Hopf Bifurcation Control of Power Systems Nonlinear Dynamics Via a Dynamic State Feedback Controller—Part II: Performance Evaluation

Editor's Notes

• #3: The SNB is the maximum operability point of the system. The stationary branch of solutions located after the SNB are not in operational region of the system. Hopf bifurcations introduce instabilities in the stationary branch of solutions before the occurrence of SNB. These instabilities are due to pairs of complex conjugate eigenvalues of the system that cross the imaginary axis to the right side of the s-plane.
• #4: First, the nonlinear dynamics of the system is constructed using analytical saturation model. Inclusion of saturation helps in capturing some nonlinear dynamics that otherwise would have been neglected from the system. Then, from the observable states of the system (angle of the generators in the multimachine case and d-component of the stator current in the single-machine case) feedbacks have been taken and the dynamics state feed back controller is designed based on the later-described methodology. However, Hopf bifurcations can lead to unstable periodic orbits which would lead to chaotic behavior and ultimately voltage collapse (from literature). Mathematically, the controller manipulates the dynamics such that the Hopf bifurcations occurring on the stationary branch of solutions of the system before SNB, will be shifted to the inoperable region of the system; in other words, the Hopf bifurcations would be “eliminated” from the operational region of the system until it reaches SNB.
• #5: The flowchart describes how the control should be applied in 6 steps: 1-derive the nonlinear dynamics of the power system with incorporation of saturation phenomena. 2-find an equilibrium for the system and analyze stability (SEP). 3-perform bifurcation analysis using MATCONT and derive the following: equilibria structure of the system after the SNB + location and number of Hopf bifurcations before the SNB. 4-if only one Hopf bifurcation, then choose one point from the equilibria structure after SNB and design the controller only with one control gain vector (K1). Recall that for single-machine case we control the d-axis current of the generators and for multimachine we control the generator angle (delta). If two Hopf bifurcations, then choose two points from the equilibria structure after SNB and design the controller with both control gain vectors (K1 and K2). 5-The controller does NOT change the equilibria structure. But it MIGHT alter the stability of the initial equilibrium point (initial SEP). So we derive the Jacobian of the controlled system, and the we calculate the numerical Jacobian at the initial EP derived at step 2. We also should add the controller equilibria to the vector of the SEP from step 2 (because we also have controller states now which also should be at equilibrium for derivation of the stationary branch of solutions for the controlled system). 6-we perform eigenvalue analysis on the Jacobian of the controlled system at the initial SEP. if the EP from step 2 with addition of the controller equilibria is still stable, then the point/points chosen from the equilibria after the SNB are feasible point/points to relocate the Hopf bifurcation(s) to. If not, we should update the points from step 4, and do it until the controlled system is stable at the initial EP.
• #6: For the single machine case, we construct all the dynamics and derive a stable equilibrium to initialize the bifurcation analysis in MATCONT. Then we derive the location of HBs, SNB and also obtain the stationary branch after the SNB for choosing feasible points to relocate the Hopf bifurcations to. Then we design the controller and evaluate the stability of the initial equilibrium. If stable, the chosen points from the stationary branch after SNB are feasible; if not, we update the chosen points. After finding the feasible points, we perform another bifurcation analysis to evaluate the control performance in terms of relocation of bifurcations. We need to apply the control to a multi-machine case to demonstrate the scalability of the framework. In this case, we assemble the system dynamics in PSAT. Then for detection of the Hopf bifurcations, we modify the CPF routine such that we get a stable equilibrium initially, detect HBs, SNB and also get the stationary branch of solutions after SNB so that we can relocate the HBs to those. Then we design the controller, and evaluate the performance of the controller through bifurcation analysis for the controlled system in MATCONT.