Hopf Bifurcation Control of Power Systems via Dynamic Feedback
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This two-part paper introduces a dynamic state feedback control law that guarantees the elimination of Hopf bifurcations (HB) before reaching the saddle-node bifurcations (SNB). Part I is devoted to the mathematical representation of the detailed system dynamics, investigation of HB and SNB theorems, and state feedback controller design. For purposes of dynamical analysis, the stable equilibria of the system is obtained. Then the control system is designed with the objective of preventing the voltage collapse before the SNB, such that the structural stability of the system is preserved in the stationary branch of the solutions. The controller aims to relocate Hopf bifurcations to the stationary branch of solutions located after SNB, eliminating the HB from normal operating region of the system. In order to evaluate the performance of the proposed controller, bifurcation analysis has been performed in Part II using single-machine and multi-machine test systems.
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Hopf Bifurcation Control of Power Systems via Dynamic Feedback
- 1. Hopf Bifurcation Control of Power System Nonlinear Dynamics Via a Dynamic State Feedback Controller: Part I Pouya Mahdavipour Vahdati and Luigi Vanfretti KTH Royal Institute of Technology Rensselaer Polytechnic Institute 1
- 2. Background • Voltage collapse from Saddle-Node Bifurcations. • Instabilities from Hopf Bifurcations. 2
- 3. Contributions • The saturation phenomenon is incorporates in the nonlinear dynamics. • A dynamic state feedback control law is for elimination of Hopf bifurcations before the saddle-node bifurcation (SNB). • The control law relocates Hopf bifurcations to stationary branch of solutions located after the SNB. 3
- 4. Overview of the Control Scheme and Contributions 4 Modeling Elimination of Two Hopf Bifurcations
- 5. Incorporation of saturation 5 • M. Despalatovi’c, M. Jadri´c, and B. Terzi´c, “Modeling of saturated synchronous generator based on steady-state operating data,” IEEE Trans. on Industry Applications, vol. 48, no. 1, pp. 62–69, 2012. • Model has been extended for power grid analysis (i.e. within a network), here SMIB is shown. • States and bifurcation parameters of the single machine system. • Results:
- 6. Controller Design • Detection of Hopf Bifurcations without analytical calculation of Eigenvalues. • System dynamics: • Jacobian at equilibrium: • Characteristic polynomial of Jacobian: • Construct H from the coefficients of characteristic polynomial: • Equivalent criteria for occurrence of Hopf bifurcations (no need to compute eigenvalues!) 6 Computationally inexpensive
- 7. Controller Design • Two Hopf bifurcations occurring at and . • Relocation of them to and located after the SNB. • The control law is formulated: • Controlled dynamics: 7 Theorem 1: The controller preserves the equilibria structure of the uncontrolled system. Theorem 2: The control law is stable for all .
- 8. Results for the single-machine system 8 Active load as bifurcation parameter Reactive load as bifurcation parameter
- 9. Conclusions • If saturation is not modeled: – Hopf bifurcations are not represented adequately. – Loadibility margins are under-estimated. • The controller successfully relocates Hopf bifurcations with the proposed algorithm to find feasible equilibria after the SNB. • The efficacy of the controller has been proved in the single-machine case. 9
Editor's Notes
- 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.
- 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.
- This is the general scheme of control. First the detailed nonlinear dynamics of the system is constructed. And then based on states that are observable (angle of the generators in the multimachine case and d-component of the stator current in the single-machine case) in practice, the control law is designed. The controller can eliminate two Hopf bifurcations from the stationary branch in our case studies. In previous studies, only one could have been controlled to prevent voltage collapse.
- The approach is based on this paper. The generator for the simulations of the single-machine case is the same as the paper above. The network equations have been rewritten to construct the single-machine system’s dynamics while incorporating saturation. The states and bifurcation parameters of the system are shown. In first bifurcation analysis which was performed in MATCONT, the bifurcation parameter is considered to be the active power consumption of the nonlinear load. In the second analysis, the bifurcation parameter is the reactive power consumption of the nonlinear load. In the first analysis (active power as bifurcation parameter), the results of our simulations show that in bifurcation analysis of the system, when we don’t consider saturation there is only one Hopf bifurcation in the stationary branch. But if we consider, two Hopf bifurcations are observed. Eigenvalues of the system are also shown. The role of integrating saturation is even bolder in the case of second analysis (reactive power as bifurcation parameter); the results of our simulations show that when we don’t consider saturation there no Hopf bifurcation in the stationary branch. But if we consider, there is one Hopf bifurcation. Eigenvalues of the system are also shown. These observations prove the role of integrating saturation phenomenon in obtaining detailed dynamics. Observe that in both cases, also the loadibility margin of the system is increased when the saturation is included in the model; in other words, the SNB occurs in higher active and reactive loadings of the system.
- Here we introduce a method for detection of Hopf bifurcations which does NOT reqiure analytical calculation of eigenvalues. Let x be the states of the system, F a nonlinear smooth function representing the dynamics and \mu the bifurcation parameter of the system. We calculate the Jacobian matrix of the system at an equilibrium like (x^e, \mu^e) and then derive the characteristic polynomial of it with P(\lambda, \mu^e). Then the we construct the matrix H which contains the coefficients of the characteristic polynomial. Then the criteria for Hopf bifurcations, which are equivalent to the criteria of Hopf bifurcation theorem, are derived as follows.
- We want to relocate the Hopf bifurcations from the stationary branch of solutions before the SNB, to the stationary branch located after the SNB, such that there are no bifurcations before the saddle-node. The control law is formulated as follows; u_{i} is the control input and y_{c} represents the controller state vector. This controller should alter the dynamics such that in case Hopf bifurcations exist before the SNB, they should be relocated to the inoperable part of the stationary solutions (mathematically) which means the system stability would be guaranteed until the singularity of the Jacobian at SNB. Dynamics of the controlled system would be as follows. Vector u(x, y_{c}) is control input vector, and h(x, y_{c}) is the controller state vector. K_{1} and K_{2} are the control gain vectors while L is constant parameter vector. There are two important theorems; one is that the controller does NOT alter the equilibria structure of the uncontrolled system. Which means that the stationary branch is the same in both controlled and uncontrolled system. The second one is that for all positive constant parameter vector the control law is stable. Proofs can be found in the paper. The flowchart describes how the control should be applied as follows: 1-derive the nonlinear dynamics of the power system with incorporation of saturation phenomena. 2-construct the system of DAEs for the model. 3-use implicit function theorem to extract ODEs from DAEs. 4-find an equilibrium for the system and analyze stability (SEP). 5-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. 6-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). 7-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 4. We also should add the controller equilibria to the vector of the SEP from step 4 (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). 8-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.