Optimal Decentralized Control of Power Systems with Guaranteed Damping S. Mojtaba Tabatabaeipour1 Abstract— An optimal decentralized control approach for damping of inter-area oscillations in power systems with a guaranteed level of damping is suggested in this paper. Widearea signals are used to supplement the actions of local controllers to improve the damping of the overall system. To guarantee a well-damped transient response, poles of the closedloop system are required to be in a prescribed region. To optimize the performance of the system, the standard quadratic performance index is considered and minimized. A heuristic iterative algorithm is proposed which allows simultaneous design of the control structure and the control gain. At the same time, regional pole-placement constraints are taken into account and thereby a prescribed level of damping is guaranteed. The control structure and the control gain are designed by minimizing the number of required communication links and maximizing the closed-loop performance. The approach also allows promoting a pre-specified fixed-structure decentralized control design. The method is demonstrated on a two-area fourmachine benchmark and the results shows the effectiveness of the method and robustness of the design to delays and parametric uncertainties.∗
I. INTRODUCTION In large-scale interconnected power systems various oscillations related to low-frequency which are typically in the range of 0.1-2 Hz might occur that can increase the risk of insecure or even unstable operation. Large scale power systems exhibit both local and inter-area oscillations. Local oscillations are related to oscillation of a single machine against the rest of the system and inter-area oscillations are related to oscillations of a group of machines against another group. In recent years, due to the increase in power demand and installment of renewable sources in remote areas, transmission of power between remote areas are inevitable. As a result the power system is operating closer to its stability margins and in many instances the grid might be operating in a highly stressed condition. In these conditions, a small disturbance can cause the weakly-damped low-frequency inter-area modes to become unstable. The conventional approach to damp inter-area modes is to use local decentralized controllers called power systems stabilizers (PSS). In tuning local PSSs, dynamics of the overall network must be carefully taken into account, otherwise the interactions of local PSSs can destabilize the overall network [1]. In order to ensure a well-damped response, a certain damping factor must be guaranteed for the overall network. This can be achieved by forcing the eigenvalues of 1 S. M. Tabatabaeipour is with the Automation and Control Group, Department of Electrical Engineering, Technical University of Denmark,email:
[email protected].∗ This work was supported through the SOSPO project by the Danish Council for Strategic Research under grant no. 11-116794.
the closed-loop system to be in a predetermined region in the complex plane [2], [3]. Recent developments in communication technologies and new measurement technologies such as synchronized phasor measurement units (PMUs) have made it possible to use remote measured signals as feedback to construct the control inputs to supplement the actions of local controllers, thereby improving the damping of the overall network. A challenging problem in designing a decentralized wide-area controller is selection of the measurements and actuators. Generally speaking, a decentralized control problem design requires two steps: the structure (information pattern) design and the controller design. A fully centralized controller offers the best performance but it requires communication of all measurements between machines in different areas. A fully decentralized controller only requires local information but it has the lowest performance. In a partially decentralized controller, some of the measurements are communicated between subsystems to achieve stability or improve the performance of the closed-loop system. A number of methods for optimal structure selection to minimize wide-area communications and control-loop interactions has been proposed in the literature. Hankel singular values combined with right half plane zeros are used in [4]. A set of modal indices is introduced in [5] to choose input/output signals such that the control loop has a large effect on a few selected modes without considerable effect on other modes. In [6] an index based on minimum variance of modal residue was used. Geometric measures of controllability and observability are used for input-output selection in [7]. In [8] based on these measures an H2 /H∞ optimal controller with regional pole-placement constraints is designed. These measures only give an index for selection of a possibly good structure of a decentralized control and the input-output selection is performed using a combinatorial procedure. Moreover, the controller is designed after selection of the structure. A method for optimal decentralized control using state feedback with a priori fixed structure was proposed in [9]. In order to minimize the number of communication links and inter-area interactions, while simultaneously optimizing the control performance, a sparsity-promoting approach based on using alternating direction method of multipliers was proposed in [10] and [11]. This approach was used in [12] and [13] to design a sparse and wide-area damping control for power systems. The method proposed in [11] and later used in [13] for PSS design only considered the H2 performance and did not
allow any pole-placement constraints to be taken into account since it is based on solving the Lyapunov and Sylvester equations. It should be noted that a main objective of PSS design is to improve the transient behavior of the closed loop system [2], which cannot be guaranteed by optimizing the H2 or the H∞ performance as in [14]. A guaranteed level of damping and acceptable transient behavior can be achieved by a suitable placement of the poles of the closed loop systems in a specified region of the complex plane [3], [2]. In this paper, we propose a new method for decentralized wide-area control where regional pole-placement constraints are taken into account and a trade-off between the performance and the number of communication links, i.e. the sparsity of the feedback gain, is achieved. A new iterative heuristic convex optimization based approach is proposed to solve the problem. To choose the controller structure, the cardinality of the feedback matrix is minimized by relaxing the problem to a re-weighted `1 norm minimization motivated by ideas from [15]. At each iteration, a linear matrix inequality (LMI) based approach is used to minimize a weighted sum of the quadratic performance subject to regional pole-placement constraints and a weighted `1 norm of the feedback gains. The method is demonstrated for decentralized optimal control of the Kundur’s two-area fourmachine system [16]. II. P OWER SYSTEM DYNAMIC MODEL
Im
θ Re
α
Fig. 1: LMI region for pole placement
function from d to z is minimized. The dynamics of the closed-loop system is given by: x(t) ˙ = (A + BK)x + B1 d, z = Cz x,
(4)
where Q1/2 . R1/2 K
Cz =
The electromechanical dynamic behavior of the power system can be described by a set of differential algebraic equations as:
Assuming that a stabilizing K exists, it is known [17] that the H2 gain of the system defined as kCz (sI − (A + BK))−1 B1 k2 is less thatn γ if there exists a matrix P = P T > 0 such that:
x(t) ˙ = f (x, y, u)
(A + BK)T P + P (A + BK) + CzT Cz < 0,
(1)
Trace(B1T P B1 )
0 = g(x, y, u), where x ∈ Rn is the dynamic state, y ∈ Rp is the algebraic state, and u ∈ Rm is the control input which includes the input to the FACT devices and generator excitations. Assuming that f (., ., .) and g(., ., .) are continuously differentiable and the nonlinear differential algebraic model is linearized around an equilibrium point (x∗ , y ∗ , u∗ ) to get a linear statespace model.
(5)
2
