newly derived as follows. The flowchart of the adjustment of control parameters and set-points is shown in Figure3, where the fault occurs at t1, the control parameters are adjusted to the optimal values at the beginning of the time-domain simulation, and the set-points are adjusted at t2 and restored to the initial valves at t3.Figure 3
Adjustment of control parameters and set-points with different periods.The time-domain simulation is based on the differential-algebraic equations (DAE) of the AC/HVDC power system in (10), where f, g, x, and y are the differential equations, algebraic equations, state variables, and algebraic variables, respectively, a is the system parameter and t is the time. The subscript 0 denotes the initial value, cp denotes the control parameter, and sp denotes the set-point.(10)x˙=fxt,yt,acp,asp,0=gxt,yt,acp,asp,xt0=x0,yt0=y0.acp and asp are taken as variables during their respective optimization periods and constants during the rest of the time. Because of different periods of optimization to acp and asp, the partial derivative equations of (10) to acp are different from that to asp. Compared with (11), the step function ε is introduced in (12) to differentiate periods of optimization.(11)x˙acp=fxxacp+fyyacp+facp,0=gxxacp+gyyacp+gacp,(12)x˙asp=fxxasp+fyyasp+faspεt−t2−εt−t3,0=gxxasp+gyyasp+gaspεt−t2−εt−t3,where fx, fy, facp, fasp, gx,gy,gacp,andgasp are the time-varying matrices for the partial derivative.Considering that multiple state variables are coupled in (11) and (12), the intermediate variables are introduced to decouple state variables. The decoupling process for the SC is given as an example. The intermediate variables ϕITM1 and ϕITM2 are defined as follows:(13)ϕITM1=Td0″Eq″−Eq′,ϕITM2=KFVE−TFVF,where Td0″ is the d-axis subtransient open-circuit time constant, Eq″ is the q-axis subtransient electromotive force, Eq′ is the q-axis transient electromotive force.The partial derivative equations of (13) with respect to a parameter to be optimized α are given in the following equation:(14)0=ϕITM1,α−Td0″Eqα″−Eqα′,0=ϕITM2,α−KFVEα+TFVFα,combining (13), and the electromagnetic equation and the excitation system model of the SC, the newly derivative equations are given in (15). With the help of the intermediate variables, each row has only one state variable, i.e., the differential equations and the state variables are decoupled.(15)dϕITM1dt=Eq′−Eq″−Xd′−Xd″Id,dϕITM2dt=VF,where Xd′ is the d-axis transient reactance, Xd″ is the d-axis subtransient reactance, and Id is the d-axis current component.All state variables of the system, including the added intermediate variables, are rowed into the diagonal matrix, the trajectory sensitivities are decoupled, and the Jacobi matrices of state variables and algebraic variables are in the upper and lower parts of the matrix, respectively. By applying the trapezoidal differentiation method for (14) and (15) with the intermediate variables, the analytical expressions of the trajectory sensitivity are newly given in (16) and (17), where U is the unit matrix, and the superscripts t and t − 1 denote the current and last moment, respectively. The right-most term is the imbalanced matrix. Most of its elements are 0, except for those with acp or asp.(16)xacptyacpt=2UΔt−fxt−fxtgytgyt−1×2UΔt+fxt−1fyt−100×xacpt−1yacpt−1+−facpt−gacptImbalabcedmatrix,(17)xasptyaspt=2UΔt−fxt−fytgxtgyt−1×2UΔt+fxt−1fyt−100×xaspt−1yaspt−1+−fasptεt−t2−εt−t3−gasptεt−t2−εt−t3Imbalancedmatrix.By modifying the imbalanced matrix and the step function in (17), trajectory sensitivities of system variables to set-points are calculated during the optimization period. After set-points are restored to the initial value, elements of the imbalanced matrix are 0, while those of the TS are not 0 and the subsequent effect of the optimization is still quantified. Some nonzero elements in the imbalanced matrix are given in (18) and (19).The nonzero elementse corresponding to the control parameters of the SC and the CCC are given in the following equation:(18)eKA=−0.5VMt+VFt+VMt−1+VFt−1−VrefTA,eTA=0.5VAt+VAt−1+KAVMt+VFt+VMt−1+VFt−1+VrefTA2,eKF=VEt,eTF=−VFt,eKpr=−IDC,ref−IDCt,eKir=−0.5IDCt+IDCt−1−IDC,ref.The nonzero elements corresponding to the set-points are given in the following equation:(19)eVref=KAεt−t2−εt−t3TA,eVf0=−εt−t2−εt−t3,eαref=−εt−t2−εt−t3,eIDC,ref=Kirεt−t2−εt−t3.It is worth noting that the matricesfx, fy, gx, and gy in (16) and (17) are the same as those in (10). Consequently, (16) and (17) are easy to be built up after one simulation just with modifying the imbalanced matrices, which are computationally more economical than the perturbation method. The expressions of fx and fy are given in (20) and (21). gx and gy are derived similarly.(20)fx=fSGxSGfSGxSCfSGxHVDCfSCxSGfSCxSCfSCxHVDCfHVDCxSGfHVDCxSCfHVDCxHVDC,(21)fy=fSGySGfSGySCfSGyHVDCfSCySGfSCySCfSCyHVDCfHVDCySGfHVDCySCfHVDCyHVDC.


## 3.1. Dynamic Modeling of the AC/HVDC Power System

In this paper, the SGs adopt the 12th order model, including the electromagnetic equation (the 3rd order), the motion equation (the 2nd order), the governor (the 3rd order), and the excitation system (the 4th order). The model of the SC is based on [35], and it includes the electromagnetic equation (the 3rd order), the motion equation (the 2nd order), and the excitation system (the 4th order), and the excitation system (the 4th order). Figure 1 shows the model of the SC, where KA and TA are the gain and the time constant of the main regulator, KF and TF are the gain and the time constant of the rate feedback, Vref is the exciter reference voltage, Vf0 is the initial value of the output voltage of the exciter, and VM, VA, VE, and VF are the voltage outputs of the measurement, the main regulator, the exciter, and the rate feedback, respectively. The model of the HVDC is described in [36], where the CCC is applied to the rectifier and the constant extinction angle (CEA) is applied to the inverter [37]. Figure 2 shows the model of the HVDC, where kpr and kir are PI parameters of the CCC, IDC,ref, and αref are the references for the direct current and the firing-delayed angle.Figure 1
Model of SC.Figure 2
HVDC transmission system.


## 3.2. Improved Analytical Expression of TS in the AC/HVDC Power System

To solve the problem that the trajectory sensitivities of system variables are often found with respect to control parameters instead of set-points without considering different periods of optimization, the improved TS is newly derived as follows. The flowchart of the adjustment of control parameters and set-points is shown in Figure3, where the fault occurs at t1, the control parameters are adjusted to the optimal values at the beginning of the time-domain simulation, and the set-points are adjusted at t2 and restored to the initial valves at t3.Figure 3
Adjustment of control parameters and set-points with different periods.The time-domain simulation is based on the differential-algebraic equations (DAE) of the AC/HVDC power system in (10), where f, g, x, and y are the differential equations, algebraic equations, state variables, and algebraic variables, respectively, a is the system parameter and t is the time. The subscript 0 denotes the initial value, cp denotes the control parameter, and sp denotes the set-point.(10)x˙=fxt,yt,acp,asp,0=gxt,yt,acp,asp,xt0=x0,yt0=y0.acp and asp are taken as variables during their respective optimization periods and constants during the rest of the time. Because of different periods of optimization to acp and asp, the partial derivative equations of (10) to acp are different from that to asp. Compared with (11), the step function ε is introduced in (12) to differentiate periods of optimization.(11)x˙acp=fxxacp+fyyacp+facp,0=gxxacp+gyyacp+gacp,(12)x˙asp=fxxasp+fyyasp+faspεt−t2−εt−t3,0=gxxasp+gyyasp+gaspεt−t2−εt−t3,where fx, fy, facp, fasp, gx,gy,gacp,andgasp are the time-varying matrices for the partial derivative.Considering that multiple state variables are coupled in (11) and (12), the intermediate variables are introduced to decouple state variables. The decoupling process for the SC is given as an example. The intermediate variables ϕITM1 and ϕITM2 are defined as follows:(13)ϕITM1=Td0″Eq″−Eq′,ϕITM2=KFVE−TFVF,where Td0″ is the d-axis subtransient open-circuit time constant, Eq″ is the q-axis subtransient electromotive force, Eq′ is the q-axis transient electromotive force.The partial derivative equations of (13) with respect to a parameter to be optimized α are given in the following equation:(14)0=ϕITM1,α−Td0″Eqα″−Eqα′,0=ϕITM2,α−KFVEα+TFVFα,combining (13), and the electromagnetic equation and the excitation system model of the SC, the newly derivative equations are given in (15). With the help of the intermediate variables, each row has only one state variable, i.e., the differential equations and the state variables are decoupled.(15)dϕITM1dt=Eq′−Eq″−Xd′−Xd″Id,dϕITM2dt=VF,where Xd′ is the d-axis transient reactance, Xd″ is the d-axis subtransient reactance, and Id is the d-axis current component.All state variables of the system, including the added intermediate variables, are rowed into the diagonal matrix, the trajectory sensitivities are decoupled, and the Jacobi matrices of state variables and algebraic variables are in the upper and lower parts of the matrix, respectively. By applying the trapezoidal differentiation method for (14) and (15) with the intermediate variables, the analytical expressions of the trajectory sensitivity are newly given in (16) and (17), where U is the unit matrix, and the superscripts t and t − 1 denote the current and last moment, respectively. The right-most term is the imbalanced matrix. Most of its elements are 0, except for those with acp or asp.(16)xacptyacpt=2UΔt−fxt−fxtgytgyt−1×2UΔt+fxt−1fyt−100×xacpt−1yacpt−1+−facpt−gacptImbalabcedmatrix,(17)xasptyaspt=2UΔt−fxt−fytgxtgyt−1×2UΔt+fxt−1fyt−100×xaspt−1yaspt−1+−fasptεt−t2−εt−t3−gasptεt−t2−εt−t3Imbalancedmatrix.By modifying the imbalanced matrix and the step function in (17), trajectory sensitivities of system variables to set-points are calculated during the optimization period. After set-points are restored to the initial value, elements of the imbalanced matrix are 0, while those of the TS are not 0 and the subsequent effect of the optimization is still quantified. Some nonzero elements in the imbalanced matrix are given in (18) and (19).The nonzero elementse corresponding to the control parameters of the SC and the CCC are given in the following equation:(18)eKA=−0.5VMt+VFt+VMt−1+VFt−1−VrefTA,eTA=0.5VAt+VAt−1+KAVMt+VFt+VMt−1+VFt−1+VrefTA2,eKF=VEt,eTF=−VFt,eKpr=−IDC,ref−IDCt,eKir=−0.5IDCt+IDCt−1−IDC,ref.The nonzero elements corresponding to the set-points are given in the following equation:(19)eVref=KAεt−t2−εt−t3TA,eVf0=−εt−t2−εt−t3,eαref=−εt−t2−εt−t3,eIDC,ref=Kirεt−t2−εt−t3.It is worth noting that the matricesfx, fy, gx, and gy in (16) and (17) are the same as those in (10). Consequently, (16) and (17) are easy to be built up after one simulation just with modifying the imbalanced matrices, which are computationally more economical than the perturbation method. The expressions of fx and fy are given in (20) and (21). gx and gy are derived similarly.(20)fx=fSGxSGfSGxSCfSGxHVDCfSCxSGfSCxSCfSCxHVDCfHVDCxSGfHVDCxSCfHVDCxHVDC,(21)fy=fSGySGfSGySCfSGyHVDCfSCySGfSCySCfSCyHVDCfHVDCySGfHVDCySCfHVDCyHVDC.


## 4. Coordinate Optimization Model and Solution Algorithm

### 4.1. Selecting the Parameters to be Optimized Based on the Sensitivity Evaluation Index

The parameters to be optimized are selected by the evaluation index of the TS. To eliminate the difference in the range and the dimension of parameters, the mean TS is introduced [28], which reflects the influence degree of parameters on the dynamic process.(22)Smean=1NUMsamp∑i=1NUMsamp∂pα,i/p0∂α/α0,where Smean is the value of the mean TS, p is one of system variables, and NUMsamp is the sampling number on the TS curves.


### 4.2. The Optimization Model

Taking the power angle of the #N SG as the reference, the objective function J is established in (23) by calculating the difference-squared integral of the power angle difference in the transient process and the steady-state power angle difference after the fault.(23)minJ=∑j=1j≠NNUMSG∫0tendδj,Nt−δj,Ntendδj,Ntend2dt,(24)δj,Nt=δjt−δNt,δj,Ntend=δjtend−δNtend,where NUMSG is the number of SGs, and tend is the finish time of the simulation, δj is the power angle of the #j SG, and δj,N is the power angle difference between the #j and the #N SGs.Considering equality and inequality constraints, the optimization model is proposed in (25), where amin and amax are the upper and lower limits of parameters.(25)minJ,s.t.x˙=fxt,yt,a,0=gxt,yt,a,a=acpasp,amin≤a≤amax.


### 4.3. Multiparameter Coordinated Optimization considering the Optimization Period

Considering the nonlinearity of the model in (25) and the advantage of the direct optimization, the interior point method is introduced, which requires the first-order and second-order gradient information of the objective function. By accumulating the TS of the objective function J to a, the first-order gradient is obtained by (26), where NUMste is the number of the simulation steps and Sδjtm,a is the TS of the power angle of the #j SG to a at the mth step.(26)∂J∂a=2Δt∑m=0NUMste∑j=1j≠NNUMSGδj,Ntm−δj,NtNUMsteδj,NtNUMste2Sδj,Ntm,a,(27)Sδj,Ntm,a=Sδj,Ntm,a−Sδj,Ntm,a.Considering that the first-order gradient in (26) is not suitable for difference periods of multiple parameters, this paper proposes the extended first-order gradient with the improved TS in (28) and (29).(28)∂J∂acp∂J∂asp=2Δt∑m=0NUMste∑j=1j≠NNUMSGδj,Ntm−δj,NtNUMsteδj,NtNUMste2Sδj,Ntm,acpSδj,Ntm,asp,(29)Sδj,Ntm,acpSδj,Ntm,asp=Sδjtm,acpSδjtm,asp−SδNtm,acpSδNtm,asp.Since the objective function is nonlinear, the Jacobian and Hessian matrices are to be defined for optimization by the interior point method. The calculation of the Hessian matrix needs a lot of storage and calculation effort. Hence the BFGS algorithm is used to derive the approximate form of the Hessian matrix in an iterative manner in (30), where gk=∂J/∂acp,k,∂J/∂asp,k, mk=gk−gk−1, uk=ak−ak−1, and k is the iteration number.(30)Gk=Gk−1+mkmkTmkTmk−Gk−1ukukTGk−1ukTGk−1uk.


### 4.4. The Evaluation Indices for the Optimization Effect

To evaluate the optimization effect, 3 indices are defined,(a)
The optimization ratio of the amplitudes of the peak and the troughλaptThe steady-state value of the power angle differenceδ(tend) after the fault is taken as the reference, the average amplitude of peak and trough of the power angle difference δapt is defined as follows:(31)δapt=1NUMpk+NUMtr∑i=1NUMpkδi,pk−δtend+∑j=1NUMtrδj,tr−δtend,where NUMpk is the number of the peak, NUMtr is the number of the trough, δi,pk is the angle of the ith peak, and δj,tr is the angle of the jth trough.λapt is defined by (32), where δapt,woop is the average amplitude of peak and trough of the power angle difference without optimization, and δapt,wop is that with optimization.(32)λapt=δapt,woop−δapt,wopδapt,woop×100%.(b)
The optimization ratio of the convergence timeλcvBy taking ±5% ofδ(tend) as the convergence range, and tcv,woop is the convergence time without optimization and tcv,wop is that with optimization, and λcv is obtained in .(33)λcv=tcv,woop−tcv,woptcv,woop×100%.(c)
The optimization ratio of the objective function valueλJTo describe the suppression effect of the proposed optimization during the whole transient process,λJ is defined with the objective function value without optimization Jwoop and that with optimization Jwop in the following equation: (34)λJ=Jwoop−JwopJwoop×100%.


### 4.5. Procedure of Optimization

Considering the relation of active power, reactive power, and power angle, it may improve the angular oscillations by optimizing control parameters and set-points of the HVDC and the SC based on the improved TS. The multiparameter coordinated optimization scheme differentiating the periods of optimization is shown in Figure4, where the words in the green font show the optimization objects and the words in the red font show the difficulties of the proposed scheme.Figure 4
Multiparameter coordinated optimization differentiating the period of optimization.The elaborate description of the optimization procedures is as follows:Firstly, based on the differential-algebraic equations of the AC/HVDC power system in (10), the trajectory sensitivities with respect to all control parameters and set-points are obtained by the time-domain simulation and the improved analytical expression of TS, as shown in (11) and (12).Secondly, the control parameters and set-points to be optimized are selected based on the mean trajectory sensitivities in (22).Thirdly, based on (28) and (30), the first-order and second-order gradient information of the objective function with respect to the control parameters and the set-points to be optimized is extended, which is required by the interior point method.Fourthly, with the gradient information, the optimization model in (25) is solved by the interior point method, and the optimal values of the control parameters and the set-points are obtained.Fifthly, the duality gap of the interior point method is checked. If the duality gap is smaller than the convergence value, which is set at 10−4 in this paper, the optimization procedures are completed. Otherwise, it returns to the first step, and the next literation continues.


### 4.6. Parameter Optimization under Scenarios with Different Faults

The optimization results for a single fault are difficult to achieve for all faults, and even the postfault response is degraded in some scenarios. To improve the applicability, the weighting value of the optimization resultsa' is derived as follows: By introducing the index of the importance degree of the node φIDN [38], the severity index of the nth scenario φSEV (n) is calculated in the following equation:(35)φSEVn=φIDNn∑i∈ΩφIDNi,where Ω is the set of the scenarios, φIDN (n) is the index of the importance degree of the fault bus in the nth scenario.By accumulating the optimization results under different scenarios with the severity index,a′ is obtained in (36), where an,op is the optimization results for the nth scenario.(36)a′=∑n∈ΩφSEVnan,op.Considering the hazard level of different faults and introducing the weighting value of the optimization results to the proposed scheme may improve its applicability for different faults.It should be noted that the proposed models are based on a given system state, but the system scenarios and fault conditions are always changing. To improve the feasibility for actual application, the optimized parameters from different states are compromised to derive a comprehensive solution to suppress the angular oscillation. The interior point method is applied to solve the proposed model. Now the heuristic methods are widely used in many existing optimization models to damp the oscillations, which is easy to implement but has drawbacks in terms of computation efficiency and the global solution [39]. Therefore, a direct optimization, e.g., the interior point method, may be more desirable.


## 4.1. Selecting the Parameters to be Optimized Based on the Sensitivity Evaluation Index

The parameters to be optimized are selected by the evaluation index of the TS. To eliminate the difference in the range and the dimension of parameters, the mean TS is introduced [28], which reflects the influence degree of parameters on the dynamic process.(22)Smean=1NUMsamp∑i=1NUMsamp∂pα,i/p0∂α/α0,where Smean is the value of the mean TS, p is one of system variables, and NUMsamp is the sampling number on the TS curves.


## 4.2. The Optimization Model

Taking the power angle of the #N SG as the reference, the objective function J is established in (23) by calculating the difference-squared integral of the power angle difference in the transient process and the steady-state power angle difference after the fault.(23)minJ=∑j=1j≠NNUMSG∫0tendδj,Nt−δj,Ntendδj,Ntend2dt,(24)δj,Nt=δjt−δNt,δj,Ntend=δjtend−δNtend,where NUMSG is the number of SGs, and tend is the finish time of the simulation, δj is the power angle of the #j SG, and δj,N is the power angle difference between the #j and the #N SGs.Considering equality and inequality constraints, the optimization model is proposed in (25), where amin and amax are the upper and lower limits of parameters.(25)minJ,s.t.x˙=fxt,yt,a,0=gxt,yt,a,a=acpasp,amin≤a≤amax.


## 4.3. Multiparameter Coordinated Optimization considering the Optimization Period

Considering the nonlinearity of the model in (25) and the advantage of the direct optimization, the interior point method is introduced, which requires the first-order and second-order gradient information of the objective function. By accumulating the TS of the objective function J to a, the first-order gradient is obtained by (26), where NUMste is the number of the simulation steps and Sδjtm,a is the TS of the power angle of the #j SG to a at the mth step.(26)∂J∂a=2Δt∑m=0NUMste∑j=1j≠NNUMSGδj,Ntm−δj,NtNUMsteδj,NtNUMste2Sδj,Ntm,a,(27)Sδj,Ntm,a=Sδj,Ntm,a−Sδj,Ntm,a.Considering that the first-order gradient in (26) is not suitable for difference periods of multiple parameters, this paper proposes the extended first-order gradient with the improved TS in (28) and (29).(28)∂J∂acp∂J∂asp=2Δt∑m=0NUMste∑j=1j≠NNUMSGδj,Ntm−δj,NtNUMsteδj,NtNUMste2Sδj,Ntm,acpSδj,Ntm,asp,(29)Sδj,Ntm,acpSδj,Ntm,asp=Sδjtm,acpSδjtm,asp−SδNtm,acpSδNtm,asp.Since the objective function is nonlinear, the Jacobian and Hessian matrices are to be defined for optimization by the interior point method. The calculation of the Hessian matrix needs a lot of storage and calculation effort. Hence the BFGS algorithm is used to derive the approximate form of the Hessian matrix in an iterative manner in (30), where gk=∂J/∂acp,k,∂J/∂asp,k, mk=gk−gk−1, uk=ak−ak−1, and k is the iteration number.(30)Gk=Gk−1+mkmkTmkTmk−Gk−1ukukTGk−1ukTGk−1uk.


## 4.4. The Evaluation Indices for the Optimization Effect

To evaluate the optimization effect, 3 indices are defined,(a)
The optimization ratio of the amplitudes of the peak and the troughλaptThe steady-state value of the power angle differenceδ(tend) after the fault is taken as the reference, the average amplitude of peak and trough of the power angle difference δapt is defined as follows:(31)δapt=1NUMpk+NUMtr∑i=1NUMpkδi,pk−δtend+∑j=1NUMtrδj,tr−δtend,where NUMpk is the number of the peak, NUMtr is the number of the trough, δi,pk is the angle of the ith peak, and δj,tr is the angle of the jth trough.λapt is defined by (32), where δapt,woop is the average amplitude of peak and trough of the power angle difference without optimization, and δapt,wop is that with optimization.(32)λapt=δapt,woop−δapt,wopδapt,woop×100%.(b)
The optimization ratio of the convergence timeλcvBy taking ±5% ofδ(tend) as the convergence range, and tcv,woop is the convergence time without optimization and tcv,wop is that with optimization, and λcv is obtained in .(33)λcv=tcv,woop−tcv,woptcv,woop×100%.(c)
The optimization ratio of the objective function valueλJTo describe the suppression effect of the proposed optimization during the whole transient process,λJ is defined with the objective function value without optimization Jwoop and that with optimization Jwop in the following equation: (34)λJ=Jwoop−JwopJwoop×100%.


## 4.5. Procedure of Optimization

Considering the relation of active power, reactive power, and power angle, it may improve the angular oscillations by optimizing control parameters and set-points of the HVDC and the SC based on the improved TS. The multiparameter coordinated optimization scheme differentiating the periods of optimization is shown in Figure4, where the words in the green font show the optimization objects and the words in the red font show the difficulties of the proposed scheme.Figure 4
Multiparameter coordinated optimization differentiating the period of optimization.The elaborate description of the optimization procedures is as follows:Firstly, based on the differential-algebraic equations of the AC/HVDC power system in (10), the trajectory sensitivities with respect to all control parameters and set-points are obtained by the time-domain simulation and the improved analytical expression of TS, as shown in (11) and (12).Secondly, the control parameters and set-points to be optimized are selected based on the mean trajectory sensitivities in (22).Thirdly, based on (28) and (30), the first-order and second-order gradient information of the objective function with respect to the control parameters and the set-points to be optimized is extended, which is required by the interior point method.Fourthly, with the gradient information, the optimization model in (25) is solved by the interior point method, and the optimal values of the control parameters and the set-points are obtained.Fifthly, the duality gap of the interior point method is checked. If the duality gap is smaller than the convergence value, which is set at 10−4 in this paper, the optimization procedures are completed. Otherwise, it returns to the first step, and the next literation continues.


## 4.6. Parameter Optimization under Scenarios with Different Faults

The optimization results for a single fault are difficult to achieve for all faults, and even the postfault response is degraded in some scenarios. To improve the applicability, the weighting value of the optimization resultsa' is derived as follows: By introducing the index of the importance degree of the node φIDN [38], the severity index of the nth scenario φSEV (n) is calculated in the following equation:(35)φSEVn=φIDNn∑i∈ΩφIDNi,where Ω is the set of the scenarios, φIDN (n) is the index of the importance degree of the fault bus in the nth scenario.By accumulating the optimization results under different scenarios with the severity index,a′ is obtained in (36), where an,op is the optimization results for the nth scenario.(36)a′=∑n∈ΩφSEVnan,op.Considering the hazard level of different faults and introducing the weighting value of the optimization results to the proposed scheme may improve its applicability for different faults.It should be noted that the proposed models are based on a given system state, but the system scenarios and fault conditions are always changing. To improve the feasibility for actual application, the optimized parameters from different states are compromised to derive a comprehensive solution to suppress the angular oscillation. The interior point method is applied to solve the proposed model. Now the heuristic methods are widely used in many existing optimization models to damp the oscillations, which is easy to implement but has drawbacks in terms of computation efficiency and the global solution [39]. Therefore, a direct optimization, e.g., the interior point method, may be more desirable.


## 5. Numerical Analysis

The program is written by the authors by using MATLAB R2021a language and ran on the computer with Intel Core i5-10400 CPU 2.9 GHz, 16 G. To validate the effectiveness of the proposed models, the IEEE-39-bus in Figure5 is applied. The AC line 2-3 is replaced with an HVDC line with the loading of 365 MW [40]. The SC is located at bus 2, i.e., the rectifier side. The parameters of the CCC and the CEA are found in [36].Figure 5
IEEE 39-bus test system with HVDC line.Four scenarios are designed,Scenario 1.
A three-phase short-circuit fault occurs at bus 26 followed by DC mono-pole blocking fault.Scenario 2.
A three-phase short-circuit fault occurs at bus 9 followed by DC mono-pole blocking fault.Scenario 3.
A three-phase short-circuit fault occurs at bus 28 followed by DC mono-pole blocking fault.Scenario 4.
A three-phase short-circuit fault occurs at bus 4 followed by DC mono-pole blocking fault.
The analysis in Sections5.1 to 5.3 based on Scenario 1 is to validate the effectiveness of the proposed model in the signal scenario, and that in Section 5.4 based on the set of scenarios is to improve the applicability for different faults.


### 5.1. Validation of the Improved Trajectory Sensitivity

To verify the accuracy of the improved TS, the trajectory sensitivities ofδ1 with respect to KF and IDC,ref (Sδ1,KF and Sδ1,IDC,ref) are calculated by the analytical method and the perturbation method, respectively. The sensitivity curves are shown in Figures 6 and 7, which almost coincide. Table 1 gives the error analysis of the two methods and shows that the maximum absolute error is only 5.22 × 10−5 and the maximum relative error is only 1.82%. Hence, the accuracy of the improved TS is verified.Figure 6
Trajectory sensitivity of the power angle of #1 SG toKF.Figure 7
Trajectory sensitivity of the power angle of #1 SG toIDC,ref.Table 1
Error analysis of the improved TS.
ErrorSensitivityMaxAverageAbsolute errorSδ1,KF5.22 × 10−51.47 × 10−5Sδ1,IDC,ref3.93 × 10−62.21 × 10−6Relative error (%)Sδ1,KF1.820.14Sδ1,IDC,ref0.260.05


### 5.2. The Critical Parameters Selected with the TS Index

To find the critical control parameters and set-points to be optimized, the evaluation index in Section4.1 is applied. The control parameters and the set-points of the excitation systems of the SC and the CCC of the HVDC are given in Table 2, along with the evaluation indices for δ1 with respect to those. The evaluation indices corresponding to KA, KF, and TF, which are 1.27 × 10−2, 0.98 × 10−2, and 0.61 × 10−2, respectively, are the greater ones among the control parameters of the SC and the CCC. The evaluation indices corresponding to Vref, Vf0, IDC,ref, and αref are 5.38 × 10−1, 1.67 × 10−3, 2.32 × 10−4, and 1.56 × 10−1, respectively. It is clear that the evaluation indices corresponding to Vref and IDC,ref are the larger two among all set-points. The evaluation indices for δ2 to δ10 are given in Tables 3 and 4, and the evaluation indices corresponding to KA, KF, TF, Vref, and IDC,ref are greater than others. So KA, KF, and TF are selected as the critical control parameters, and Vref and IDC,ref are selected as the critical set-points. The control parameters and the set-points marked by the ( ) are not to be optimized in Tables 2–4.Table 2
Calculation of evaluation indices forδ1 based on TS.
ParameterSmeanControl parameterSCKA1.27 × 10−2(TA)4.41 × 10−5KF0.98 × 10−2TF0.61 × 10−2CCC(Kpr)5.24 × 10−5(Kir)4.52 × 10−4Set-pointSCVref5.38 × 10−1(Vf0)1.67 × 10−3CCC(αref)2.32 × 10−4IDC,ref1.56 × 10−1Table 3
The TS indices forδ2 to δ5.
ParameterδSG2δSG3δSG4δSG5KA1.47 × 10−21.47 × 10−21.47 × 10−21.47 × 10−2(TA)4.83 × 10−55.05 × 10−54.98 × 10−55.11 × 10−5KF1.14 × 10−21.16 × 10−21.18 × 10−21.21 × 10−2TF0.64 × 10−20.67 × 10−20.69 × 10−20.71 × 10−2(Kpr)5.03 × 10−55.02 × 10−55.03 × 10−55.04 × 10−5(Kir)4.29 × 10−44.20 × 10−44.16 × 10−44.33 × 10−4Vref5.96 × 10−15.98 × 10−16.03 × 10−16.07 × 10−1(Vf0)1.69 × 10−31.70 × 10−31.69 × 10−31.71 × 10−3(αref)2.35 × 10−42.35 × 10−42.36 × 10−42.36 × 10−4IDC,ref1.60 × 10−11.60 × 10−11.60 × 10−11.60 × 10−1Table 4
The TS indices forδ6 to δ10.
ParameterδSG6δSG7δSG8δSG9δSG10KA1.47 × 10−21.48 × 10−21.34 × 10−21.44 × 10−21.41 × 10−2(TA)4.87 × 10−55.27 × 10−54.41 × 10−54.45 × 10−54.22 × 10−5KF1.16 × 10−21.25 × 10−20.78 × 10−20.98 × 10−20.87 × 10−2TF0.68 × 10−20.73 × 10−20.49 × 10−20.56 × 10−20.52 × 10−2(Kpr)5.06 × 10−55.03 × 10−55.17 × 10−55.03 × 10−55.11 × 10−5(Kir)4.20 × 10−44.29 × 10−44.21 × 10−44.20 × 10−44.14 × 10−4Vref6.01 × 10−16.11 × 10−15.44 × 10−15.67×10−15.62 × 10−1(Vf0)1.71 × 10−31.69 × 10−31.69 × 10−31.69 × 10−31.69 × 10−3(αref)2.36 × 10−42.37 × 10−42.31 × 10−42.33 × 10−42.32 × 10−4IDC,ref1.59 × 10−11.61 × 10−11.56 × 10−11.57 × 10−11.57 × 10−1


### 5.3. Suppression to the Oscillations in the Signal Scenario

In this section, different optimization cases are designed,Case 1.
Optimization toIDC,ref.Case 2.
Optimization toVref and IDC,ref.Case 3.
Optimization toKA, KF, and TF.Case 4.
Optimization toKA, KF, TF, and Vref.Case 5.
Optimization toKA, KF, TF, Vref, and IDC,ref (the optimization scheme proposed by this paper).
Cases1 and 2 are compared to reflect the suppression effect of the SC on the angular oscillations and the effectiveness of coordination optimization for active power and reactive power. The comparison of Cases 2 and 3 is to investigate the difference between optimization to control set-points and optimization to control parameters. By designing different optimization cases, the effectiveness of the proposed optimization is verified.
The initial values are shown in Table5. Considering the time delay of the fault diagnosis, the stability control device, and the adjustment to the parameters in the actual engineering [41, 42], the adjustment of set-points starts at 70 ms after the DC blocking and continues for 1 s before returning to the initial value. In order to illustrate the optimization effect of parameters with small sensitivities, the optimization to Kpr and Kir of the CCC is designed in Section 5.3.6.Table 5
The initial values and ranges of the parameters.
ParameterInitial valueRangeKA200[100, 400]KF1[0.5, 0.5]TF (s)0.1[0.05, 0.5]Vref (p.u.)1.0484[0.9, 1.1]IDC,ref (p.u.)0.9125[0.85, 1]


#### 5.3.1. Optimization toIDC,ref

Considering the short-time overload capability of the nonfault pole, the upper limit ofIDC,ref is set to 1 p.u. [33]. Keeping other parameters constant, IDC,ref is optimized after the DC blocking, and the optimal value is 0.9995 p.u.. The curves of δ2,1 and δ5,1 with and without optimization are shown in Figure 8, and the analysis is given in Table 6. Compared with the objective function value without optimization, it is reduced by 1.66% with optimization. The optimization of IDC,ref improves the amplitude of the oscillations but has little effect on the convergence time.Figure 8
Power angle difference between SGs with and without optimization toIDC,ref. (a) Power angle difference between #2 and #1 SGs. (b) Power angle difference between #5 and #1 SGs.
(a)(b)Table 6
Effect analysis of optimization toIDC,ref.
δapt,woop (°)δapt,wop (°)λapt (%)δ2,15.245.161.53δ5,15.905.772.06JwoopJwopλJ(%)40.4439.771.66


#### 5.3.2. Optimization toVref and IDC,ref

In this case,Vref and IDC,ref are optimized together after the DC blocking, and the optimal values are 0.9001 p.u. and 0.9342 p.u. respectively. As shown in Figure 9, the amplitude and the convergence time of the oscillations are improved, and the analysis is given in Table 7. λapt and λcv of δ2,1 are 7.78% and 13.64%, and those of δ5,1 are 8.86% and 9.40%. J is reduced from 40.44 to 39.30. The above results verify the suppression effect of the optimization to Vref and IDC,ref.Figure 9
Power angle difference between SGs with and without optimization toVref and IDC,ref. (a) Power angle difference between #2 and #1 SGs. (b) Power angle difference between #5 and #1 SGs.
(a)(b)Table 7
Effect analysis of optimization toVref and IDC,ref.
δapt,woop (°)δapt,wop (°)λapt (%)δ2,15.244.837.78δ5,15.905.388.86tcv,woop (s)tcv,wop (s)λcv (%)δ2,133.2228.6913.64δ5,138.5134.899.40JwoopJwopλJ (%)40.4439.302.81To verify the suppression effect of the SC on the angular oscillation, the analysis of the optimization toIDC,ref in Section 5.3.1 and that to Vref with IDC,ref is compared in Figure 10. λapt and λcv of δ2,1 are increased by 6.25% and 13.64%, respectively, and those of δ5,1 are increased by 6.80% and 9.40%, respectively. Based on the above analysis, it is clear that the coordinated optimization to Vref with IDC,ref further improves the oscillations.Figure 10
Comparison of optimization toIDC,ref and optimization to Vref with IDC,ref.


#### 5.3.3. Optimization toKA, KF, and TF

With the set-points constant, the optimization toKA, KF, and TF is implemented, and the optimal values are 211, 0.5002, and 0.4956 s respectively. As shown in Figure 11, the convergence time of δ5,1 is advanced from 38.51 s to 35.36 s and that of δ7,1 is advanced from 35.02 s to 31.84 s, and the amplitude is also improved. As given in Table 8, λapt and λcv of δ5,1 are 5.48% (0.32°) and 8.17% (3.15 s), respectively, and those of δ7,1 are 3.58% (0.22°) and 9.17% (3.18 s), respectively. J is reduced by 1.77% with the optimization.Figure 11
Power angle difference between SGs with and without optimization toKA, KF, and TF. (a) Power angle difference between #5 and #1 SGs. (b) The power angle difference between #7 and #1 SGs.
(a)(b)Table 8
Effect analysis of optimization forKA, KF, and TF.
δapt,woop (°)δapt,wop (°)λapt (%)δ5,15.905.585.48δ7,16.175.953.58tcv,woop (s)tcv,wop (s)λcv (%)δ5,138.5135.368.17δ7,135.0231.849.17JwoopJwopλJ (%)40.4439.721.77Compared with the simulation results of optimization toVref and IDC,ref in Section 5.3.2, λapt and λcv of δ5,1 are reduced by 3.38% and 1.23%, respectively, and λJ is reduced by 1.04%. The suppression effect of optimization to KA, KF, and TF is weaker than that to Vref and IDC,ref.


#### 5.3.4. Optimization toKA, KF, TF, and Vref

Control parameters and set-points of the SC are optimized together to suppress the oscillations, and the optimal values are 201, 0.5005, 0.0614 s, and 0.9004 p.u., respectively. The amplitude and the convergence time are improved in Figure12, and Table 9 gives the analytic data of the simulation results. Λapt and λcv of δ5,1 are 17.46% (1.03°) and 15.53% (5.98 s), respectively, and those of δ7,1 are 18.48% (1.14°) and 11.94% (4.18 s), respectively. Compared with optimization to IDC,ref in Section 5.3.1, λapt and λcv of δ5,1 are increased by 15.4% and 15.53%, respectively, and λJ is increased by 12.15%. It is found that the suppression effect of the optimization to KA, KF, TF, and Vref is better than that to IDC,ref.Figure 12
Power angle difference between SGs with and without optimization toKA, KF, TF, and Vref. (a) Power angle difference between #5 and #1 SGs. (b) The power angle difference between #7 and #1 SGs.
(a)(b)Table 9
Effect analysis of optimization forKA, KF, TF, and Vref.
δapt,woop (°)δapt,wop (°)λapt (%)δ5,15.904.8717.46δ7,16.175.0318.48tcv,woop (s)tcv,wop (s)λcv (%)δ5,138.5132.5315.53δ7,135.0230.8411.94JwoopJwopλJ (%)40.4434.8513.81Figure13 shows the comparison of the simulation results in this section and those in Section 5.3.3. The effect of the coordinated optimization to KA, KF, TF, and Vref is better than that to KA, KF, and TF. λapt and λcv of δ5,1 are increased by 11.98% and 7.36%, respectively, and those of δ7,1 are increased by 14.9% and 2.77%, respectively. And λJ is increased by 12.04%.Figure 13
Comparison of optimization toKA, KF, and TF and optimization to KA, KF, TF, and Vref.


#### 5.3.5. Optimization toKA, KF, TF, Vref and IDC,ref

In this section, the proposed scheme is implemented. The optimal values ofKA, KF, TF, Vref, and IDC,ref are 201, 0.5007, 0.0517 s, 0.9004 p.u. and 0.9998 p.u., respectively. As shown in Figure 14, the power angular oscillation is sharper without the optimization. To clearly describe the effect of the scheme, the analytic data of the simulation results are given in Table 10. λapt and λcv of δ2,1 are 17.63% (0.92°) and 22.97% (7.63 s), respectively, and those of δ3,1 are 14.46% (0.75°) and 18.96% (5.98 s), respectively. And λJ is 15.11%, which is the largest one among the 5 groups of cases.Figure 14
Power angle difference between SGs with and without coordinated optimization toKA, KF, TF, Vref, and IDC,ref. (a) Power angle difference between #2 and #1 SGs. (b) Power angle difference between #3 and #1 SGs.
(a)(b)Table 10
Effect analysis of coordinated optimization to the CCC and the SC.
δapt,woop (°)δapt,wop (°)λapt (%)δ2,15.244.3217.63δ3,15.174.4214.46tcv,woop (s)tcv,wop (s)λcv (%)δ2,133.2225.5922.97δ3,131.5425.5618.96JwoopJwopλJ (%)40.4434.3315.11With the comparison of Tables6–9, the effect of the coordinated optimization proposed in this paper is better than that of the optimization to control parameters or set-points. Compared with the simulation results in Section 5.3.2, λapt and λcv of δ2,1 are increased by 9.85% and 9.33%, respectively, and λJ is increased by 12.3%. The above analysis validates the effectiveness of the proposed scheme for oscillation suppression.


#### 5.3.6. Optimization toKpr and Kir of the CCC

As shown in Section5.2, the mean trajectory sensitivities of Kpr and Kir are smaller than other parameters. To illustrate the optimization effect of parameters with smaller sensitivities, the optimization to Kpr and Kir is designed. The initial values of Kpr and Kir are 1.0989 and 91.575, respectively, and the range are (0.9890, 1.2088) and (82.4175, 100.7325), respectively. The optimal values are 0.9902 and 82.4323. As shown in Figure 15 and Table 11, the optimization effect is slight, and there is almost no effect on the convergence time. And λJ is 0.79%, which is smaller than the previous 5 groups of cases.Figure 15
Power angle difference between SGs with and without optimization toKpr and Kir of the CCC. (a) Power-angle difference between #2 and #1 SGs. (b) Power-angle difference between #5 and #1 SGs.
(a)(b)Table 11
Effect analysis of coordinated optimization toKpr and Kir of the CCC.
δapt,woop (°)δapt,wop (°)λapt (%)δ2,15.245.200.76δ5,15.905.850.84JwoopJwopλJ (%)40.4440.120.79


### 5.4. The Oscillations Suppression for Scenarios with Different Faults

In actual engineering, the fault scenarios are complex. Considering the research focus of this paper, the set of fault scenarios is simplified to four scenarios, i.e.,Scenarios 1 to 4. φIDN of buses 9, 21, 26, and 28 are 3.7, 1.8, 4, and 2.4, respectively [43]. The severity indices of four scenarios calculated by (35) are 0.311, 0.151, 0.336, and 0.202, respectively. According to the sensitivity evaluation indices, the parameters to be optimized for the four scenarios are the same, i.e. KA, KF, TF, Vref, and IDC,ref. The proposed scheme is implemented in Scenarios 2 to 4, and the optimization results for different scenarios (an, op and the weighting value a') are shown in Table 12. As shown in Table 13, a' has a more acceptable effect for different scenarios when compared with an,op. Consequently, the applicability of the proposed scheme is verified with the severity indices of different faults.Table 12
The optimization under scenarios with different faults.
Scenarioan,opKAKFTF (s)Vref (p.u.)IDC,ref (p.u.)12010.50070.05170.90040.99982200.38280.50100.050.90260.99993200.94050.50530.05160.91510.99994199.78380.50130.05550.90170.9999a'200.61240.50180.05180.90430.9998Table 13
The comparison of the suppression effect ofa' and aω,op.
Scenarioλofv (%)an,opa′115.1114.57220.6420.56318.7318.76413.1813.06


## 5.1. Validation of the Improved Trajectory Sensitivity

To verify the accuracy of the improved TS, the trajectory sensitivities ofδ1 with respect to KF and IDC,ref (Sδ1,KF and Sδ1,IDC,ref) are calculated by the analytical method and the perturbation method, respectively. The sensitivity curves are shown in Figures 6 and 7, which almost coincide. Table 1 gives the error analysis of the two methods and shows that the maximum absolute error is only 5.22 × 10−5 and the maximum relative error is only 1.82%. Hence, the accuracy of the improved TS is verified.Figure 6
Trajectory sensitivity of the power angle of #1 SG toKF.Figure 7
Trajectory sensitivity of the power angle of #1 SG toIDC,ref.Table 1
Error analysis of the improved TS.
ErrorSensitivityMaxAverageAbsolute errorSδ1,KF5.22 × 10−51.47 × 10−5Sδ1,IDC,ref3.93 × 10−62.21 × 10−6Relative error (%)Sδ1,KF1.820.14Sδ1,IDC,ref0.260.05


## 5.2. The Critical Parameters Selected with the TS Index

To find the critical control parameters and set-points to be optimized, the evaluation index in Section4.1 is applied. The control parameters and the set-points of the excitation systems of the SC and the CCC of the HVDC are given in Table 2, along with the evaluation indices for δ1 with respect to those. The evaluation indices corresponding to KA, KF, and TF, which are 1.27 × 10−2, 0.98 × 10−2, and 0.61 × 10−2, respectively, are the greater ones among the control parameters of the SC and the CCC. The evaluation indices corresponding to Vref, Vf0, IDC,ref, and αref are 5.38 × 10−1, 1.67 × 10−3, 2.32 × 10−4, and 1.56 × 10−1, respectively. It is clear that the evaluation indices corresponding to Vref and IDC,ref are the larger two among all set-points. The evaluation indices for δ2 to δ10 are given in Tables 3 and 4, and the evaluation indices corresponding to KA, KF, TF, Vref, and IDC,ref are greater than others. So KA, KF, and TF are selected as the critical control parameters, and Vref and IDC,ref are selected as the critical set-points. The control parameters and the set-points marked by the ( ) are not to be optimized in Tables 2–4.Table 2
Calculation of evaluation indices forδ1 based on TS.
ParameterSmeanControl parameterSCKA1.27 × 10−2(TA)4.41 × 10−5KF0.98 × 10−2TF0.61 × 10−2CCC(Kpr)5.24 × 10−5(Kir)4.52 × 10−4Set-pointSCVref5.38 × 10−1(Vf0)1.67 × 10−3CCC(αref)2.32 × 10−4IDC,ref1.56 × 10−1Table 3
The TS indices forδ2 to δ5.
ParameterδSG2δSG3δSG4δSG5KA1.47 × 10−21.47 × 10−21.47 × 10−21.47 × 10−2(TA)4.83 × 10−55.05 × 10−54.98 × 10−55.11 × 10−5KF1.14 × 10−21.16 × 10−21.18 × 10−21.21 × 10−2TF0.64 × 10−20.67 × 10−20.69 × 10−20.71 × 10−2(Kpr)5.03 × 10−55.02 × 10−55.03 × 10−55.04 × 10−5(Kir)4.29 × 10−44.20 × 10−44.16 × 10−44.33 × 10−4Vref5.96 × 10−15.98 × 10−16.03 × 10−16.07 × 10−1(Vf0)1.69 × 10−31.70 × 10−31.69 × 10−31.71 × 10−3(αref)2.35 × 10−42.35 × 10−42.36 × 10−42.36 × 10−4IDC,ref1.60 × 10−11.60 × 10−11.60 × 10−11.60 × 10−1Table 4
The TS indices forδ6 to δ10.
ParameterδSG6δSG7δSG8δSG9δSG10KA1.47 × 10−21.48 × 10−21.34 × 10−21.44 × 10−21.41 × 10−2(TA)4.87 × 10−55.27 × 10−54.41 × 10−54.45 × 10−54.22 × 10−5KF1.16 × 10−21.25 × 10−20.78 × 10−20.98 × 10−20.87 × 10−2TF0.68 × 10−20.73 × 10−20.49 × 10−20.56 × 10−20.52 × 10−2(Kpr)5.06 × 10−55.03 × 10−55.17 × 10−55.03 × 10−55.11 × 10−5(Kir)4.20 × 10−44.29 × 10−44.21 × 10−44.20 × 10−44.14 × 10−4Vref6.01 × 10−16.11 × 10−15.44 × 10−15.67×10−15.62 × 10−1(Vf0)1.71 × 10−31.69 × 10−31.69 × 10−31.69 × 10−31.69 × 10−3(αref)2.36 × 10−42.37 × 10−42.31 × 10−42.33 × 10−42.32 × 10−4IDC,ref1.59 × 10−11.61 × 10−11.56 × 10−11.57 × 10−11.57 × 10−1


## 5.3. Suppression to the Oscillations in the Signal Scenario

In this section, different optimization cases are designed,Case 1.
Optimization toIDC,ref.Case 2.
Optimization toVref and IDC,ref.Case 3.
Optimization toKA, KF, and TF.Case 4.
Optimization toKA, KF, TF, and Vref.Case 5.
Optimization toKA, KF, TF, Vref, and IDC,ref (the optimization scheme proposed by this paper).
Cases1 and 2 are compared to reflect the suppression effect of the SC on the angular oscillations and the effectiveness of coordination optimization for active power and reactive power. The comparison of Cases 2 and 3 is to investigate the difference between optimization to control set-points and optimization to control parameters. By designing different optimization cases, the effectiveness of the proposed optimization is verified.
The initial values are shown in Table5. Considering the time delay of the fault diagnosis, the stability control device, and the adjustment to the parameters in the actual engineering [41, 42], the adjustment of set-points starts at 70 ms after the DC blocking and continues for 1 s before returning to the initial value. In order to illustrate the optimization effect of parameters with small sensitivities, the optimization to Kpr and Kir of the CCC is designed in Section 5.3.6.Table 5
The initial values and ranges of the parameters.
ParameterInitial valueRangeKA200[100, 400]KF1[0.5, 0.5]TF (s)0.1[0.05, 0.5]Vref (p.u.)1.0484[0.9, 1.1]IDC,ref (p.u.)0.9125[0.85, 1]


### 5.3.1. Optimization toIDC,ref

Considering the short-time overload capability of the nonfault pole, the upper limit ofIDC,ref is set to 1 p.u. [33]. Keeping other parameters constant, IDC,ref is optimized after the DC blocking, and the optimal value is 0.9995 p.u.. The curves of δ2,1 and δ5,1 with and without optimization are shown in Figure 8, and the analysis is given in Table 6. Compared with the objective function value without optimization, it is reduced by 1.66% with optimization. The optimization of IDC,ref improves the amplitude of the oscillations but has little effect on the convergence time.Figure 8
Power angle difference between SGs with and without optimization toIDC,ref. (a) Power angle difference between #2 and #1 SGs. (b) Power angle difference between #5 and #1 SGs.
(a)(b)Table 6
Effect analysis of optimization toIDC,ref.
δapt,woop (°)δapt,wop (°)λapt (%)δ2,15.245.161.53δ5,15.905.772.06JwoopJwopλJ(%)40.4439.771.66


### 5.3.2. Optimization toVref and IDC,ref

In this case,Vref and IDC,ref are optimized together after the DC blocking, and the optimal values are 0.9001 p.u. and 0.9342 p.u. respectively. As shown in Figure 9, the amplitude and the convergence time of the oscillations are improved, and the analysis is given in Table 7. λapt and λcv of δ2,1 are 7.78% and 13.64%, and those of δ5,1 are 8.86% and 9.40%. J is reduced from 40.44 to 39.30. The above results verify the suppression effect of the optimization to Vref and IDC,ref.Figure 9
Power angle difference between SGs with and without optimization toVref and IDC,ref. (a) Power angle difference between #2 and #1 SGs. (b) Power angle difference between #5 and #1 SGs.
(a)(b)Table 7
Effect analysis of optimization toVref and IDC,ref.
δapt,woop (°)δapt,wop (°)λapt (%)δ2,15.244.837.78δ5,15.905.388.86tcv,woop (s)tcv,wop (s)λcv (%)δ2,133.2228.6913.64δ5,138.5134.899.40JwoopJwopλJ (%)40.4439.302.81To verify the suppression effect of the SC on the angular oscillation, the analysis of the optimization toIDC,ref in Section 5.3.1 and that to Vref with IDC,ref is compared in Figure 10. λapt and λcv of δ2,1 are increased by 6.25% and 13.64%, respectively, and those of δ5,1 are increased by 6.80% and 9.40%, respectively. Based on the above analysis, it is clear that the coordinated optimization to Vref with IDC,ref further improves the oscillations.Figure 10
Comparison of optimization toIDC,ref and optimization to Vref with IDC,ref.


### 5.3.3. Optimization toKA, KF, and TF

With the set-points constant, the optimization toKA, KF, and TF is implemented, and the optimal values are 211, 0.5002, and 0.4956 s respectively. As shown in Figure 11, the convergence time of δ5,1 is advanced from 38.51 s to 35.36 s and that of δ7,1 is advanced from 35.02 s to 31.84 s, and the amplitude is also improved. As given in Table 8, λapt and λcv of δ5,1 are 5.48% (0.32°) and 8.17% (3.15 s), respectively, and those of δ7,1 are 3.58% (0.22°) and 9.17% (3.18 s), respectively. J is reduced by 1.77% with the optimization.Figure 11
Power angle difference between SGs with and without optimization toKA, KF, and TF. (a) Power angle difference between #5 and #1 SGs. (b) The power angle difference between #7 and #1 SGs.
(a)(b)Table 8
Effect analysis of optimization forKA, KF, and TF.
δapt,woop (°)δapt,wop (°)λapt (%)δ5,15.905.585.48δ7,16.175.953.58tcv,woop (s)tcv,wop (s)λcv (%)δ5,138.5135.368.17δ7,135.0231.849.17JwoopJwopλJ (%)40.4439.721.77Compared with the simulation results of optimization toVref and IDC,ref in Section 5.3.2, λapt and λcv of δ5,1 are reduced by 3.38% and 1.23%, respectively, and λJ is reduced by 1.04%. The suppression effect of optimization to KA, KF, and TF is weaker than that to Vref and IDC,ref.


### 5.3.4. Optimization toKA, KF, TF, and Vref

Control parameters and set-points of the SC are optimized together to suppress the oscillations, and the optimal values are 201, 0.5005, 0.0614 s, and 0.9004 p.u., respectively. The amplitude and the convergence time are improved in Figure12, and Table 9 gives the analytic data of the simulation results. Λapt and λcv of δ5,1 are 17.46% (1.03°) and 15.53% (5.98 s), respectively, and those of δ7,1 are 18.48% (1.14°) and 11.94% (4.18 s), respectively. Compared with optimization to IDC,ref in Section 5.3.1, λapt and λcv of δ5,1 are increased by 15.4% and 15.53%, respectively, and λJ is increased by 12.15%. It is found that the suppression effect of the optimization to KA, KF, TF, and Vref is better than that to IDC,ref.Figure 12
Power angle difference between SGs with and without optimization toKA, KF, TF, and Vref. (a) Power angle difference between #5 and #1 SGs. (b) The power angle difference between #7 and #1 SGs.
(a)(b)Table 9
Effect analysis of optimization forKA, KF, TF, and Vref.
δapt,woop (°)δapt,wop (°)λapt (%)δ5,15.904.8717.46δ7,16.175.0318.48tcv,woop (s)tcv,wop (s)λcv (%)δ5,138.5132.5315.53δ7,135.0230.8411.94JwoopJwopλJ (%)40.4434.8513.81Figure13 shows the comparison of the simulation results in this section and those in Section 5.3.3. The effect of the coordinated optimization to KA, KF, TF, and Vref is better than that to KA, KF, and TF. λapt and λcv of δ5,1 are increased by 11.98% and 7.36%, respectively, and those of δ7,1 are increased by 14.9% and 2.77%, respectively. And λJ is increased by 12.04%.Figure 13
Comparison of optimization toKA, KF, and TF and optimization to KA, KF, TF, and Vref.


### 5.3.5. Optimization toKA, KF, TF, Vref and IDC,ref

In this section, the proposed scheme is implemented. The optimal values ofKA, KF, TF, Vref, and IDC,ref are 201, 0.5007, 0.0517 s, 0.9004 p.u. and 0.9998 p.u., respectively. As shown in Figure 14, the power angular oscillation is sharper without the optimization. To clearly describe the effect of the scheme, the analytic data of the simulation results are given in Table 10. λapt and λcv of δ2,1 are 17.63% (0.92°) and 22.97% (7.63 s), respectively, and those of δ3,1 are 14.46% (0.75°) and 18.96% (5.98 s), respectively. And λJ is 15.11%, which is the largest one among the 5 groups of cases.Figure 14
Power angle difference between SGs with and without coordinated optimization toKA, KF, TF, Vref, and IDC,ref. (a) Power angle difference between #2 and #1 SGs. (b) Power angle difference between #3 and #1 SGs.
(a)(b)Table 10
Effect analysis of coordinated optimization to the CCC and the SC.
δapt,woop (°)δapt,wop (°)λapt (%)δ2,15.244.3217.63δ3,15.174.4214.46tcv,woop (s)tcv,wop (s)λcv (%)δ2,133.2225.5922.97δ3,131.5425.5618.96JwoopJwopλJ (%)40.4434.3315.11With the comparison of Tables6–9, the effect of the coordinated optimization proposed in this paper is better than that of the optimization to control parameters or set-points. Compared with the simulation results in Section 5.3.2, λapt and λcv of δ2,1 are increased by 9.85% and 9.33%, respectively, and λJ is increased by 12.3%. The above analysis validates the effectiveness of the proposed scheme for oscillation suppression.


### 5.3.6. Optimization toKpr and Kir of the CCC

As shown in Section5.2, the mean trajectory sensitivities of Kpr and Kir are smaller than other parameters. To illustrate the optimization effect of parameters with smaller sensitivities, the optimization to Kpr and Kir is designed. The initial values of Kpr and Kir are 1.0989 and 91.575, respectively, and the range are (0.9890, 1.2088) and (82.4175, 100.7325), respectively. The optimal values are 0.9902 and 82.4323. As shown in Figure 15 and Table 11, the optimization effect is slight, and there is almost no effect on the convergence time. And λJ is 0.79%, which is smaller than the previous 5 groups of cases.Figure 15
Power angle difference between SGs with and without optimization toKpr and Kir of the CCC. (a) Power-angle difference between #2 and #1 SGs. (b) Power-angle difference between #5 and #1 SGs.
(a)(b)Table 11
Effect analysis of coordinated optimization toKpr and Kir of the CCC.
δapt,woop (°)δapt,wop (°)λapt (%)δ2,15.245.200.76δ5,15.905.850.84JwoopJwopλJ (%)40.4440.120.79


## 5.3.1. Optimization toIDC,ref

Considering the short-time overload capability of the nonfault pole, the upper limit ofIDC,ref is set to 1 p.u. [33]. Keeping other parameters constant, IDC,ref is optimized after the DC blocking, and the optimal value is 0.9995 p.u.. The curves of δ2,1 and δ5,1 with and without optimization are shown in Figure 8, and the analysis is given in Table 6. Compared with the objective function value without optimization, it is reduced by 1.66% with optimization. The optimization of IDC,ref improves the amplitude of the oscillations but has little effect on the convergence time.Figure 8
Power angle difference between SGs with and without optimization toIDC,ref. (a) Power angle difference between #2 and #1 SGs. (b) Power angle difference between #5 and #1 SGs.
(a)(b)Table 6
Effect analysis of optimization toIDC,ref.
δapt,woop (°)δapt,wop (°)λapt (%)δ2,15.245.161.53δ5,15.905.772.06JwoopJwopλJ(%)40.4439.771.66


## 5.3.2. Optimization toVref and IDC,ref

In this case,Vref and IDC,ref are optimized together after the DC blocking, and the optimal values are 0.9001 p.u. and 0.9342 p.u. respectively. As shown in Figure 9, the amplitude and the convergence time of the oscillations are improved, and the analysis is given in Table 7. λapt and λcv of δ2,1 are 7.78% and 13.64%, and those of δ5,1 are 8.86% and 9.40%. J is reduced from 40.44 to 39.30. The above results verify the suppression effect of the optimization to Vref and IDC,ref.Figure 9
Power angle difference between SGs with and without optimization toVref and IDC,ref. (a) Power angle difference between #2 and #1 SGs. (b) Power angle difference between #5 and #1 SGs.
(a)(b)Table 7
Effect analysis of optimization toVref and IDC,ref.
δapt,woop (°)δapt,wop (°)λapt (%)δ2,15.244.837.78δ5,15.905.388.86tcv,woop (s)tcv,wop (s)λcv (%)δ2,133.2228.6913.64δ5,138.5134.899.40JwoopJwopλJ (%)40.4439.302.81To verify the suppression effect of the SC on the angular oscillation, the analysis of the optimization toIDC,ref in Section 5.3.1 and that to Vref with IDC,ref is compared in Figure 10. λapt and λcv of δ2,1 are increased by 6.25% and 13.64%, respectively, and those of δ5,1 are increased by 6.80% and 9.40%, respectively. Based on the above analysis, it is clear that the coordinated optimization to Vref with IDC,ref further improves the oscillations.Figure 10
Comparison of optimization toIDC,ref and optimization to Vref with IDC,ref.


## 5.3.3. Optimization toKA, KF, and TF

With the set-points constant, the optimization toKA, KF, and TF is implemented, and the optimal values are 211, 0.5002, and 0.4956 s respectively. As shown in Figure 11, the convergence time of δ5,1 is advanced from 38.51 s to 35.36 s and that of δ7,1 is advanced from 35.02 s to 31.84 s, and the amplitude is also improved. As given in Table 8, λapt and λcv of δ5,1 are 5.48% (0.32°) and 8.17% (3.15 s), respectively, and those of δ7,1 are 3.58% (0.22°) and 9.17% (3.18 s), respectively. J is reduced by 1.77% with the optimization.Figure 11
Power angle difference between SGs with and without optimization toKA, KF, and TF. (a) Power angle difference between #5 and #1 SGs. (b) The power angle difference between #7 and #1 SGs.
(a)(b)Table 8
Effect analysis of optimization forKA, KF, and TF.
δapt,woop (°)δapt,wop (°)λapt (%)δ5,15.905.585.48δ7,16.175.953.58tcv,woop (s)tcv,wop (s)λcv (%)δ5,138.5135.368.17δ7,135.0231.849.17JwoopJwopλJ (%)40.4439.721.77Compared with the simulation results of optimization toVref and IDC,ref in Section 5.3.2, λapt and λcv of δ5,1 are reduced by 3.38% and 1.23%, respectively, and λJ is reduced by 1.04%. The suppression effect of optimization to KA, KF, and TF is weaker than that to Vref and IDC,ref.


## 5.3.4. Optimization toKA, KF, TF, and Vref

Control parameters and set-points of the SC are optimized together to suppress the oscillations, and the optimal values are 201, 0.5005, 0.0614 s, and 0.9004 p.u., respectively. The amplitude and the convergence time are improved in Figure12, and Table 9 gives the analytic data of the simulation results. Λapt and λcv of δ5,1 are 17.46% (1.03°) and 15.53% (5.98 s), respectively, and those of δ7,1 are 18.48% (1.14°) and 11.94% (4.18 s), respectively. Compared with optimization to IDC,ref in Section 5.3.1, λapt and λcv of δ5,1 are increased by 15.4% and 15.53%, respectively, and λJ is increased by 12.15%. It is found that the suppression effect of the optimization to KA, KF, TF, and Vref is better than that to IDC,ref.Figure 12
Power angle difference between SGs with and without optimization toKA, KF, TF, and Vref. (a) Power angle difference between #5 and #1 SGs. (b) The power angle difference between #7 and #1 SGs.
(a)(b)Table 9
Effect analysis of optimization forKA, KF, TF, and Vref.
δapt,woop (°)δapt,wop (°)λapt (%)δ5,15.904.8717.46δ7,16.175.0318.48tcv,woop (s)tcv,wop (s)λcv (%)δ5,138.5132.5315.53δ7,135.0230.8411.94JwoopJwopλJ (%)40.4434.8513.81Figure13 shows the comparison of the simulation results in this section and those in Section 5.3.3. The effect of the coordinated optimization to KA, KF, TF, and Vref is better than that to KA, KF, and TF. λapt and λcv of δ5,1 are increased by 11.98% and 7.36%, respectively, and those of δ7,1 are increased by 14.9% and 2.77%, respectively. And λJ is increased by 12.04%.Figure 13
Comparison of optimization toKA, KF, and TF and optimization to KA, KF, TF, and Vref.


## 5.3.5. Optimization toKA, KF, TF, Vref and IDC,ref

In this section, the proposed scheme is implemented. The optimal values ofKA, KF, TF, Vref, and IDC,ref are 201, 0.5007, 0.0517 s, 0.9004 p.u. and 0.9998 p.u., respectively. As shown in Figure 14, the power angular oscillation is sharper without the optimization. To clearly describe the effect of the scheme, the analytic data of the simulation results are given in Table 10. λapt and λcv of δ2,1 are 17.63% (0.92°) and 22.97% (7.63 s), respectively, and those of δ3,1 are 14.46% (0.75°) and 18.96% (5.98 s), respectively. And λJ is 15.11%, which is the largest one among the 5 groups of cases.Figure 14
Power angle difference between SGs with and without coordinated optimization toKA, KF, TF, Vref, and IDC,ref. (a) Power angle difference between #2 and #1 SGs. (b) Power angle difference between #3 and #1 SGs.
(a)(b)Table 10
Effect analysis of coordinated optimization to the CCC and the SC.
δapt,woop (°)δapt,wop (°)λapt (%)δ2,15.244.3217.63δ3,15.174.4214.46tcv,woop (s)tcv,wop (s)λcv (%)δ2,133.2225.5922.97δ3,131.5425.5618.96JwoopJwopλJ (%)40.4434.3315.11With the comparison of Tables6–9, the effect of the coordinated optimization proposed in this paper is better than that of the optimization to control parameters or set-points. Compared with the simulation results in Section 5.3.2, λapt and λcv of δ2,1 are increased by 9.85% and 9.33%, respectively, and λJ is increased by 12.3%. The above analysis validates the effectiveness of the proposed scheme for oscillation suppression.


## 5.3.6. Optimization toKpr and Kir of the CCC

As shown in Section5.2, the mean trajectory sensitivities of Kpr and Kir are smaller than other parameters. To illustrate the optimization effect of parameters with smaller sensitivities, the optimization to Kpr and Kir is designed. The initial values of Kpr and Kir are 1.0989 and 91.575, respectively, and the range are (0.9890, 1.2088) and (82.4175, 100.7325), respectively. The optimal values are 0.9902 and 82.4323. As shown in Figure 15 and Table 11, the optimization effect is slight, and there is almost no effect on the convergence time. And λJ is 0.79%, which is smaller than the previous 5 groups of cases.Figure 15
Power angle difference between SGs with and without optimization toKpr and Kir of the CCC. (a) Power-angle difference between #2 and #1 SGs. (b) Power-angle difference between #5 and #1 SGs.
(a)(b)Table 11
Effect analysis of coordinated optimization toKpr and Kir of the CCC.
δapt,woop (°)δapt,wop (°)λapt (%)δ2,15.245.200.76δ5,15.905.850.84JwoopJwopλJ (%)40.4440.120.79


## 5.4. The Oscillations Suppression for Scenarios with Different Faults

In actual engineering, the fault scenarios are complex. Considering the research focus of this paper, the set of fault scenarios is simplified to four scenarios, i.e.,Scenarios 1 to 4. φIDN of buses 9, 21, 26, and 28 are 3.7, 1.8, 4, and 2.4, respectively [43]. The severity indices of four scenarios calculated by (35) are 0.311, 0.151, 0.336, and 0.202, respectively. According to the sensitivity evaluation indices, the parameters to be optimized for the four scenarios are the same, i.e. KA, KF, TF, Vref, and IDC,ref. The proposed scheme is implemented in Scenarios 2 to 4, and the optimization results for different scenarios (an, op and the weighting value a') are shown in Table 12. As shown in Table 13, a' has a more acceptable effect for different scenarios when compared with an,op. Consequently, the applicability of the proposed scheme is verified with the severity indices of different faults.Table 12
The optimization under scenarios with different faults.
Scenarioan,opKAKFTF (s)Vref (p.u.)IDC,ref (p.u.)12010.50070.05170.90040.99982200.38280.50100.050.90260.99993200.94050.50530.05160.91510.99994199.78380.50130.05550.90170.9999a'200.61240.50180.05180.90430.9998Table 13
The comparison of the suppression effect ofa' and aω,op.
Scenarioλofv (%)an,opa′115.1114.57220.6420.56318.7318.76413.1813.06


## 6. Conclusion

To suppress the angular oscillations among SGs in the AC/HVDC power system, this paper proposes a coordinated optimization of the parameters and set-points of the constant current controller of the HVDC and the exciter of the SC based on the improved trajectory sensitivity. To differentiate periods of optimization, an improved analytical model of the trajectory sensitivity is proposed by introducing the step function to quantify the influence of the parameters on the angular oscillations. Based on the indication indices calculated by a mean trajectory sensitivity, the critical control parameters and the critical set-points of the constant current controller of the HVDC and the excitation system of the SC are newly decided to be optimized. With the objective of the minimal deviation of power angle differences, a coordinated multiparameter optimization model considering the periods of optimization is proposed and solved with the interior point method. Testing the proposed model in different scenarios reveals a suppression effect on the angular oscillations.Some conclusions are found(1)
The accuracy of the improved TS is validated by comparing it with the perturbation method. According to the numerical analysis, the maximum absolute error is only 5.22 × 10−5, and the maximum relative error is only 1.82%.(2)
Based on the improved analytical expression of the TS, the proposed optimization scheme is suitable for control parameters and set-points with different periods of the optimization.(3)
Optimization toKA, KF, TF, and Vref can further improve the oscillations and shorten the convergence time, which has a better effect than optimization to KA, KF, and TF.(4)
Optimization toVref with IDC,ref has a better effect on the oscillations than optimization to KA, KF, and TF.(5)
By introducing the severity index of different scenarios and calculating the weighting value of the optimization results, the applicability of the proposed optimization model is verified.(6)
Compared with independent optimization, optimization to the HVDC and the SC has a better suppression effect.(7)
Compared with optimization to control parameters only, optimization of both control parameters and set-points has a better suppression effect. The adjustment range of set-points of the HVDC is reduced.In the following study, oscillation suppression with parameter optimization at the sending and the receiving ends of the HVDC may be studied based on the proposed algorithm.



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