Interaction analysis between induction motor loads and STATCOM in weak grid using induction machine model

Abstract

Static synchronous compensators (STATCOM) can be used as a reactive power compensation for induction motor (IM) loads due to its effective control and good compensation. Terminal voltage control (TVC) in a STATCOM has a great influence on voltage dynamics which is a significant concern in a system with many IM loads. This paper investigates the interaction between IM loads and TVC in a STATCOM under weak grid conditions from the viewpoint of active and reactive power flow. A corresponding induction machine model is proposed, based on which the interaction mechanism between IM loads and TVC in a STATCOM can be intuitively understood. It is shown that the negative damping component provided by TVC in a STATCOM can lead to system oscillation instability. Grid strength and the inertia constant of the induction machine affect the extent of such interaction. Time-domain simulation results of IM loads connected to an infinite system through a long transmission line, with STATCOM compensation implemented in MATLAB/Simulink, validate the correctness of the analyses.

1 Introduction

Induction machines are an important element in power systems. About 60%–70% of industrial energy consumption is due to induction motor (IM) loads [1]. Owing to the reactive power consumption characteristics of IM loads in power systems, reactive compensation devices is necessary. For effective control and good compensation performance [2, 3], static synchronous compensators (STATCOM) are widely employed for reactive power compensation. Research on induction machines and STATCOMs is mostly concerned about low voltage ride through characteristics [4, 5], improvement of voltage sag due to starting of high power induction motor [6] and stabilizing of sub-synchronous resonance [7]. However, the interaction between IM loads and a STATCOM has not gained enough attention. Actually, terminal voltage control (TVC) in a STATCOM has a great influence on voltage dynamics which is a significant concern in a system with heavy IM loads. This phenomenon has not been studied extensively. Furthermore, long distance transmission, intensive use of transmission and the increasing number of IM loads make the system overstressed [8]. The AC grid strength which is described by the short circuit ratio (SCR) becomes weak. Since the bus voltage is more sensitive to TVC under weak grid conditions, interactions between IM loads and a STATCOM will be strengthened. Therefore, it is necessary to analyze the complex interaction between IM loads and TVC in a STATCOM under the weak grid conditions.

Although previous work has rarely focused on this topic, there is still some similar research about the interaction between IM loads and grid-connected equipment with voltage control. Earlier studies found that the interaction between IM loads and the excitation system of synchronous generators would lead to system oscillations [9,10,11]. Time domain simulations, bifurcation theory and modal analysis are used in those studies to uncover the factors that affect the interaction. Nowadays, with the development of renewable energy, HVDC, FACTs, etc., more and more power electronics equipped with intricate control strategies are integrated into power system, making the power system dynamic more complicated [12]. Thus, the interactions between power electronics and IM loads tend to attract more attention [13,14,15]. In [13], the voltage oscillation between IM loads and DFIGs in power distribution networks is reported. The studies of [14, 15] show that the interaction of IM loads and a droop-controlled voltage source converter (VSC) in a microgrid can lead to low-frequency and medium-frequency oscillations. All this research has focused on the interaction between IM loads and grid-connected equipment with voltage control, and resulted progress in this area. However, there are few reports about the interaction analysis of IM loads and a STATCOM. Therefore, this paper concentrates on that. In order to make it easy to comprehend the internal mechanism of the interaction from physical viewpoint, the flow of active and reactive power is considered. Based on this, a corresponding induction machine model is proposed and used to reveal the internal mechanism.

The paper is structured as follows. Section 2 introduces the basic understanding for interaction. Section 3 first presents the proposed induction machine model, then the STATCOM and the network are modeled. Section 4 develops the modal analysis. Section 5 investigates the effects of TVC in the STATCOM, the grid strength and the inertia constant of the induction machine on system stability. Simulation studies are carried out in Sect. 6 to validate the above analyses. Section 7 summarizes the conclusions.

2 Basic understanding for interaction between IM loads and STATCOM

Figure 1 shows an IM load connected to an infinite grid through a long transmission line with a STATCOM compensation. The STATCOM and IM are connected to the point of common coupling (PCC). A fixed compensation capacitor for the IM is also located at the PCC bus.

Fig. 1

Induction motor loads connected to infinite grid through long transmission line with STATCOM compensation

In Fig. 1, U _dcref and U _dc are the voltage reference value and the measured voltage at the DC capacitor; U _tref and U _t are the terminal voltage reference and measured voltage; i _dref and i _qref are the current reference with respect to d axis and q axis; i _d and i _q are the corresponding measured values of the current in the filter inductor; E _d and E _q are the outputs of the PI regulator; E is the internal voltage vector of the VSC.

The corresponding single line diagram of the system is shown in Fig. 2. U _g is the magnitude of infinite grid voltage; L ₃ is the equivalent inductance of the transformer and transmission line; P _e  + jQ _e are active and reactive power injected into the IM; U _t and θ _ut are the terminal voltage magnitude and phase; C _f and R _d are fixed compensation capacitance and resistance; P _ec  + jQ _ec are active and reactive power flowing from the fixed capacitor to the PCC bus; P _eg  + jQ _eg are active and reactive power delivered to PCC bus by the infinite grid; P _e2 + jQ _e2 are active and reactive power sent to PCC bus by the STATCOM; L ₁ and R ₁ are filter inductance and resistance in the STATCOM.

Fig. 2

Single line diagram of system

We assume that the public coordinate viz. d axis is in phase with the infinite grid voltage. For clearly comprehension of the system, the power P _eg  + jQ _eg and P _ec  + jQ _ec (shown in Fig. 2) are combined into the power P _e1 + jQ _e1, that is P _e1 + jQ _e1 = (P _eg  + jQ _eg ) + (P _ec  + jQ _ec ). Thus, based on the flow of active and reactive power in Fig. 2, the block diagram of the overall system is shown in Fig. 3.

Fig. 3

Block diagram of overall system

It can be seen from the Fig. 3 that the input electric power of IM consists of two parts: ① the power from network, which is affected by the terminal voltage vector, the transmission line and the characteristics of capacitor; ② the power from the STATCOM, which is closely related to the terminal voltage magnitude and the corresponding controller.

The interaction between the IM and the STATCOM becomes pronounced under weak grid conditions. On one hand, the power injected into the IM comes from the network and the STATCOM. If the network operates under weak grid conditions then a large amount of reactive power is provided by the STATCOM, and strong interaction between IM and STATCOM occurs. While on the other hand, also under weak grid conditions, the dynamics of terminal voltage magnitude are mainly affected by TVC in the STATCOM, which leads to the fact that the active power transferred from the network to IM relies on the dynamic performance of TVC. Thus, the interaction between the IM loads and STATCOM, which includes the reactive power exchange between IM and STATCOM and the effects of TVC on IM-received active power, make it necessary to investigate the internal mechanism.

3 System modelling for analysis

Based on the basic understanding of the interaction shown in Section 2, this section focuses on the system modelling which will be modeled from the viewpoint of active and reactive power flow. One thing that should be noted is that the problems of concern have a relatively slow timescale, which means only quasi steady state of the IM, power electronics equipment and network is considered, while transient processes are ignored [16]. The modal analysis shown in Section 4 will justify this simplification.

3.1 Proposed induction machine model

For the timescale we are concerned with, IM loads in most power system analyses are treated as variable impedances, which is a good approach for understanding IM loads separately. However, as mentioned in [17], the conventional IM model output variables, like the “slip” in a certain bus, lack physical significance in a power system. Moreover, this conventional model does not lay stress on the voltage magnitude. Actually, voltage stability in IM loads, when they cover an extensive area supplied by a large power system, is an issue of significant concern, and is related to the reactive power balancing. Thus, in order to avoid these defects and to accommodate the above mentioned interaction analysis, this paper proposes an IM model suitable for in-depth analysis.

Based on Fig. 3, the proposed IM model selects both active and reactive power flowing into the IM to be the inputs. The outputs are the terminal voltage phase and magnitude. It is noteworthy that the terminal voltage magnitude and phase are not determined only by the IM since the input active and reactive power are supplied from the external grid. However, the information of terminal voltage vector can be deduced from the characteristics of the IM. Therefore, the terminal voltage vector can be selected as the interface between the IM and the external grid. The IM model is derived as follows.

The steady-state equivalent circuit of an IM is shown in Fig. 4. x _sσ and x _rσ are the stator and rotor leakage reactances; x _m is the magnetizing reactance; s _slip is the slip; R _r is the rotor reactance.

Fig. 4

Steady state equivalent circuit of induction machine

The active and reactive power consumed by the IM depend on the slip and the terminal voltage magnitude as shown in Fig. 4. From the modelling perspective mentioned above, active power and reactive power are chosen as the model inputs, and the linearized relationship between P _e , Q _e and s _slip , U _t based on Fig. 4 is given by:

$$ \left[ {\begin{array}{*{20}c} {\Delta U_{t} } \\ {\Delta s_{slip} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {K_{1} } & {K_{2} } \\ {K_{3} } & {K_{4} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\Delta P_{e} } \\ {\Delta Q_{e} } \\ \end{array} } \right] $$

(1)

The formulae for the coefficients are given in Appendix A.

Rotor motion is described by the following differential equation:

$$ \Delta P_{e} - \Delta P_{m} = 2H\frac{{{\text{d}}\Delta \omega_{r} }}{{{\text{d}}t}} $$

(2)

where ΔP _e and ΔP _m are the deviation of electrical and mechanical power; Δω _r is the deviation of rotor speed; H is the inertia constant.

During the swings, the frequency of stator voltage deviates from the synchronous frequency. This is reflected in the following linearized equation:

$$ \Delta \omega_{s} = \frac{1}{{1 - s_{slip0} }}\Delta \omega_{r} + \frac{{\omega_{s0} }}{{1 - s_{slip0} }}\Delta s_{slip} $$

(3)

where Δω _s and Δs _slip are the deviations of the terminal voltage frequency and slip; ω _s0 and s _slip0 are their respective initial states.

The integral of stator frequency is the terminal voltage phase from the physical point of view. The linearized relation is given by:

$$ \omega_{s0} \Delta \omega_{s} = \frac{{{\text{d}}\Delta \theta_{ut} }}{{{\text{d}}t}} $$

(4)

where Δθ _ut is the deviation of the terminal voltage phase.

Based on the equations above, the linearized induction machine model is illustrated in Fig. 5. The motor convention makes ΔP _e positive and ΔP _m negative. The mechanical power is assumed to be constant in this paper.

Fig. 5

Block diagram of linearized induction machine model

3.2 STATCOM model

The basic function of a STATCOM is to inject reactive power to systems, which is realized using a voltage source converter. The control system of a VSC consists of: ① an inner current loop to control the filter inductor current and limit the valve current during disturbances; ② a phase locked loop (PLL) to detect the phase of the terminal voltage and establish the dq0 coordinate; ③ a direct-voltage control loop; ④ a terminal voltage control loop to make the PCC voltage follow the reference. The detailed control system is illustrated in Fig. 1.

On account of the timescale we are concerned with, the fast dynamic control loops in the VSC are ignored, and the following assumptions are made:

• 1) The inner current control loops are ignored, due to their fast response, which means the current through the filter inductor is equal to the current reference.

• 2) The phase locked loop is also ignored, so the PLL can instantaneously detect the phase of terminal voltage.

• 3) The DC voltage is constant and the STATCOM does not provide active power, so the DC voltage controller is ignored and the d axis current reference is zero.

Therefore, on the timescale of this modeling, only the TVC has a significant effect on system stability. The selection of the controller and parameters is based on the following considerations. Both an integral controller and a proportional integral controller can be employed in terminal voltage control, and the selection of the control parameters depends on the grid environment [18]. For example, in [19], a single VSC connects to an infinite grid and it is found that the terminal voltage controller’s integral gain can be increased to make the system operate at its theoretical minimum SCR limit. In [20], it is derived from a VSC system that a faster outer voltage control loop will produce a less robust system. In [21], based on an entire MMC-VSC system, tuning the terminal voltage controller (TVC) parameters shows that the integral gain of the ac voltage controller needs to increase to achieve stability requirements. From the above, this paper selects an integral controller for investigation and the range of integral gain is between 20 and 120 [22]. The TVC is described as follows:

$$ i_{qref} = ( - U_{tref} + U_{t} )\frac{{k_{ivt} }}{s} $$

(5)

Under these assumptions, the active and reactive power (viz., P _e2 and Q _e2) flowing from the STATCOM to the PCC bus are given by:

$$ \left\{ \begin{array}{ll} P_{e2} = 0 \hfill \\ Q_{e2} = - U_{t} ( - U_{tref} + U_{t} )\frac{{k_{ivt} }}{s} \hfill \\ \end{array} \right. $$

(6)

3.3 Network model

The active and reactive power, namely P _eg and Q _eg , flowing from the infinite grid to PCC bus are:

$$ \left\{ \begin{array}{ll} P_{eg} = \frac{{U_{t} U_{g} }}{{x_{3} }}\sin ( - \theta_{ut} ) \hfill \\ Q_{eg} = \frac{{U_{t} U_{g} \cos ( - \theta_{ut} ) - U_{t}^{2} }}{{x_{3} }} \hfill \\ \end{array} \right. $$

(7)

where x ₃=ω _s L ₃ represents the equivalent reactance of the transformer and transmission line; ω _s is the frequency of the infinite grid.

The fixed compensation capacitor provides a contribution to the total reactive power. The consumed active power is ignored as the resistance R _d in capacitor is very small. The power from capacitor is:

$$ \left\{ \begin{array}{ll} P_{ec} = 0 \hfill \\ Q_{ec} = - \omega_{s} C_{f} U_{t}^{2} \hfill \\ \end{array} \right. $$

(8)

In summary, the power transferred to the IM is:

$$ \left\{ \begin{aligned} P_{e} = \underbrace {{P_{eg} \,+\, P_{ec} }}_{{P_{e1} }} + P_{e2} \hfill \\ Q_{e} = \underbrace {{Q_{eg} + Q_{ec} }}_{{Q_{e1} }} + Q_{e2} \hfill \\ \end{aligned} \right. $$

(9)

where P _e1 and Q _e1 defined above are the sum of the power from the transmission line and the capacitor.Linearizing (6)–(9) around an operating point gives:

$$ \Delta P_{e} = K_{5} \Delta U_{t}\,{+}\,K_{6} \Delta \theta_{ut} $$

(10)

$$ \Delta Q_{e} = K_{7} \Delta U_{t} + K_{8} \Delta \theta_{ut} $$

(11)

The detailed expressions of K ₅ ~ K ₈ are given in Appendix A.

4 Modal analysis

Modal analysis is a well-developed tool to determine the dominant mode that governs the system stability. Figure 6 is the eigenvalue spectrum of the power system under weak grid conditions (viz. the SCR is 1.3) with different integral gains (k _ivt ) of TVC in the STATCOM. In order to ensure accuracy, the calculation of the pole-zero map is based on the detailed mathematic model which includes the VSC control loops and the fast dynamics of network. The parameters are shown in Table 1. In Fig. 6, the dominant pole moves from the right half plane to the left with increasing k _ivt . That means the system changes from unstable to stable. Therefore, TVC in the STATCOM has a crucial impact on the dominant mode. Moreover, the frequencies of the dominant eigenvalue in all the cases are around 10.5 rad/s (1.6 Hz), and the damping is close to 0 when k _ivt is small. The participation factors of state variables shown in Table 2 suggest that the dominant mode is mainly influenced by the IM and the TVC parameters. The direct voltage control (DVC) loop and the PLL do not mainly participate in shaping of the dominant mode and hence they can be ignored in the following analysis.

Fig. 6

Eigenvalue spectrum of system with different TVC integral gain k _ivt

Table 1 System parameters for modal analysis

Table 2 Participation of state variables in mode (k _ivt  = 120)

The eigenvalue spectrum and participation factors indicate that there is highly dynamic interaction between the IM and TVC in the STATCOM, and also that TVC mainly affects the damping. Therefore, similarly to the model used by Heffron and Phillips [23], the system model can be simplified as follows.

Considering (10), (11) and the linearized IM model shown in Fig. 5, the system small-signal analysis model is shown in Fig. 7. By regarding Δθ _ut as the input, ΔP _e and Δs _slip as the outputs, the shaded area shown in Fig. 7 can be simplified into transfer functions G ₁(s) and G ₂(s).

Fig. 7

Small-signal analysis model

Because the initial state of slip s _slip0 is close to zero and the initial state of terminal voltage frequency ω _s0 is close to 1 p.u., the small-signal analysis model shown in Fig. 7 can be transformed into Fig. 8. G ₁(s) = Δθ _ut /ΔP _e , G ₂(s) = Δθ _ut /Δs _slip . Detailed expressions for these transfer functions are given in the Appendix A.

Fig. 8

Simplified analysis model for analyzing effect of TVC in STATCOM on IM

From Fig. 8, the rotor speed ω _r accelerates or decelerates when there is an unbalance between IM input and output power. Then small variations of stator speed Δω _s change the terminal voltage phase θ _ut which influences the feedback electric power P _e and slip s _slip through G ₁(s) and G ₂(s) respectively. Hence, the system stability is determined by G ₁(s) and G ₂(s).

5 Stability analyses

This section elaborates several aspects of the small-signal stability of the system based on the simplified analysis model developed above. TVC in the STATCOM can influence the stability of the IM through its contribution to the damping component of the system. Meanwhile, the grid strength increases or decreases this interaction, as will be further discussed below. The effect of the inertia constant representing the mass of induction machine is also investigated.

5.1 Effect of TVC on system stability

The characteristics of G ₁(s) and G ₂(s) are closely related to the integral gain of TVC in the VSC. With constant grid conditions, G ₁(s) and G ₂(s) are investigated with different integral gains. Since small-signal stability is due to insufficient damping of oscillations to a great extent, we focus on the damping torque provided by G ₁(s) and G ₂(s) together. G(s) denotes the transfer function from the stator speed to electromagnetic power:

$$ G(s) = 2H\omega_{s0} G_{2} (s) + \frac{{\omega_{s0} }}{s}G_{1} (s) $$

(12)

The effect of TVC on system stability is investigated through the magnitude and phase frequency characteristics of the transfer function G(s).

Bode plots of G(s) with different integral gains of TVC are given in Fig. 9a. Corresponding phasor diagrams of G(s) is illustrated in Fig. 9b. Due to the motor convention, the damping component is negative when G(s) is in phase with Δω _s and vice versa. The negative damping region indicates that the electromagnetic power increases when the stator speed increases, and the change of electromagnetic power is adverse to stator speed stability. On the other hand, the positive damping region indicates that the electromagnetic power decreases as the stator speed increases, and the change of electromagnetic power is beneficial for stator speed stability. As shown in Fig. 9a, when the integral gain of TVC shifts from 30 to 120, the phase frequency response corresponding to the rotor oscillation frequency, namely the shadowed part of the Bode plots, increases and finally becomes larger than 90°. Correspondingly, the damping component at the rotor oscillation frequency changes from negative to positive. Moreover, the magnitude frequency response also increases as the integral gain increases.

Fig. 9

The effect of TVC on system stability, (a) Bode plots of G(s) with different integral gain of TVC, (b) phasor diagram of G(s) at the rotor oscillation frequency

In summary, the damping component increases with the increase of the integral gain. Note that the integral gain cannot increase without limit due to constraints of coordinated control. The PLL and the DVC loop restrain the upper limit of integral gain of TVC. It is quite a complex question that is beyond the scope of this paper.

5.2 Effect of grid strength on system stability

The effect of the grid strength on system stability is to increase or decrease the interaction between IM loads and TVC in the STATCOM. A stronger grid has a greater influence on the dynamics of terminal voltage magnitude. This makes the system more stable and defends against small-signal disturbances.

Figure 10 illustrates the variation of the real part of the minimum system eigenvalue, as well as the corresponding damping ratio, with respect to different integral gains and SCRs, with the SCR representing the grid strength. The shaded area in the figure corresponds to negative real part of the minimum eigenvalue and therefore positive damping ratio which indicates stable operation region. The damping ratio scale in this figure is an approximate calculation. It can be seen from Fig. 10 that the curve moves down with increasing grid strength, which means the system can tolerate a larger range of integral gain.

Fig. 10

Real part of minimum system eigenvalue and corresponding damping ratio with respect to different integral gains and SCRs

5.3 Effects of inertial constant on system stability

The inertia of induction machines is generally close to 0.5 s [8, 24]. In order to investigate the effects of the inertia constant on system stability, an open loop transfer function has been derived based on Fig. 8. G _open (s) denotes the open loop transfer function and is given below

$$ G_{open} (s) = \frac{{ - G_{1} (s)}}{{2Hs\left( {\frac{s}{{\omega_{s0} }} - G_{2} (s)} \right)}} $$

(13)

The inertia constant H only appears in the denominator of (13). Thus, the phase of the open loop transfer function is constant with respect to the inertia constant, while increasing the inertia constant will decrease the amplitude frequency gain and this stabilizes the system. In addition to changing the system stability, the inertia constant also affects the oscillation frequency which can be observed in the simulation results.

Figure 11 shows the effect of the induction machine inertia on the Bode plot of G _open (s) where k _ivt is 50. These Bode plots are sufficient to evaluate the system stability because there is no right-half-plane pole of G _open (s). It is clear from the Bode plots that the phase frequency responses are identical with varying inertia constant. As the inertia constant decreases, the point where the amplitude of G _open (s) intersects with 0 dB shifts from left to the right, and the phase corresponding to this point gradually passes through the value of 180°. That means the system becomes more unstable as the inertia constant decreases.

Fig. 11

Bode plots of open loop transfer function with varied inertia constant

6 Simulation results

Based on the topology shown in Fig. 1, time-domain simulation was conducted in MATLAB/Simulink to validate the analyses described in the previous section. Parameters have been listed in Table 1.

Figure 12 compares induction machine rotor speed responses for a STATCOM equipped with TVC and constant q axis current control. The IM rotor speed is unstable with TVC and stable without it. Interaction between IM loads and TVC in the STATCOM may lead system to instability in a weak grid.

Fig. 12

Comparison of rotor speed responses for STATCOM equipped with TVC and constant q axis current control (SCR = 1.3)

Figure13 is the rotor speed responses of induction machine with different integral gain of terminal voltage control. A small disturbance in grid occurs at 6 s. It can be seen from the figure that with the increase of integral gain, the stability of the system gets better. Simulation results in Fig. 13 are in accordance with the conclusion from Fig. 9. As a whole, these results illustrate that in weak grid, the damping of system oscillations is affected by terminal voltage control. Selecting suitable TVC parameters in such cases needs careful consideration.

Fig. 13

Rotor speed responses of induction machine with different integral gains used for terminal voltage control

Figure 14 shows the influence of integral gain on rotor speed stability for different grid strengths. With integral gain of 60 in Fig. 4a the system is stable with SCR = 1.6, while with integral gain of 30 in Fig. 4b the system is unstable with the same SCR. With increasing grid strength, the system stability gets better, but rotor speed oscillations still last for several seconds. Overall, the simulation results are in accordance with the theoretical analysis.

Fig. 14

Rotor speed responses compared with different grid strengths

Figure 15 shows the rotor speed responses with different inertia constants. When the inertia constant is 0.5 the system is unstable after a disturbance in grid, while for higher inertial constants it is stable. These simulation results support the conclusion obtained from Fig. 10.

Fig. 15

Rotor speed responses with varying inertia constant

7 Conclusion

This paper analyzed the interaction between induction motor loads and a STATCOM in a weak grid. An induction machine model was proposed to investigate the internal mechanism that causes this interaction. The results show that the stability of such systems, containing IM loads and a STATCOM, under weak grid conditions is determined by the terminal voltage controller parameters. As a consequence, it is necessary to reconsider the parameters of the VSC terminal voltage controller in systems containing substantial IM loads. Overall conclusions are as follows.

• 1) Terminal voltage control has a negative damping effect on stability of such systems. Decreasing the integral gain of the TVC deteriorates the system stability.

• 2) The grid strength determines the extent of interaction. With decreasing grid strength, as measured by the short circuit ratio, the negative damping component provided by the TVC increases. Therefore, the integral gain parameter of the TVC in a weak grid should be larger than that in a stiff grid for stable operation.

• 3) The inertia constant of the induction machine determines the frequency range of the interaction. Increasing the inertia constant decreases the oscillation frequency and improves the system stability.

Appendix A

Detailed formulae for linearized IM model coefficients and transfer function are as follows:

$$ \begin{aligned} K_{1} &= \frac{{U_{t0} (P_{e0}^{2} - Q_{e0}^{2} )}}{{2P_{e0} (P_{e0}^{2} + Q_{e0}^{2} )}} \\ &\quad -\frac{{U_{t0} }}{{2s_{slip0} }}\frac{{(R_{r}^{2} - s_{slip0}^{2} L_{rr}^{2} )\left( {R_{r}^{2} + s_{slip0}^{2} L^{\prime}L_{rr} } \right)L_{ss} }}{{(R_{r}^{2} + s_{slip0}^{2} L_{rr}^{2} )\left( {2P_{e0} s_{slip0} L^{\prime}L_{rr} L_{ss} - Q_{e0} R_{r} L_{m}^{2} } \right)}} \\ \end{aligned} $$

(A1)

$$ \begin{aligned} K_{2} &= \frac{{U_{t0} Q_{e0} }}{{(P_{e0}^{2} + Q_{e0}^{2} )}}\\ &\quad +\frac{{U_{t0} }}{2}\frac{{(R_{r}^{2} - s_{slip0}^{2} L_{rr}^{2} )L_{m}^{2} R_{r} }}{{(R_{r}^{2} + s_{slip0}^{2} L_{rr}^{2} )(2P_{e0} s_{slip0} L^{\prime}L_{rr} L_{ss} - Q_{e0} R_{r} L_{m}^{2} )}} \\ \end{aligned} $$

(A2)

$$ K_{3} = - \frac{{R_{r}^{2} L_{ss} + s_{slip0}^{2} L^{\prime}L_{rr} L_{ss} }}{{2P_{e0} s_{slip0} L^{\prime}L_{rr} L_{ss} - Q_{e0} R_{r} L_{m}^{2} }} $$

(A3)

$$ K_{4} = \frac{{R_{r} s_{slip0} L_{m}^{2} }}{{2P_{e0} s_{slip0} L^{\prime}L_{rr} L_{ss} - Q_{e0} R_{r} L_{m}^{2} }} $$

(A4)

$$ K_{5} = - \frac{{U_{g0} \sin (\theta_{ut0} )}}{{x_{3} }} \, $$

(A5)

$$ \, K_{6} = - \frac{{U_{t0} U_{g0} \cos (\theta_{ut0} )}}{{x_{3} }} $$

(A6)

$$ K_{7} =\frac{{U_{g0} \cos (\theta_{ut0} ) - 2U_{t0} }}{{x_{3} }}{ + }\frac{{k_{ivt} }}{s}(U_{tref} - 2U_{t0} ) - 2\omega C_{f} U_{t0} $$

(A7)

$$ K_{8} = - \frac{{U_{t0} U_{g0} \sin (\theta_{t0} )}}{{x_{3} }} $$

(A8)

$$ G_{1} (s) = \frac{{K_{6} + K_{2} (K_{5} K_{8} - K_{7} K_{6} )}}{{1 - (K_{1} K_{5} + K_{2} K_{7} )}} $$

(A9)

$$ \begin{aligned} G_{2} (s) &= \frac{{(K_{5} K_{8} - K_{7} K_{6} )(K_{2} K_{3} - K_{4} K_{1} )}}{{1 - (K_{1} K_{5} + K_{2} K_{7} )}} \\ &\quad + \frac{{(K_{3} K_{6} + K_{4} K_{8} )}}{{1 - (K_{1} K_{5} + K_{2} K_{7} )}} \\ \end{aligned} $$

(A10)