Sensitivity Analysis

To verify whether or not an NL model suffices to represent WTG in power sys- tem analysis, sensitivity analysis related to several parameters is performed.

Reactive-power set-point

Qi is the injected reactive power at bus D ; thus, Qref = Qi for the WTG

and QN L = −Qi for the NL. In Table 2.4, the loading and the complex pair

of eigenvalues at the HB point are presented when Qi is varied from −0.2 to

+0.2 pu. The values separated by the slash correspond to the WTG and the NL, respectively. Observe that using the NL, fosc is an upper bound for the

base case. Moreover, using the NL, estimation of the HB point suffices, i.e., the values are close to the base case, just a little higher or lower depending on whether the reactive power is injected or absorbed.

WTG inertia

The inertia has minimal influence on the eigenvalues. The inertia is varied from 0.5 to 10 s. Dominant eigenvalue trajectories show almost no change. A slight decrease in loading is observed when the inertia is increased. Note that inertia is more influential during transient operation. The rotational motion

Figure 2.27: Dominant eigenvalues pathway using the WTG model for vwind = 12 m/s and vwind = 10 m/s.

Figure 2.28: Dominant eigenvalues pathway using the NL model for PN L= −0.2980 p.u. and PN L = −0.1723 p.u.

Table 2.4: HB point sensitivity with respect to Qi Qi Loading at HB point µ 0.2 1.039 / 1.045 ±j7.10 / ± j7.17 0.1 1.007 / 1.010 ±j7.88 / ± j8.01 0 0.972 / 0.972 ±j8.60 / ± j8.69 -0.1 0.935 / 0.932 ±j9.06 / ± j9.37 -0.2 0.897 / 0.889 ±j9.42 / ± j9.83

of a WTG acts as a filter of wind speed fluctuations. Consequently, if the gen- erator has a lower inertia, its power output would present more fluctuations, causing more oscillations in the system’s synchronous generators.

Static load’s parameters

Load modeling is complex because it is dependent on patterns of people’s behavior which are difficult to define. So far, it has been considered as an exponential load model with pv = qv = 0 (static constant power load). Due to load-model uncertainty, three new loads are used: an induction motor (IM) at half load, an IM at full load, and a room air conditioner (RAC) [41, 42]. In Table 2.5, the loading at the HB point and its associated pair of complex eigenvalues are presented. The results show a great agreement between the WTG and NL models.

Network’s parameters

Generally, wind power generation is located either on the sub-transmission or distribution side. In these cases, the network has an important resistance which may impact the analysis. The parameters of line 1 (Figure 2.23), which represent the sub-transmission side, will be doubled in order to account for a weaker connection to the grid. The resistance of line 2, distribution side, will be varied between 0.05 and 0.15 p.u. Once again, the results (Table 2.6) show a good agreement between the WTG and NL models.

Table 2.5: HB point and load parameters dependence

Load pv qv Loading at HB point µ

Base Case 0 0 0.972 / 0.972 ±j8.60 / ± j8.69 IM half load 0.2 1.6 1.185 / 1.187 ±j7.86 / ± j7.78 IM full load 0.1 0.6 1.068 / 1.069 ±j8.23 / ± j8.26 RAC 0.5 2.5 1.499 / 1.503 ±j7.52 / ± j7.16

Table 2.6: HB point and network parameters dependence R1+ jX1 R2+ jX2 Loading at HB point µ 0.03+j0.1 0.05+j0.1 1.101 / 1.101 ±j8.31 / ± j8.28 0.03+j0.1 0.1+j0.1 0.972 / 0.972 ±j8.60 / ± j8.69 0.03+j0.1 0.15+j0.1 0.861 / 0.861 ±j9.04 / ± j9.42 0.06+j0.2 0.05+j0.1 0.895 / 0.897 ±j9.30 / ± j9.08 0.06+j0.2 0.1+j0.1 0.808 / 0.809 ±j9.61 / ± j9.63 0.06+j0.2 0.15+j0.1 0.730 / 0.731 ±j9.80 / ± j10.28

Active and reactive power controllers’ parameters

The tuning of PI controllers can be considered an art which requires ex- perience and a detailed knowledge of the system under control. There are many tuning rules but usually the parameters must be adjusted through trial and error simulations [43]. Thus, proportional and integral parame- ters of the active and reactive power controllers are varied to observe the incidence of each of them on the system’s eigenvalues. To consider this variation, a multiplicative factor is considered as KP,i = f actor × KP,i,base

and KI,i = f actor × KI,i,base, where i = {1, 2, 3, 4}. KP,i,base and KI,i,base

correspond to the proportional and integral gains for the base case. It is observed that eigenvalues are practically insensitive to KI1, KI3, KP 1 and

KP 3 variations which are related to the slow loop of the active and reactive

power controllers. With respect to the parameters of the fast loop of the con- trollers, two cases are considered. In the first one, KI2 = f actor × KI2,base

and KP 2 = f actor × KP 2,base while KI4 and KP 4 are kept at their base

values (Figure 2.29). In the second case, KI4 = f actor × KI4,base and

KP 4 = f actor × KP 4,base while KI2 and KP 2 are kept at their base val-

Figure 2.29: Eigenvalues sensitivity with respect to KP 2 and KI2.

frequency of the modes. Higher parameters do not considerably decrease the frequency of the modes with respect to the base case. In the second case, an opposite behavior is observed. Lower parameters can decrease the oscillation frequency of the modes, making the system more stable. Parameters above those of the base case do not considerably modify the system’s eigenvalues. Remember that these parameters are related to the fast loop of the reactive power control and the set-point is set to zero. Thus, just an adjustment of these parameters can be beneficial for the system’s stability without requir- ing more power capability of the WTG. The loading at the HB point is 0.962 p.u. when factor=0.1. For other factor values, the loading at the HB point remains approximately the same (0.972 p.u.).

Note that when the eigenvalues’ pathway differs from that of the base case, due to a parameter adjustment, some WTG’s state variables, such as x2 and x4, participate in the unstable modes. Note that KP 2, KP 4, KI2

and KI4 do not change the equilibrium point having just incidence in the

eigenvalues pathway. In addition, a limit pathway of the eigenvalues has been observed at which the proportional and integral gains of the active and reactive power controllers do not change the system eigenvalues at all. For example, when KP 4 = 10 p.u. and KP 4 = 500 p.u., the same eigenvalues’

pathway is found. This limit pathway is located close to the base case path- way. This phenomenon has to be investigated in a future research. Note that the negative load model is a good approximation of this limit behavior.

With respect to the NL model, its validity for representing the power coming from the wind in power system stability analysis depends on the parameters of the active and reactive power controllers. For the base case, an NL model suffices, but when the controllers’ parameters are changed, especially those associated with the fast loop of the reactive power controller, an NL model gives a poor estimate of the system’s stability.

In summary,

a. Modal analysis is insensitive to HD variations (wind generator inertia).

b. Dominant eigenvalue pathways for the WTG and NL cases are similar when the base case parameters are used for the fast loop of the active and reactive power controllers. For any other value of these parameters, the validity of the NL model must be checked.

c. For the base case parameters of the active and reactive power controllers, the NL gives a good estimate of the system HB point.

d. When the parameters of the active and reactive power controllers are changed, the loading at the HB point does not change dramatically and the NL model makes a good estimation of it. However, the oscillation fre- quency may differ considerably depending on the controllers’ parameters.