Frequency-limited pseudo-optimal rational Krylov algorithm for power system reduction
Introduction
Today’s power system is a large network of interconnected power apparatus like generators, lines, and buses that covers a large geographical territory. There is a growing trend to facilitate further interconnections with neighboring systems, and thus the size of the interconnected power system network is likely to continue to increase. The mathematical representation of these large-scale power system networks can easily reach several thousands of differential equations. This poses a challenge for fast and efficient simulation, analysis, and control system design for these large-scale power systems despite a significant growth in the storage and computational capabilities in recent time [1]. Model order reduction (MOR) offers a solution to the problem by providing a reduced order model (ROM), which enables fast simulation and control system design without significantly affecting the accuracy. MOR is generally referred to as “dynamic equivalency” in the power system literature [2].
The analysis of a complete power system network with every subtle detail is neither practical nor required. In the MOR of power systems, the power system is first partitioned according to the importance [1], [2], [3], [4], [5], [6]. The portion of the power system under investigation, which contains the important variables, constitutes the study area, and it is mathematically described by a detailed nonlinear model. Note that this study area is not reduced. On the other hand, the portion whose effect on the analysis in the study area is only of the interest is mathematically described by a linear model, and it constitutes the external area; see Fig. 1. MOR is applied to the linear model of the external area. For instance, not only that a linear model suffice in the small-signal stability analysis and damping controller design, it can be further reduced using MOR techniques without a significant loss of accuracy [28].
The coherency-based MOR methods have been historically employed to obtain a dynamically equivalent ROM [6], [7], [8]. The response of coherent generators is similar to a particular set of inputs. The first step in the coherency-based MOR techniques is to identify and group the coherent set of generators and construct a lumped system. A ROM is then obtained from the lumped model by exploiting the physical properties of electrical machines connected to the power system network. The dependence on physical properties restricts the flexible applicability of these methods. Recently, an increasing interest in MOR techniques which rely on the mathematical properties instead of the physical properties of the power system apparatus is shown by the power system community [4], [9]. For instance, balanced truncation and moment matching have been successfully used in power system reduction, showing some promising results [10], [11], [12].
The power systems exhibit local and interarea oscillations in the frequency region between 0.8–2 Hz and 0.1–0.7 Hz, respectively [13]. These are associated with the poorly damped modes of the power system model and are often called “electromechanical or critical modes”. These modes are crucial for small-signal stability analysis and for the damping controller design. Therefore, these modes must be preserved in the ROM to retain the oscillatory behavior of the original model. The frequency response of the ROM should closely match that of the original system within 0.1-2 Hz. The importance of good frequency-domain accuracy within 0.1–2 Hz has been recognized consistently in the literature; see for instance [5], [14]. In [5], the ROM interpolates the original system at and around zero frequency to effectively capture these oscillations in the frequency-domain. In [14], it is suggested to retain the critical modes in the ROM to preserve the oscillation associated with these modes.
The preservation of slow and poorly damped modes in the ROM is beneficial from the damping controller design perspective [14], and it also improves the accuracy of the ROM in the time-domain [4], [9], [15]. Most of the MOR algorithms used for power system reduction like balanced truncation [16] and moment matching [5] do not have modal preservation property. It is customary to increase the order of ROM in these algorithms in a hope to capture the poorly damped critical modes of the original system in the ROM. This popular belief has recently been refuted in [4], and it is argued that there is no guarantee to capture these modes in the ROM by increasing the order. It is further shown that the quality of the ROM can be improved by preserving the slow and poorly damped modes instead of increasing its order [4]. In [15], an -MOR algorithm is proposed for power systems that includes modal preservation as a cost function of its optimality criteria. The algorithm does preserve the electromechanical modes in the ROM, but the first-order optimality conditions (as defined in [17], [18]) of -MOR are no longer satisfied with this heuristic modification in [18]. It gives good frequency and time domain accuracy, but the excessive computational cost associated with the particle swarm optimization technique [19] makes it unsuitable for large-scale systems. In [20], the power system reduction is considered as a finite-frequency MOR problem with an additional constraint that the electromechanical modes of the systems are preserved in the ROM. The algorithm is computationally efficient, but the ROM of acceptable accuracy is not that compact because it uses modal truncation to preserve critical modes. The order of ROM should be significantly larger than the number of modes to be preserved. The accuracy in the specified frequency region is obtained by using frequency-dependent extended realization of the original system. MOR is applied to this extended realization, and the ROM is obtained via an inverse transformation. In [21], the optimal frequency-limited -MOR problem is considered, and an algorithm is proposed, which generates an optimal ROM. The algorithm requires the solution of Lyapunov equations and linear matrix inequalities (LMIs) to find the optimal ROM, which is not feasible in a large-scale setting. In [22], the problem is described as bi-tangential Hermite interpolation, which can be solved in a computationally efficient way. However, the original system is required to be converted into pole-residue form, which is again computationally not feasible in a large-scale setting. Moreover, both the algorithm [21], [22] are iterative algorithms with no guarantee on the convergence, and they do not have a modal preservation property.
In this paper, we consider the same problem of [20] and propose a computationally efficient MOR algorithm that ensures a good accuracy in the specified frequency region with explicit modal preservation. Unlike [20], a fairly compact ROM can be obtained using the proposed algorithm, and the order of ROM can even be equal to the number of modes to be preserved. The algorithm uses a moment matching approach based on the parametrized family of ROM [23] and generates a ROM which satisfies a subset of the first-order optimality conditions for frequency-limited -MOR problem [22]. Unlike [21], [22], the proposed algorithm is iteration-free and does not requires the solutions of large-scale Lyapunov equations, LMIs, and pole-residue form. The performance of the proposed algorithm is tested by considering benchmark power system reduction problems.
Section snippets
Preliminaries
Consider an -machine, k-bus system as the external area which is connected to p-buses of the study area via p-tie lines. The external area can be described by the following second-order classical model [35] used for power system reduction for , i.e.,, and are the inertial coefficient, damping coefficient, rotor angle, angular
Main work
The analytical damping controller design procedures like LQG and result in a controller whose order is greater than or equal to that of the power system model. To obtain a lower order controller, a ROM of the original model is first sought using MOR [27]-[29]. It is stressed in [14] that the ROM should preserve the critical modes and the frequency-domain behavior of the original system over the frequencies associated with the critical modes. These modes are generally poorly damped and can
Applications
In this section, we demonstrate the applications of FLPORK on three interconnected power system models. The first model is an interconnection of the IEEE 145-bus 50-machine system with the New England Test System-New York Power System (NETS-NYPS) 68-bus 16-machine system. The second model is the interconnection of the Northeastern Power Coordinating Council (NPCC) 140-bus 48-machine system with the IEEE 145-bus 50-machine system. The third model is the IEEE 145-bus 50-machine system. We first
Conclusion
In this paper, a frequency-limited MOR technique is proposed which yields a ROM which not only satisfies a subset of the first-order optimality conditions of the problem but also preserves the electromechanical modes of the power system. The proposed algorithm can generate an accurate ROM with the desired modes, which ensures a good accuracy in the desired frequency interval. The proposed algorithm is applicable to large-scale systems and hence can be used for fast dynamic
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
The first author would like to thank M. A. Pai of the University of Illinois at Urbana-Champaign, Urbana, USA for explaining his work [5], [35], and answering several questions related to power system modeling for MOR done in his students’ work [38], [39]. This work was supported by National Natural Science Foundation of China under Grant (No. 61873336, 61873335), and supported in part by 111 Project (No. D18003).
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