6.1. Description of the Studied System
In order to test the effectiveness and robustness of the proposed strategy, the WSCC system was used. This system contains three machines, nine buses, three transformers, three loads and six transmission lines. All system machines were equipped with PSS and AVR. A single line diagram of the WSCC is depicted in
Figure 4. As shown in this figure, a renewable source was added to the original system at bus number 2.
The generator data, as well as the parameters of the excitation systems which were adopted in this section, are tabulated in
Table 2 and
Table 3, respectively. Note that all system data are taken from [
26].
From Equation (74), it can be seen that the model of a single machine with PSS and AVR has eight differential equations, in which four nonlinearities can be examined. These nonlinearities appear in the equations related to state variables , and . For a power network with m machines, the number of nonlinearities which will lead to fuzzy rules is . For example, for a three-machine power network, the number of fuzzy rules is 212. However, it is very difficult to deal with this large number of fuzzy rules and to establish adequate fuzzy controllers for the power network. In order to overcome this problem, a new strategy for reducing the number of fuzzy rules is applied. To do this, the studied power network is firstly subdivided into m independent subsystems, where m is the number of generating units. Each sub-system is equivalent to a generating unit in series with the Thevenin equivalent as seen from this unit. Therefore, a sub-system can be modeled as a voltage source Vth in series with a resistance Rth and a reactance Xth. Parameters Vth, Rth and Xth are calculated after resolution of the load flow problem at various renewable energy penetration levels (REL). In this way, each sub-system can be treated uniquely, thereby reducing the number of nonlinearities.
The second step of this strategy aims to represent these sub-systems by TSF models. To this end, the significances of the four nonlinearities, i.e., , , and , of each sub-system are ranked based on their limits and variation ranges from one REL to another. Note that limits and of nonlinearities are calculated using Equations (75)–(78). Nonlinearities with non-significant variations are assumed to be uncertain parameters. This makes it possible to apply the proposed UFC in a more effective way. Note that the controller gains of the UFC laws are determined using the LMI-based framework.
The main steps for the implementation of the proposed strategy are presented in
Figure 5. In the figure, switches
are used to switch from one operating condition to another.
6.3. Implementation of the Fuzzy Logic Controller
In this study, the WSCC system is decomposed into SMIB systems. Then, a fuzzy decentralized control is applied for each machine in the presence of AVR and PSS regulators. The block structure of the SMIB system incorporating the proposed controllers is depicted in
Figure 6.
Let
and
. Thus, the values of the premise variables are as tabulated in
Table 6.
Referring to
Table 6, it can be noted that the variation ranges of
N1 and
N2, from one case to another, are small, as are the differences between their minima and maxima. Consequently, they are considered to be uncertain parameters, and only nonlinearities
N3 and
N4 are retained.
Nonlinearities
N1 and
N2 are transformed into uncertainties using Equations (79) and (80).
where
where
Values
α1m,
α2m,
α1r and
α2r for the three machines and under the three investigated cases are tabulated in
Table 7.
In this study,
is adopted, which leads to the creation of 2
2 rules. Therefore, the closed-loop system as shown in
Figure 7 can be described by the following equation.
where
;
denotes the identity matrix of order 8.
can be rewritten as follows:
where
can be rewritten by the following equation. Note that
Be is a certain and constant vector.
The controller gains are calculated using the relaxed stability LMI conditions for various operating conditions of the power grid with the aim of ensuring the global stability of the closed-loop system.
6.4. LMIs Results
By resolving inequality (31) using LMI tools where
is expressed by Equation (32), matrix
can be determined. This will make it possible to calculate the controller gains by applying the following equation, derived from Equation (28).
Therefore, the controller gains Khi, for all machines are as follows. Note that returns a square diagonal matrix with the elements of vector V on the main diagonal.
Machine 1:
Kh1 = diag([−20.0841; −6.3786; 6.5044; 0.0054; 17.6024; −0.0717; −0.0005; 0.6927])
Kh2 = diag([−15.5438; −6.4481; 6.7043; 0.0055; 17.7752; −0.0719; −0.0005; 0.6869])
Kh3 = diag([−20.2411; −6.4473; 6.7237; 0.0055; 17.7867; −0.0722; −0.0005; 0.6864])
Kh4 = diag([−15.4935; −6.4211; 6.7269; 0.0054; 17.7249; −0.0721; −0.0005; 0.6883])
Kh1 = diag([−47.5461; −17.4030; 16.6695; 0.0112; 48.9141; −0.0498; −0.0017; 0.7061])
Kh2 = diag([−43.6529; −17.6256; 17.1635; 0.0111; 49.4496; −0.0499; −0.0017; 0.7056])
Kh3 = diag([−48.1855; −17.5658; 17.1148; 0.0111; 49.3019; −0.0499; −0.0018; 0.7025])
Kh4 = diag([−43.3297; −17.5041; 17.0545; 0.0111; 49.1423; −0.0500; −0.0018; 0.7003])
Kh1 = diag([−106.6596; −41.0408; 40.5350; 0.0205; 115.5535; −0.0285; −0.0014; 0.7218])
Kh2 = diag([−104.2250; −41.6401; 41.7002; 0.0203; 117.0049; −0.0284; −0.0014; 0.7230])
Kh3 = diag([−108.3986; −41.4404; 41.5048; 0.0202; 116.4695; −0.0286; −0.0014; 0.7189])
Kh4 = diag([−103.2000; −41.2640; 41.3033; 0.0202; 115.9994; −0.0287; −0.0015; 0.7156])
Machine 2:
Kh1 = diag([−338.6814; −71.1361; 112.3736; 0.1210; 477.5795; 0.0045; 0.0023; 0.7392])
Kh2 = diag([−334.0817; −71.5548; 113.4050; 0.1203; 480.0420; 0.0046; 0.0023; 0.7403])
Kh3 = diag([−341.2589; −71.6018; 113.4861; 0.1204; 480.3293; 0.0045; 0.0023; 0.7397])
Kh4 = diag([−334.3295; −71.6225; 113.4874; 0.1204; 480.3763; 0.0045; 0.0023; 0.7392])
Kh1 = diag([−486.3532; −112.3096; 103.5166; 0.1248; 761.9326; 0.0048; 0.0023; 0.7354])
Kh2 = diag([−478.9745; −112.7118; 104.2701; 0.1242; 764.2205; 0.0048; 0.0023; 0.7363])
Kh3 = diag([−489.1084; −112.7933; 104.3534; 0.1243; 764.7052; 0.0048; 0.0023; 0.7358])
Kh4 = diag([−479.3030; −112.7768; 104.3349; 0.1243; 764.5921; 0.0048; 0.0023; 0.7353])
Kh1 = diag([−386.1554; −90.9984; 76.7770; 0.0930; 622.1562; 0.0045; 0.0021; 0.7315])
Kh2 = diag([−378.6095; −91.3404; 77.4429; 0.0924; 624.0694; 0.0045; 0.0021; 0.7327])
Kh3 = diag([−388.6921; −91.4230; 77.5151; 0.0924; 624.5182; 0.0045; 0.0021; 0.7320])
Kh4 = diag([−378.9129; −91.4099; 77.4975; 0.0924; 624.3996; 0.0045; 0.0021; 0.7314])
Machine 3:
Kh1 = diag([−55.0533; −8.5458; 3.4539; 0.0044; 111.0722; 0.0197; 0.0064; 0.6825])
Kh2 = diag([−49.7522; −8.6696; 3.5643; 0.0044; 112.6474; 0.0201; 0.0065; 0.6841])
Kh3 = diag([−55.9195; −8.6353; 3.5505; 0.0044; 112.1611; 0.0199; 0.0065; 0.6817])
Kh4 = diag([−49.3622; −8.6072; 3.5373; 0.0044; 111.7751; 0.0199; 0.0065; 0.6797])
Kh1 = diag([−76.8399; −13.7223; 4.7242; 0.0056; 157.0066; 0.0106; 0.0031; 0.6922])
Kh2 = diag([−70.3368; −13.9090; 4.8679; 0.0056; 159.0961; 0.0108; 0.0032; 0.6940])
Kh3 = diag([−77.9120; −13.8686; 4.8528; 0.0056; 158.5171; 0.0107; 0.0032; 0.6919])
Kh4 = diag([−69.9745; −13.8326; 4.8413; 0.0056; 158.1775; 0.0107; 0.0032; 0.6906])
Kh1 = diag([−54.8479; −8.3011; 3.5848; 0.0041; 105.3049; 0.0164; 0.0052; 0.6849])
Kh2 = diag([−48.2928; −8.4332; 3.7099; 0.0041; 106.9853; 0.0168; 0.0054; 0.6869])
Kh3 = diag([−55.8142; −8.4013; 3.6976; 0.0041; 106.5875; 0.0167; 0.0053; 0.6846])
Kh4 = diag([−47.9430; −8.3743; 3.6839; 0.0041; 106.2313; 0.0166; 0.0053; 0.6826])
6.5. Nonlinear Time Simulations and Discussion
In order to test the applicability and robustness of the proposed UFC, a severe disturbance (described below) is applied to the system. Firstly, a high level of penetration of renewable energy occurs at
t0 = 1 s. Then, the amount of renewable energy is reduced after 20 milliseconds, i.e., at
t = 1.02 s. The nonlinear simulation results obtained using the UFC together with AVR and PSS are compared with the case when only conventional AVR and PSS (AVR+PSS) are applied.
Figure 8,
Figure 9,
Figure 10,
Figure 11 and
Figure 12 show the system responses corresponding to rotor angle variation, speed deviation, internal voltage deviation, field voltage and the output voltages of PSSs, respectively. From
Figure 8, it is clear that the rotor angles of the machines converge much more rapidly upon the desired values when the proposed UFC is applied as opposed to conventional AVR and PSS. Moreover, it can be seen that the convergence time for the rotor angle of G3 is much greater than those of generators G1 and G2. This is due to the high inertia constant of generator G3 compared to G1 and G2, as shown in
Table 2. The desired rotor angles of generators G1, G2 and G3 are 1.0251, 1.2109 and 0.13836 radians, respectively.
Figure 9 shows that the variations of angular speed with the fuzzy controller are more efficient than those with conventional regulators. In fact, it can be clearly seen that the oscillations of the fuzzy regulator are greater than those of the conventional regulator at the moment of application of the disturbances. This proves that the proposed fuzzy-based regulator responds significantly to disturbances, and is more sensitive to any fault than conventional regulators. The significant response of the proposed controller can be also observed in
Figure 10, where the internal voltages underwent significant variations during the disturbances. For this reason, the field voltages of the machines contained pulses.
Figure 10 and
Figure 11 also show that the proposed UFC provides good performance and achieves a rapid convergence to equilibrium in the system in comparison with the AVR and PSS regulators, despite its sensitivity to disturbances.
Figure 12 shows that the output signals
Vpss of the fuzzy regulator reach their steady state values faster than the signals generated by the conventional PSS regulator and with the minimum number of oscillations. Moreover, the significant reaction of the fuzzy controller during variations in operating conditions can be seen, which proves its efficiency.
Figure 13 illustrates the variations of the fuzzy control signals related to the three machines. Significant oscillations were observed when the disturbance was applied. This proves that the regulator responds effectively following a change in the system operating conditions.
Therefore, the proposed TSF modeling approach could achieve accurate representations of the studied multimachine system with a reduced number of fuzzy rules. Indeed, the nonlinear time-domain simulation proved that the proposed UFC works effectively at the moment of the application of disturbances, which is not the case for conventional stabilizers. This is due to the fact that the UFC output signal acts on all state variables, while conventional controller signals act directly on the excitation system.