Dynamic equivalence by an optimal strategy

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Abstract

Due to the curse of dimensionality, dynamic equivalence remains a computational tool that helps to analyze large amount of power systems’ information. In this paper, a robust dynamic equivalence is proposed to reduce the computational burden and time consuming that the transient stability studies of large power systems represent. The technique is based on a multi-objective optimal formulation solved by a genetic algorithm. A simplification of the Mexican interconnected power system is tested. An index is used to assess the proximity between simulations carried out using the full and the reduced model. Likewise, it is assumed the use of information stemming from power measurements units (PMUs), which gives certainty to such information, and gives rise to better estimates.

Highlights

► Dynamic equivalents are reviewed. ► The dyamic equivalence is formulated as a multi-objective optimization problema. ► Applications to an actual power network are presented.

Introduction

One way to speed up the dynamic studies of currently interconnected power systems without significant loss of accuracy is to reduce the size of the system model by means of dynamic equivalents. The dynamic equivalent is a simplified dynamic model used to replace an uninterested part, known as an external part, of a power system model. This replacement aims to reduce the dimension of the original model while the part of interest remains unchanged [1], [2], [3], [4], [5], [6].

The phrases “Internal system” (IS) and “external system” (ES) are used in this paper to describe the area in question, and the remaining regions, respectively. Boundary buses and tie lines can be defined in each IS or ES. It is usually intended to perform detailed studies in the IS. However, the ES is important to the extent where it affects IS analyses.

The equivalent does not alter the transient behavior of the part of the system that is of concern and greatly reduces the dimension of the network, reducing computational time and effort [4], [7], [8]. The dynamic equivalent also can meet the accuracy in engineering, achieving effective, rapid and precise stability analysis and security controls for large-scale power system [4], [8]. However, the determination of dynamic equivalents may also be a time consuming task, even if performed off-line. Moreover, several dynamic equivalents may be required to represent different operating conditions of the same system. Therefore, it is important to have computational tools that automate the procedure to evaluate the dynamic equivalent [7].

Ordinarily, dynamic equivalents can be constructed following two distinct approaches: (i) reduction approach, and (ii) identification approach. The reduction approach is based on an elimination/aggregation of some components of the existing model [4], [5], [9]. The two mostly found in the literature are known as modal reduction [6], [10] and coherency based aggregation [2], [11], [12]. The identification approach is based on either parametric or non-parametric identification [13], [14]. In this approach, the dynamic equivalent is determined from online measurements by adjusting an assumed model until its response matches with measurements. Concerning the capability of the model, the dynamic equivalent obtained from the reduction approach is considerably more reliable and accurate than those set up by the identification approach, because it is determined from an exact model rather than an approximation based on measurements. However, the reduction-based equivalent requires a complete set of modeling data (e.g. model, parameters, and operating status) which is rarely available in practice, in particular the generators’ dynamic parameters [5], [13], [15], [16].

On the other hand, due to the lack of complete system data, and/or frequently variations of the parameters with time, the importance of estimation methods is revealed noticeably. Especially, on-line model correction aids for employing adaptive controllers, power system stabilizers (PSS) or transient stability assessment. The capability of such methods has become serious rival of the old conventional methods (e.g. the coherency [11], [12] and the modal [6], [10] approaches). The equivalent estimation methods have spread, because it can be estimated founded on data measured only on the boundary nodes between the study system and the external system. This way, without any need of information from the external system, estimation process tries to estimate a reduced order linear model, which is replaced for the external part. Evidently, estimation methods can be used, in presence of perfect data of the network as well to compute the equivalent by simulation and/or model order reduction [15].

Sophisticated techniques have become interesting subject for researchers to solve identification problems since 90s. For example, to obtain a dynamic equivalent of an external subsystem, an optimization problem has been solved by the Levenberg–Marquardt algorithm [17]. Artificial neural networks (ANN) are the most prevalent method between these techniques because of its high inherent ability for modeling nonlinear systems, including power system dynamic equivalents [15], [18], [19], [20], [21], [22], [23], [24], [25].

Power system real time control for security and stability has promoted the study of on-line dynamic equivalent technologies, which progresses in two directions. One is to improve the original off-line method. The mainstream approach is to obtain equivalent model based on typical operation modes and adjust equivalent parameters according to real time collected information [4]. Distributed coordinate calculation based on real time information exchanging makes it possible to realize on-line equivalence of multi-area interconnected power system in power market environment [4]. Ourari et al. [26] developed the slow coherency theory based on the structure preservation technique, and integrated dynamic equivalence into power system simulator Hypersim, verifying the feasibility of on-line computation from both computing time and accuracy [27].

Prospects of phasor measurement technique based on global positioning system (GPS) applied in transient stability control of power system are introduced in Ref. [4]. Using real data collected by phasor measurement unit (PMU), with the aid of GPS and high-speed communication network, online dynamic equivalent of interconnected power grid may be achieved [4].

In this paper, the dynamic equivalence problem is formulated by two objective functions. An evolutionary optimization method based on genetic algorithms is used to solve the problem.

Section snippets

Proposition

The main objective of this paper is the external system's model order reduction of an electrical grid, preserving only the frontier nodes. That is, those nodes of the external system directly linked to nodes of the study system. At such frontier nodes, fictitious generators are allocated. The external boundary is defined by the user. Basically, it is composed by a set of buses, which connect the external areas to the study system. There is not restriction about this set. Different operating

Results

In this case, the decision variables, x, are eight parameters per each equivalent generator: {xd,xd,xq,xq,Td0,Tq0,H,D}. In this paper, for five equivalent generators, there are 40 parameters to be estimated.

Likewise, in this case, a random change in the load of all buses gives rise to the transient behavior. A normal distribution with zero mean is utilized to generate the increment (decrement) in all buses. The variation is limited to a maximum of 50%. The disturbance lasts for 0.12 s and

Conclusions

Undoubtedly, the power system equivalents’ calculation remains a useful strategy to handle the large amount of data, calculations, information and time, which represent the transient stability studies of modern power grids. The proposed approach is founded on a multi-objective formulation, solved by a genetic algorithm, where the objective functions weight independently each operating condition taken into account. The use of information stemming from PMUs helps to improve the estimated

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