Offshore wind farm usually adopts permanent magnet synchronous motor (PMSM) as the main generation type due to its relative higher power rating compared with that of tradition doubly-fed induction generator (DFIG). As is shown in Figure 1, one single wind farm usually consists of up to 16 feeders which are connected to collection bus, for each feeder in wind farm, up to 12 WTGs are connected in series in each feeder. A typical simplified PMSM wind farm is selected to conduct aggregation simulation research in this paper, six feeders are linked to 35 kV collection bus with three or four WTG connected in series in one single feeder, altogether 34 WTGs are included in this wind farm.

As is shown in Figure 2, the working interval within the full wind speed range for PMSM can be divided into four operation modes, which are I-low constant speed mode, II-MPPT mode, III-high constant speed mode and IV-constant power mode respectively, x-axis represents the wind speed v, and y-axis refers to rotor speed ω _r.

When wind speed locates within [v _cutin, v ₁], the purpose is to control the rotor speed to stabilize at a low constant speed ω _{r_min}. With the increase of wind speed, PMSM then enters the MPPT stage, PMSM will operate at a particular rotor speed so as to catch the maximum power from wind, both tip speed ratio λ and rotor speed will be controlled at their optimal value λ _opt and ω _{r_opt.} When wind speed exceeds v ₂, rotor speed reaches the rated value ω _{r_max}, PMSM will then operate at high constant speed ω _{r_max} until exceeding the rated power, afterwards the pitch angle regulator will be activated and PMSM will operate at constant power mode.

Each PMSM WTG in the target research wind farm adopts traditional Vector Control (VC) strategy to decouple the control of WTG active power and reactive power, the control diagram for PMSM rotor side converter (RSC) and grid side converter (GSC)is shown in Figures 3, 4 respectively.

PMSM RSC adopts rotor flux linkage oriented vector control strategy, the control purpose of this control strategy is to control RSC active power and reactive power independently. As is shown in Figure 3, two cascaded control loops are included in RSC control strategy, the control object of active power control loop is PMSM rotor speed ω _r, so as to track the MPPT point given a specific wind speed. For reactive power control loop, the d-axis reference current i _sd ^* is set to zero, so that the PMSM electromagnetic torque can only be influenced by q-axis stator current i _sq, which is exactly the output of rotor speed closed loop.

The control diagram of PMSM GSC is shown in Figure 4, GSC is aimed at controlling dc-link voltage and balancing active power through dc-link capacitor, the GSC d axis reference current is obtained from the output of dc-link voltage regulator and the q axis reference current is get via reactive power regulator.

Based on the principles listed above, the equivalent parameters of aggregated PMSM WTG can be calculated with Eq. 1, 2, where a total of m WTGs with the same generation type are aggregated into one single WTG. { S e q = ∑ i = 1 m S i = m S n X s - e q = x s m r s - e q = r s m B c _ e q = m B c (1) { H t _ e q = H t H g _ e q = H g K s _ e q = K s D s _ e q = D s S T _ e q = m S T Z T _ e q = Z T m (2)

S _n is the rated capacity of one single WTG, x _s, r _s and B _c are the reactance, resistance and admittance of the WTG; m is the total number of WTGs in the same cluster; S _T is the rated capacity of the transformer; Z _T is the impedance of the transformer; H _t and H _g is the inertia time constant of the wind turbine and generator.

As for the aggregation of wind speed, there are a variety of aggregation methods to calculate the aggregated wind speed for wind farm. When the element of wake effect is not taken in account, then the aggregated wind speed is the same as the speed for single WTG: v e q = v (3)

However, when taking wake effect into consideration, then the calculation of aggregated wind speed is based on the power curve of WTG, firstly the power of each WTG in wind farm needs to be calculated and then get the average power, so that the aggregated wind speed can be drawn from the average wind speed as well as the given WTG power curve, assuming that the number of WTGs that will be aggregated is m, then the output power can be calculated as: P m = f ( v m ) (4)

Based on the power equation in Eq. 4, the aggregated wind speed can be calculated from: v e q = 1 m ∑ i = 1 m f − 1 ( v k ) (5)

For offshore PMSM wind farm, the ground capacitance is usually 20–25 times than that of overhead lines. So it cannot be ignored in the SMA analysis. Active power equivalent principle is applied in the aggregation of offshore submarine cable, as is shown in Figure 5, S _i and U _i represent the output power and terminal voltage of the ith WTG on the feeder respectively, C _i and Z_i represent the capacitance and impedance of the ith cable line on the feeder respectively, U _pcc represents the voltage at the PCC point. And S _eq and U _eq represent the equivalent output power and terminal voltage of the aggregated WTG separately, C _eq and Z _eq represent the equivalent capacitance and impedance of the aggregated cable line.

Based on the power equivalent principle, in order to calculate SMA submarine cable parameter, the power loss of the impedance on the nth cable at the end of the feeder, and the power loss of the impedance of the mth cable on the feeder can be expressed as: { Δ S n = ( S n − j 1 2 C n U n 2 ) 2 Z n / U n 2 Δ S m = ( ∑ i = m n − 1 ( S i − j 1 2 ( C i + 1 + C i ) U i 2 + Δ S i + 1 ) − j 1 2 C n U n 2 ) 2 Z m U m 2 (6)

Based on Eq. 6, the power loss of all submarine cables on one feeder, and the power loss on the equivalent circuit can be expressed as: { Δ S z total = Δ S 1 + Δ S 2 + … + Δ S n Δ S z e q = ( S e q − j 1 2 C e q U e q 2 ) 2 Z e q U e q 2 (7)

The expression of equivalence impedance Z _eq can be calculated based on the power equivalent principle of ΔS _ztotal = ΔS _zeq: Z e q = Δ S z total ( S e q − j 1 2 C e q U e q 2 ) 2 U e q 2 (8)

Besides, according to the principle that the reactive power consumed on the capacitor before and after the cable line equivalence is the same, the equivalent capacitor can be expressed as: C eq = ( C n U n 2 + C n U n − 1 2 + C n − 1 U n − 1 2 + … + C 2 U 2 2 + C 1 U 2 2 + C 1 U 1 2 ) ( U e q 2 + U 1 2 ) (9)

Equations 1–9 are the main calculation algorithm for PMSM wind farm single machine aggregation. When referring multi machine aggregation, wind speed for each WTG is selected as the main grouping index, all WTGs in a wind farm are grouped into four groups according to the WTG operation mode so as to minimize the aggregation error, since the operation mode is directly related to the wind speed and wind speed can be well estimated and calculated through various prediction algorithm, so it is easy to group one single wind farm based on wind speed for each WTG.

Simulation verification is firstly carried out between detailed model and SMA model under the wind speed of 7, 9, and 11 m/s respectively. A three-phase short circuit fault occurs in simulation and the fault duration is 0.625 s.

Figure 6A shows the comparison between the detailed model and SMA model under the wind speed of 7 m/s. The first panel is the voltage RMS value, indicating that grid voltage drops to 20% of the normal value and lasts for 625 ms, the rest panels are output active power, reactive power and current. Figure 6B is the average error curve of active power, reactive power and current at the PCC point. Based on Figure 6A and Figure 6B, it can be seen that the two curves overlap well with each other in steady state. Although there is a deviation in the dynamic process, it is not very large, which means that the SMA model can reflect the dynamic process of the detailed model.

Similar conclusions can also be found in Figure 7, 8, where the wind speed are 9 and 11 m/s respectively.

In order to investigate the aggregation accuracy of SMA model, we define the average error E _x as an important index to quantify the aggregation accuracy, as is shown as: E x = 1 n ∑ i = 1 n ( X e q i − X i ) X N × 100 % (10) where x is the parameter to be investigated, n is the number of sampling points, the subscript eq indicates the equivalent parameter of the aggregation model while N indicates the rated value. As is shown from Tables 1–3, the average error comparison before, during and after grid fault are listed respectively for further analysis.

It can be seen from Table 1 that for different wind speeds, when the collection line is a cable line, the results of the wind farm aggregation model match well with the detailed wind model, the average error of voltage, current and active power is within 0.12%, while the error of reactive power is within 0.05%, which can fully prove the correctness of the aggregation method proposed in this paper. As the wind speed increases, it can be seen that the magnitude of the error is also increasing. The reason is that the output parameter is increased with the increase of wind speed, but the base value used in the evaluation method is unchanged, which results in the increase of average error, but the increase is still within the normal range.

Comparing Table 1, Table 2, and Table 3, it can be seen that the average error during grid fault process is larger than that before and after grid fault. The main reason is that the line parameters are calculated at the rated power for the convenience of simulation, during grid fault process, the output active power is less than that in steady state, which causes the loss on the line of the aggregated model to be different from detailed model.

Wind speed of each wind turbine generator is randomly set to compare the aggregation accuracy between SMA and MMA algorithm. Wind speed for each WTG is selected as the main grouping index, as is shown in Figure 2, the wind speed of wind turbine generator determines the operation mode, so all wind turbine generators running in the same operation mode are grouped together to be aggregated into one WTG.

Simulation results are shown in Figure 9 to compare the aggregation accuracy between detailed model, MMA model and SMA model. A three-phase symmetrical short-circuit fault occurs in the simulation at t = 10 s, the amplitude of grid voltage drop is 20% of the rated voltage. The first panel of Figure 9A is the grid voltage while the rest panels are output active power, reactive power and current. Figure 9B shows the error curve of PCC point voltage RMS, output active power, output reactive power and output current. Combining the two figures, it can be seen that the MMA model has higher accuracy than SMA model, MMA can better reflect the dynamic process of the detailed model in the dynamic process.

From the above simulation results, although the response trend of SMA model, MMA model and detailed model are basically the same, there are still dynamic errors during the whole process, so that a quantitative analysis is still needed to calculate the error before and during grid fault. The results are shown in the following table.

Comparing Table 4 and Table 5, it can be seen that the MMA model has higher accuracy than SMA model. When wind turbines are in different operation modes, the SMA model is not sufficient enough to characterize the entire wind farm, the average error between MMA model and the detailed model are smaller than that between SMA model and the detailed model, showing that grouping wind farm according to wind speed can effectively improve the aggregation accuracy.