3.6 Case Study
3.6.1 Optimal wide area control with limited measurements
The wide area controller performance with limited states is demonstrated by consid- ering the New England 39-bus system. The New England 39-bus system is shown in Fig. 3.4. The system consists of 10 generators. Each generator is equipped with PSS. The exciters are of the IEEE-ST1A type. The generators are represented by
means of the two-axis model and the network is represented by means of the phasor model. Loads are represented by means of a constant impedance model. The detailed dynamic and small signal modelings of system components are explained in [50]. The information about the line, generator data and PSS parameters of the New England 39-bus system are provided in [53]. The base case power flow result is obtained from the bus data provided in [53]. The eigenvalues of the open loop system are shown in Fig. 3.5. It can be observed that some of the open loop system eigenvalues are lying behind the 10% damping line.
Figure 3.5: Eigenvalues of the open loop system.
In this particular study, the wide area controller with limited states is designed for two different cases. For the first case, it is assumed that PMUs are installed at all the generator buses. However, none of the PMU is equipped with non-linear state estimator. For the second case, each PMU is assumed to be equipped with non-linear state estimator. However, there are only limited number of generator buses where PMUs are placed. For both the cases, same number of states are considered. In the conventional LQR control design, the values of Q matrix are assigned based upon the participating states of inter-area modes of the system. High values are assigned to the most participating states and low values are assigned to the less participating states. The R matrix is chosen as identity matrix to share the control cost burden equally to all the generators.
Case 1: PMUs are Placed at All the Generator Buses without Non-linear State Estimator
For this particular case, state information is available for all the generators. However, for each generator, only the frequency and rotor angle states are observable. There- fore, in the structurally constrainedH2-norm optimization proposed, large penalty is applied for the entries in the columns (of state feedback gain matrix) corresponding to the states other than the frequency and rotor angles states of the generators. For the entries in the columns corresponding to the frequency and rotor angles states of the generators no penalty is applied (i.e., penalty factor is set to zero). The structure of the corresponding Case 1 state feedback gain matrix is shown in Fig. 3.6.
The closed loop eigenvalues of system for the ideal case and Case 1 are plotted in Fig. 3.7. Here, the “ideal case” refers to the scenario in which all the system states are observable. For the ideal case, the state feedback controller is designed by means of conventional LQR optimization. Compared to the ideal case, the eigenvalues of the closed loop system in Case 1 slightly move towards to the imaginary axis. However, all the eigenvalues are lying inside the 10% damping line.
States
Generator
Figure 3.6: Structure of the feedback gain matrix with limited states in Case 1.
Figure 3.7: Comparison of closed loop eigenvalues in Case 1 with the closed loop eigenvalues for the feedback controller with all the states.
Figure 3.8: Dynamic responses of generator speeds (COI referred) in Case 1 for the outage of Line 14-15.
The effectiveness of the controller based upon only frequency and rotor angle states of all generators is further verified in time domain. The time domain simulation of system is carried out by considering the outage of Line 14-15. The time variations of the rotor speeds (w.r.t COI) of Generators 1 and 10 with and without state feedback controllers are plotted in Fig. 3.8. The performance of the state feedback controller designed for Case 1 is to some extent inferior to the performance of the ideal state feedback controller. However, significant improvement over the open loop system is still observed.
Case 2: PMUs are Placed at Limited Number of Generator Buses with Non-linear State Estimator
For this particular case, state information is available only for limited number of generators. However, all the states of a generator (for which state information is available) are observable. The proposed structurally constrained H2-norm optimiza- tion technique is used to obtain the required feedback gain matrix structure. Here, PMUs are assumed to be placed near Generators 2, 5, 7, 8, and 9. The structure of
the corresponding Case 2 state feedback gain matrix is shown in Fig. 3.9.
Generator
States
Figure 3.9: Structure of the feedback gain matrix with limited states in Case 2.
Figure 3.10: Comparison of closed loop eigenvalues in Case 2 with the closed loop eigenvalues for the feedback controller with all the states.
Figure 3.11: Dynamic responses of generator speeds (COI referred) in Case 2 for the outage of Line 14-15.
The closed loop eigenvalues of system for the ideal case and Case 2 are plotted in Fig. 3.10. The time variations of the rotor speeds (w.r.t COI) of Generators 1 and
10 with and without state feedback controllers are plotted in Fig. 3.8 (for the outage of the same line). It can be easily observed that the performance of the wide area controller designed for Case 2 is far superior compared to the performance of the wide area controller designed for Case 1.