Strong Resonance Investigation and Suppression in PMSG integrated Power Systems

Permanent magnet synchronous generators (PMSGs) with full converters have been widely used in wind power generation due to its superior flexibility and controllability. However, under some circumstance, the oscillation modes of PMSG (POMs) may excite strong resonance with the electromechanical oscillation modes (EOMs) of the power system that degrades the power system small signal stability. In this paper, A two-open-loop subsystem model is firstly derived to analyze the oscillation modes. Then the POMs are investigated with modal analysis, the relationship between POMs and related controllers are clarified. On this basis, the strong resonance between PMSG and the external power system is revealed and identified. Furthermore, a five-step parameter tuning method is proposed to relocate the position of POM as well as suppress the strong resonance. Both modal analysis and time-domain simulations validate the effectiveness of the proposed method.


Introduction
In the last several decades, wind power generation grows very fast over the world due to the requirements of carbon reduction. To achieve better controllability, various control schemes have been proposed in the wind power generation to improve dynamic performance [1][2][3][4][5]. Full converter-based wind power generation (FCWG) [1,6,7], which uses full scale converters to connect the variable speed wind generators (VSWGs) with the power grid, becomes very popular these years due to the wide application and cost reduction of power electronics. Among FCWG, the direct-driven permanent magnet synchronous generator (PMSG) is a very promising wind generation technology, which may dominate the wind power market in the next decade.
However, unlike the conventional synchronous generators (CSGs), the integration of wind power generators does bring problems, such as inertia reduction, dynamic interaction complexity, protection inaccuracy [6,[8][9][10]. The unexpected concern on power system stability also draws great attention in this field [6,11,12]. Since FCWG has completely different physical structure from CSGs, its impact mechanism on power system small signal stability needs to be further clarified. Early works in this field have made some valuable progress. A model reduction strategy proposed in [13] can reduce the model complexity and improve the computational efficiency of a large-scale power system with increasing penetration of wind power generation. [14] presents a stationary αβframe impedance model to predict the stability impact of PLL and its coupling effect. The mechanism of the system instability conducted in [15] uses "positive feedback effect" between the electrical subsystem and the control subsystem to explain the dynamic process of instability brought by power converters.
As for other published works, the impact of the integration of FCWG on power system small signal stability with can be assessed from two aspects, i.e., the power flow changes and the dynamic interaction. [6] performed modal analysis to evaluate the overall impact, whilst [11] and [16] investigate the impact of VSWGs on small signal stability by using damping torque analysis, which can be a considerable improvement in understanding the overall impact.
In this paper, the strong modal resonance that threats the power system small signal stability is studied. A fivestep parameter optimization method is proposed to tune the open-loop oscillation mode of PMSG. Since the POM tuning objective function is quite complex, which cannot be obtained analytically, the particle swarm optimization (PSO) algorithm is applied to optimize controller parameters of PMSG. PSO algorithm can evaluate the quality of the solution through fitness and has demonstrated its advantages in solving practical problems such as easy implementation, high precision and fast convergence [17].
The rest of this paper is organized as follows. Section 2 introduces the modeling and linearization of PMSG and the external power system. Section 3 presents the strong modal resonance between PMSG and the external power system. A POM tuning method is also proposed to optimize the controller parameters to suppress the strong modal resonance. Section 4 gives a case study to demonstrate the proposed optimization method, while Section 5 concludes this paper.

Linearization of PMSG
A typical PMSG with full converters is shown in Fig. 1. There are four main parts: 1) The permanent magnet synchronous generator; 2) The machine side controller (MSC) and the associated control system; 3) The DClink, the grid side converter (GSC) and the associated control system; 4) The synchronous reference frame phase-locked loop (SRF-PLL) which keeps the synchronization with the power system.
The linearization equations for PMSG under x-y coordinates system can be expressed as below Hence, the transfer function of PMSG can be obtained,

Linearization of External Power System
The state space model for N generators in the rest of the power system can be written as The network equation is V is the SG terminal current injection and bus voltage at the connecting point; I ,V is the output current and voltage at the PCC for PMSG, which is modeled as a current source; Hence, the network equation can be rewritten as Hence, the linearization equations of the external power system can be expressed as where ) .

Closed-loop Model of Entire Power System
Combine Eq(1) and Eq (7)

Strong Modal Resonance
Based on the modeling in Section 2, the total system can be divided into two subsystems. The bus voltage variation at PCC can be regarded as the input of PMSG, and the output is current injection variation that interacts with the external power system. Denote g as the critical electromechanical oscillation mode (EOM) of the open-loop power system. Denote h as the open loop-mode of PMSG (POM) which has the largest impact on the critical EOM of the power system. The distance between g and h is h g . Based on Eq(1) and Eq (7), the transfer function of PMSG and the power system can be expressed as Denote ̂g and ̂h as the solutions of Eq(9) corresponding to g and h . According to open-loop modal resonance theory proposed in [18], when d , i.e. h g , ̂g and ̂h tend to move the right side and the left side of g and h . The direction of movement of ̂g and ̂h depends on the relative location of g and h . For example, if h is at the right of g , the system electromechanical mode may move to the right side, the PMSG mode may move to the left side, and vice versa, as shown in Fig. 2.
From the discussion above, it can be concluded that the relative position of POM and EOM may lead to strong modal resonance, and one of the closed-loop modes may deteriorate.

Resonance Suppression with POM Tuning
To reduce the negative effect on system small signal stability, the POM tuning method is proposed to move the POM away from EOM. There are five major steps in POM tuning: The objective function () i fx is proposed to minimize the critical electromechanical power oscillation in the external power system, which can be measured at critical bus. Once the minimal () i fx is achieved, the power oscillation is suppressed at the optimal level. Since the objective function is not explicit, the traditional PSO algorithm in [17]. is implemented to solve this constrained optimization problem.

Case Study
An example system is built as shown in Fig. 3. A PMSG is connected to the single machine infinite bus (SMIB) system.
The five-step tuning is demonstrated as follows. Fig. 3. PMSG-SMIB system Step 1: Identify EOM Consider the SMIB system, the open loop EOM is identified to be -0 3148 4 5190i through modal analysis.

Step 2: Identify POMs
Modal analysis is carried out on the open loop PMSG model with power flow profile at PCC, and thus the typical modes can be identified by employing participation factors with associated variables, as shown in Table 1.

Step 3: Resonance Mode Identification
From Table 1, it is easy to find that the EOM is very close to the one of the POMs, i e POM8 (the PLL mode) Then POM tuning is to optimize POM8

Step 4: Parameter optimization
The PSO program in [17] is used to optimize the error function is the angular speed variation from the steady-state value , which is used to judge the dynamic performance of the power system. The objective of this constrained optimization problem is to search the optimal parameters of so as to find the minimum of the objective function in the time period ( , ) with the time step of 0.01s.
After POM tuning, the optimal parameters of the PLL controller are identified to be By changing the parameters of the PLL controller to the optimal parameters. The PLL mode moves from -0 3145 4 5138i to -0.4461+3.8346i, and thus the closed-loop EOM moves from ̂g -0 1664 4 3213i to ̂g -0.3769+4.4679i, the damping ratio increases twice from 3.85% to 8.41% indicating that the modal resonance is suppressed.

Step 5: Simulation verification
Time domain simulations are performed to verify the effectiveness A large disturbance simulation is performed in the PMSG-SMIB system At t 0 1s, a three-phase to earth short circuit occurs at Bus , and subsequently clears after 100ms The simulation results are shown in It can be seen that the modal resonance is avoided after POM tuning. The dynamic performance of the power system is improved, and the small signal stability is enhanced. In this paper, the strong modal resonance between PMSG and the external power system is analyzed. A POM tuning based on parameter optimization is proposed to suppress the negative impact of modal resonance on power system small signal stability. Once the objective POM is identified, the constrained optimization problem can be determined to tune the related controller parameters. The PSO algorithm is employed to optimize the dynamic performance error function and achieve a larger damping ratio of the closed-loop power system. The five-step case study has demonstrated the effectiveness of the proposed method. Therefore, better damping characteristics of EOM can be achieved with the proposed POM tuning method without auxiliary damping equipment installed, which is more economical for system operators.