Research on Control Inertia and Stability of PMSG

Abstract

With the large-scale development and application of offshore wind resources, the proportion of powered electronic equipment in power systems is increasing, which leads to the reduction of traditional mechanical inertia and system oscillation. To solve the above problems, this paper takes the grid connected converter of an offshore full power fan as the research object, deduces the “inertia” expression and physical significance of its inner and outer loop controller, and compares it with the mechanical inertia motion equation to obtain a unified mathematical expression. On this basis, combined with the control bandwidth and Rouse criterion, the discrimination method of system stability is proposed, and the correlation of the internal and external control parameters of the full power fan grid connected converter and its influence on the grid connected stability of offshore fans are explained according to the derived judgment expression. Finally, the correctness and scope of application of the theoretical analysis and judgment method are verified by PSCAD/EMTDC time-domain simulation software.

1. Instruction

The resource of wind is rich in China. Distributed wind power generation has the advantage of small volume and being close to the load center, making wind power an essential part of the distribution network in the future [1,2]. However, large-scale wind power would reduce the equivalent inertia of the distribution network, causing oscillation problems [3,4]. Recently, oscillation instability in different frequency ranges has occurred in many wind farms at home and abroad [5,6,7], which brings significant challenges to the safe and stable operation of the power system and has caused substantial economic losses and negative effects on society. Based on this, it is urgent to investigate the system oscillation problem caused by the Grid Connected Converter (GCC).

The Permanent Magnet Synchronous Generator (PMSG) with multiple magnetic poles does not need to rely on a gearbox for transmission, reducing maintenance costs. PMSG is widely used in distributed generation (DG) for this reason. The research shows that the various oscillation problems of PMSG mostly originate from changes in system parameters, and it is considered to be oscillatory instability caused by small disturbances [8]. At present, two methods are mainly adopted to analyze the small-signal stability: the small-signal average model and the impedance method. For example, reference [9] adopted the harmonic state-space (HSS) modeling approach to study the instability mechanism of the interconnected system. Reference [10] built impedance models of PMSG and HVDC rectifiers based on the harmonic linearization method and analyzed the possible unstable operating range of the system from the perspective of the frequency domain. Reference [11] adopted eigenvalue analysis theory to study the influence of parameters of the converter controllers and the short-circuit ratio of voltage source converter-based HVDC oscillatory modes. Reference [12] constructed a closed-loop frequency domain model of modular multilevel converter (MMC) and analyzed the effect of a zero-sequence circulating current controller on the stability of MMC-HVDC. Reference [13] utilized the state space method to model the dominant mode of PMSG. Reference [14] derived the impedance model of the grid-connected renewable energy system (GCRES). Reference [15] mainly focused on the impact of wind turbine grid-side controller’s parameters on its AC-side impedance characteristics. Reference [16] studied the influence of wind turbine output power and DC converter’s control parameters on system stability based on mode analysis.

However, the impedance method proposed in the above literature cannot accurately determine the stability margin of the GCRES with PMSG, and the solution of the full-order state matrix model is complicated. In particular, when the number of nodes is large, the full-order state matrix model cannot be solved.

To solve these problems, this paper proposes a reduced-order model to directly clarify the impact of PMSG controller parameters on small-signal stability based on the theoretical analysis of damping characteristics and inertia hysteresis in the controller. Lessons are drawn from the unit oscillation model and dynamics analysis research in a circuit and synchronous generator (SG) system. Since the dynamic characteristics of the PMSG are associated with the inertia of the controller, the stability limit consisting of the PI gains is derived. The derived stability limit is computationally simple and does not require a full-order model of the PMSG. Therefore, it can be used to approximately assess the instability risk of the PMSG caused by the inertia of the controller more conveniently. In addition, derivation of the stability limit reveals why the PMSG may become unstable due to the inappropriate selection of the controller’s parameters. Furthermore, the effectiveness of the proposed methodology is verified through PSCAD/EMTDC time-domain simulation.

Compared with existing works in the concerned research field, the major contributions of this paper can be summarized as follows:

(1)

We propose a reduced-order model to directly clarify the impact of PMSG controller parameters on small-signal stability based on the theoretical analysis of damping characteristics and inertia hysteresis in the controller.

(2)

The stability limit consisting of the PI gains is derived based on the inertia of the controller. The derived stability limit is computationally simple and does not require a full-order model of the PMSG. Therefore, it can be used to approximately assess the instability risk of the PMSG caused by the inertia of the controller more conveniently.

2. Inertia Analysis of Mechanical Motion

According to the macrophysics theory, inertia is the inherent property of an object by which it continues in its existing state of rest or uniform motion in a straight line. It represents the resistance degree of an object to changes in motion state. The classic mechanical motion shows that different objects have different inertia. The motion equilibrium point of a coupled system composed of different objects is not synchronized with the equilibrium point of the controlled state quantity, which will cause the coupled system to oscillate.

Similar to the mechanical motion, the control of power electronic components also has inertia, which manifests as the output lags the input. For example, inductive elements have magnetic inertia, which manifests as current lags voltage by π/2. Capacitive elements have electric field inertia, which manifests as voltage lags current by π/2 [17], as shown in Figure 1. Thus, when the second derivative of the current is equal to 0 near an equilibrium point of the system, the first derivative of the current is not equal to 0. This results in electromagnetic oscillation.

According to the Kirchhoff’s current law, the mathematical expression of a second-order circuit composed of the inductor and the capacitor can be derived from Figure 1, shown as follows:

$$\frac{d^2\Delta X}{d{t}^2}+G_1\frac{d\Delta X}{dt}+G_2\Delta X=0,$$

(1)

Equation (1) reflects the dynamic response of the state variable ΔX at the equilibrium point, as well as the inertia and damping characteristics of the system, where G₁ = 1/$LC$, G₂ = 1/$C$.

3. Derivation and Analysis of PMSG Controller Inertia

Due to the influence of inductive components, capacitance, and control delay, the inner loop current of the PMSG-GCC exhibits inertia characteristics, which may cause the system with PMSG to oscillate. The physical nature of this phenomenon is the hysteresis effect of feedback control. A typical single PMSG model considered in this paper consists of a closed-loop (outer and inner loop) controller and an LC filter, as depicted in Figure 2, where L_g and C_f are ac-side filter inductance and capacitor, respectively, and V_dc is the dc-side voltage. Phase voltages are expressed as v_a, v_b, and v_c, while phase currents are i_a, i_b, and i_c, and m_a, m_b, and m_c are the modulating (reference) signals for the pulse width modulation (PWM).

When the system is three-phase balanced, the synchronous generator voltage loop equation expression under the d-q axis is obtained as follows:

$$\{\begin{array}{l}{v}_{ds}=-R{i}_{ds}-{\omega }_r{\Psi }_{qs}+\frac{d{\Psi }_{ds}}{dt}\\ v_{qs}=-R{i}_{qs}+{\omega }_r{\Psi }_{ds}+\frac{d{\Psi }_{qs}}{dt}\end{array},$$

(2)

where ω_r is the electrical angular velocity of a synchronous generator; Ψ_ds and Ψ_qs are the d-axis and q-axis flux linkage, respectively.

When R ≈ 0, Ψ_ds, and Ψ_qs can be expressed as follows:

$$\{\begin{array}{l}{\Psi }_{ds}=-L_{ts}{i}_{ds}+L_{dm}(I_f-i_{ds})=-L_d{i}_{ds}+{\Psi }_r\\ {\Psi }_{qs}=-(L_{ts}+L_{dm})i_{qs}=-L_q{i}_{qs}\end{array},$$

(3)

where Ψ_r is rotor flux linkage; L_d and L_q are the stator d-axis and q-axis inductance, respectively.

The d-axis and q-axis cross-coupling variables can be regarded as disturbance terms, and mathematical compensation could be carried out in the control strategy when the control relationship between Δν_S and Δi_S is studied [18].

When PMSG is controlled using a zero d-axis current control strategy (ZDC), i_ds equals 0, and electromagnetic power output by PMSG can be obtained as follows:

$$P_e=\frac{3}{2}p{\Psi }_r{i}_{qs}{\omega }_r=\frac{3}{2}p{\Psi }_r{i}_s{\omega }_r,$$

(4)

It can be seen from Figure 3 that the relationship between the deviations of the external loop output and the power is obtained as follows:

$$k_{p_1}\Delta P_e+\frac{k_{i1}}{s}\Delta P_e=\Delta i_{sref},$$

(5)

where k_p1 and k_i1 are the proportional gain coefficient and integral constant, respectively, of the external loop PI controller. Take the derivative on both sides of Equation (5) as follows:

$$k_{p1}\frac{d\Delta P_e}{dt}+k_{i1}\frac{d\Delta P_e}{dt}=\frac{d\Delta i_{sref}}{dt},$$

(6)

Equation (7) can be obtained from the closed-loop control logic block diagram of PMSG.

$$(Z_l+k_{p2})/k_{i2}\frac{d\Delta i_s}{dt}+\Delta i_s=k_{p2}/k_{i2}\frac{d\Delta i_{sref}}{dt}+\Delta i_{sref},$$

(7)

where k_p2 and k_i2 are the proportional gain coefficient and integral constant, respectively, of the inner loop PI controller; Z_l is the impedance of the ac line, and Z_l = R + ω_rL_g.

Combining Equations (6)–(8) can be derived as follows,

$$\frac{(Z_l+k_{p2})}{k_{i2}}\frac{d\Delta i_s}{dt}+\Delta i_s=\frac{k_{p2}}{k_{i2}}(k_{p1}\frac{d\Delta P_e}{dt}+k_{i1}\Delta P_e)+(k_{p1}\Delta P_e+\frac{k_{i1}}{s}\Delta P_e),$$

(8)

Equation (8) can be simplified as follows:

$$\frac{d^2\Delta i_s}{d{t}^2}+G_3\frac{d\Delta i_s}{dt}+G_4\Delta i_s=0,$$

(9)

where:

$$\{\begin{array}{l}{G}_3=\frac{\epsilon (k_{p2}/k_{i2}+k_{p1}/k_{i1}-1/k_{i1})}{(Z_l+k_{p2}-\epsilon k_{p1}{k}_{p2})/(k_{i1}{k}_{i2})}\\ G_4=\frac{\epsilon }{(Z_l+k_{p2}-\epsilon k_{p1}{k}_{p2})/(k_{i1}{k}_{i2})}\\ \epsilon =\frac{3{\omega }_r{\Psi }_r}{2}\end{array},$$

(10)

Comparing Equations (1) and (9), the inertia equation considering damping could describe the characteristics of the output of the PMSG-GCC controller. Due to the inductance and capacitor affection and the controller delay, the current reference produced by the active power displays “inertia” characteristics similar to the synchronous generator, and it provides an oscillation mode for the system. As a result, there will be a certain deviation in the actual output, and the deviation signal of the controller cannot be converted entirely into control action in time. This is reflected in the inertia of the controller [19]. The analysis shows that the hysteresis effect could be expressed via the second-order equation. Hence, the PMSG control output characteristics are basically consistent with the inertia equation (Equation (10)).

According to the automatic control theory, after a state variable is disturbed, if the motion characteristics of the state variable are bounded and convergent, the system is stable. On the contrary, if the motion characteristics of the state variable are unbounded or divergent, the system will be unstable. If the motion characteristics of the state variable are bounded but not convergent, it is a critical steady-state. According to the Rouse–Hurwitz criterion, the system is stable when the coefficients G3 > 0 and G4 > 0. Based on this, the following criterion can be obtained:

$$\{\begin{array}{l}\frac{\epsilon (k_{p2}/k_{i2}+k_{p1}/k_{i1}-1/k_{i1})}{(Z_l+k_{p2}-\epsilon k_{p1}{k}_{p2})/(k_{i1}{k}_{i2})}>0 \mathrm{stable}\\ \frac{\epsilon (k_{p2}/k_{i2}+k_{p1}/k_{i1}-1/k_{i1})}{(Z_l+k_{p2}-\epsilon k_{p1}{k}_{p2})/(k_{i1}{k}_{i2})}<0 \mathrm{unstable}\end{array},$$

(11)

According to the automatic control principle, the PI parameter of PSMG-GCC usually satisfies the stable condition represented in Equation (11). The Equation (12) can thus be obtained as follows,

$$\{\begin{array}{l}{k}_{p1}<\frac{Z_l+k_{p2}}{\epsilon k_{p2}}, \mathrm{stable}\\ k_{p1}>\frac{Z_l+k_{p2}}{\epsilon k_{p2}}, \mathrm{unstable}\end{array},$$

(12)

The following conclusions can be drawn from the above formulas:

(1)

The stability of PMSG is related to the impedance of the AC line and the setting value of the controller.

(2)

When the inner loop parameter k_p2 increases, the inner loop control bandwidth increases and the stability threshold of the outer loop parameter k_p1 decreases with other conditions remaining the same.

To sum up, the stability criterion proposed in Equation (12) can estimate the stability domain of the outer-loop control parameters according to the inner-loop control parameters of PMSG and provide a reference for engineering applications.

4. Simulation Results

4.1. Effectiveness of Proposed Stability Criterion

This paper proposes a stability criterion to judge whether the PMSG system is stable. To verify the effectiveness of the proposed stability criterion, three cases representing the different stability situations of the PMSG system with different values of k_p1 and dominant characteristic root were assessed, and a comparative analysis is conducted based on these scenarios accordingly.

Specifically, in

Table 1. The parameters of PMSG outer loop control.

Case	kp1	λi	Bond	Stability Criterion
1	0.772	−5.84 + j227.21	1.187	Stable
2	1.186	−0.28 + j215.78	1.187	Critically stable
3	1.224	3.13 + j185.47	1.187	Unstable

, Bond is the stable boundary value calculated via Equation (12). Comparing the value of k_p1 and Bond, the stability of the PMSG system can be judged, and the stability criterion results of the system stability using the proposed method is shown in Table 1. In addition, the oscillation modes of the full-order model associated with the PMSG were computed to confirm the stability limit assessed above, and the eigenvalue of the full-order PMSG model with the poorest damping was λ_i. It can be observed that if k_p1 > Bond, the stability criterion shows that the PMSG is unstable, and λ_i is in the right half of the complex plane, which provides the same conclusion. Otherwise, k_p1 < Bond, the PMSG is assessed as stable through the stability criterion, and λ_i is in the left half of the complex plane. Moreover, when k_p1 = Bond, the stability limit shows that the PMSG is in the vicinity of critical stability. However, the eigenvalue λ_i does not pass through the imaginary, which indicates that the criterion is conservative.

The results of the nonlinear simulation are presented in Figure 4. During the simulation, after the PMSG ran stably for 0.1 s, we reduced the output of the wind farm by 0.03 p.u. The active power output by the PMSG system is illustrated in Figure 4.

It can be seen from Figure 4 that when the value of k_p1 is 0.772, the system is in a stable operation state. When k_p1 = 1.186, the system oscillates with equal amplitude, that is, the system is in a critically stable state, and the value of Bond calculated via Equation (12) is 1.186. As the value of k_p1 further increases (k_p1 = 1.224), the system diverges and becomes unstable. This indicates that as the value of k_p1 increases, the real part of the dominant characteristic root of the system decreases. When k_p1 exceeds the value of Bond, the real part of the dominant characteristic root is greater than 0, and the system becomes unstable. This result is the same as the judgment result obtained using the proposed method in this paper.

4.2. Stable Domain

According to the proposed stability criterion (Equation (12)), the stability boundary of the PMSG system with respect to the parameters k_p1 and k_p2 can be obtained. Based on this, the reference value of the stability domain of the PMSG system can be determined, as shown in Figure 5; the orange line is the stable boundary derived from the full-order model of PMSG, and the blue line is the stable limit calculated by the proposed stability criterion. Figure 5 also indicates that the stable limit calculated by the proposed method is within the stable limit derived from the full-order model, which confirms the correctness of the computation results presented in Table 1.

The area A and B in Figure 5 represent the parameter setting range within which the system is stable and unstable, respectively. The main reason for the system instability is that the value of k_p1 is set too large in area B.

Furthermore, the correctness of the stability limit is validated according to the trajectories of the oscillation modes on the complex plane, as shown in Figure 6. It can be seen that when the parameter k_p1 increases from 0.772 to 1.224, the poorest damping eigenvalue for the full-order PMSG model λ_i moves to the right half of the complex plane.

Generally speaking, when designing PMSG control parameters, we first determine the control bandwidth of the inner loop and then set the control bandwidth and related parameters of the outer loop according to the bandwidth and parameters of the inner loop [20]. The transfer function of the PMSG inner loop current control can be expressed as:

$$T_i(s)=V_{dc}\frac{k_{p2}/L_g\cdot s+k_{i2}/L_g}{s^2+(V_{dc}{k}_{p2}+2R)/L_g\cdot s+V_{dc}{k}_{i2}/L_g},$$

(13)

Let s = jω; also, the transfer function T_i(s) amplitude is set to 0.707, and the corresponding frequency ω is the bandwidth of the inner loop current control, which can be written as ω = ω_i = 2πf_i. The relationship between the parameters of the inner and outer loop can be expressed as follows:

$$\{\begin{array}{l}{k}_{p1}=L_g{\omega }_i^2/V_{dc}\cdot (\sqrt{1+4{\xi }_i^4}-2{\xi }_i^2)\\ k_{i1}=\frac{1}{V_{dc}\cdot (\sqrt{\sqrt{1+4{\xi }_i^4}-2{\xi }_i^2})/2L_g{\xi }_i{\omega }_i^{}}\end{array},$$

(14)

where ξ_i is the damping ratio, V_dc is the voltage value of the system DC side, and k_p1 and k_i1 are the reference values for the outer loop parameters calculated by the inner loop control bandwidth. The range of the inner loop bandwidth f_i was set as 50~280 Hz during the simulation process in this paper.

The above cases study evaluates the correctness of the small-signal stability limit derived in Equation (12), demonstrating that the gains of the PI controller have an impact on the stability of the PMSG. Furthermore, the proposed stability limit can be applied to the selection assessment of the gains of the closed-loop controller for the PMSG.

5. Conclusions

This paper investigated the small-signal stability issue of the PMSG connected to an AC grid based on the inertia analysis of the controller. According to the analysis of damping characteristics and inertia hysteresis in the closed-loop controller, a simple method to detect the instability risk posed by the PMSG controller’s gains was developed on the basis of the SG’s rotor motion and circuit model. Additionally, the analysis and stability limit presented in the study revealed that an inappropriate choice of parameters might provide instability risk to the grid-connected PMSG. In addition, the proposed method enhanced our understanding of the essential factors regarding the combined effect of damping characteristics and inertia hysteresis in the closed-loop controller on PMSG small-signal stability. According to the theoretical analysis and some cases study based on PMSG, the following conclusions were obtained:

(1)

The control stability of the PMSG system is related to the impedance of AC side line Z_l and the setting value of the controller.

(2)

When other conditions remain unchanged, as the inner loop parameter k_p2 increases, the inner loop control bandwidth increases, and the stability threshold of the outer loop parameter k_p1 decreases.

(3)

The proposed stability limit can be applied to the selection assessment of the gains of the closed-loop controller for the PMSG.