First International Conference on Electrical Systems PCSE’05 May 9-11 2005, O. E. Bouaghi Univ. Algeria.

Optimal Power Flow Using Partitioning technique *

**

Belkacem Mahdad*, Tarek Bouktir**, Kamel Srairi*

Department of Electrical Engineering, University of Biskra, Algeria. Department of Electrical Engineering, University of Oum El Bouaghi, Algeria. Email:[email protected]

Abstract In this paper we present an approach to parallelizing power flow problem (OPF) that is suitable for large-sized power systems. To reduce the computational burden, and to gain qualitative insight, it is often useful to partition the original model of a large-scale power system into several reduced sub-networks. This paper proposes a partitioning method for identifying groups of strongly connected sub-Networks in an interconnected power system grid. The method exploits concepts of the recursive spectral bisection technique from the graph theory; here we will explore partitioning applications in the optimal power flow. This approach has the advantage that load buses, as well as generator buses, are identified within the resulting decomposed areas. The IEEE 30-bus system case is presented here to demonstrate the feasibility of the proposed technique.

Keywords: Graph partitioning, network/graph, Optimal Power Flow, Fiedler vector. Parallelizing power flow.

I.

INRODUCTION

available algorithms. However it is also possible for each utility to have a different OPF implementation for its area. We will apply the concept of the recursive spectral graph bisection presented in [2, 5, 6] as our basis to formulate the network decomposition scheme. The RSB method is proven in [5] to be optimal among various graphpartitioning techniques. We test the algorithm in IEEE 30-buses to demonstrate the performance of the method in preserving the optimal cost of generation for the decomposed sub-networks.

PROBLEM FORMULATION:

II.

a) Basic Graph Theory The objective of graph partitioning is to separate the graph’s vertices into a predetermined number sub-graphs, in which each sub-graph has an equal number of vertices and the cutset links among these sub-graphs are minimised (fig.1). 1

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The possibility of operating the power system at the minimal cost while satisfying specified transmission line power flow control constraints and security constraints is one of main current issues in stretching transmission capacity by technology deployment. Several papers have been published, Work in [1] explain various important applications of graph partitioning: VLSI circuit layout, image processing, solving spare linear systems, computing fill-reducing orderings for sparse matrices, and distributing workloads for parallel computation, Pothen, Simon, and Liou introduce an approach in [2] to partition the input graph using the spectral information of the Laplacian matrix, L; this technique is referred to as recursive spectral bisection (RSB). Miroslav Fiedler proved several properties of the second smallest eigenvalue and its corresponding eigenvector of a Laplacian matrix in his famous works [3] and [4]. Many of the previously mentioned researches identify only generator buses within an area. The advantage of this method is that it provides partitions indicating both generators and load buses within a coherent area. In contrast to the previous works that focused on partitioning the network in dynamic models, this paper will focus on network partitioning applications in the optimal power flow (OPF). We solve the optimal power flows for each region and coordinate the multiple OPFs through an iterative update on constraint Lagrange multipliers. Naturally, the OPFs solved in each region can be implemented with the fastest

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Figure 1: Graph Partitioning of the original model of a network into three reduced sub-networks.

Laplacian matrix L that this matrix L should satisfy: L = M.Mt =D–A

(1)

where M is the incidence matrix of graph G, A is the adjacency matrix of G, and D is the diagonal matrix whose ith entry is the degree of the vertex vi (1

Optimal Power Flow Using Partitioning technique *

**

Belkacem Mahdad*, Tarek Bouktir**, Kamel Srairi*

Department of Electrical Engineering, University of Biskra, Algeria. Department of Electrical Engineering, University of Oum El Bouaghi, Algeria. Email:[email protected]

Abstract In this paper we present an approach to parallelizing power flow problem (OPF) that is suitable for large-sized power systems. To reduce the computational burden, and to gain qualitative insight, it is often useful to partition the original model of a large-scale power system into several reduced sub-networks. This paper proposes a partitioning method for identifying groups of strongly connected sub-Networks in an interconnected power system grid. The method exploits concepts of the recursive spectral bisection technique from the graph theory; here we will explore partitioning applications in the optimal power flow. This approach has the advantage that load buses, as well as generator buses, are identified within the resulting decomposed areas. The IEEE 30-bus system case is presented here to demonstrate the feasibility of the proposed technique.

Keywords: Graph partitioning, network/graph, Optimal Power Flow, Fiedler vector. Parallelizing power flow.

I.

INRODUCTION

available algorithms. However it is also possible for each utility to have a different OPF implementation for its area. We will apply the concept of the recursive spectral graph bisection presented in [2, 5, 6] as our basis to formulate the network decomposition scheme. The RSB method is proven in [5] to be optimal among various graphpartitioning techniques. We test the algorithm in IEEE 30-buses to demonstrate the performance of the method in preserving the optimal cost of generation for the decomposed sub-networks.

PROBLEM FORMULATION:

II.

a) Basic Graph Theory The objective of graph partitioning is to separate the graph’s vertices into a predetermined number sub-graphs, in which each sub-graph has an equal number of vertices and the cutset links among these sub-graphs are minimised (fig.1). 1

0.8

The possibility of operating the power system at the minimal cost while satisfying specified transmission line power flow control constraints and security constraints is one of main current issues in stretching transmission capacity by technology deployment. Several papers have been published, Work in [1] explain various important applications of graph partitioning: VLSI circuit layout, image processing, solving spare linear systems, computing fill-reducing orderings for sparse matrices, and distributing workloads for parallel computation, Pothen, Simon, and Liou introduce an approach in [2] to partition the input graph using the spectral information of the Laplacian matrix, L; this technique is referred to as recursive spectral bisection (RSB). Miroslav Fiedler proved several properties of the second smallest eigenvalue and its corresponding eigenvector of a Laplacian matrix in his famous works [3] and [4]. Many of the previously mentioned researches identify only generator buses within an area. The advantage of this method is that it provides partitions indicating both generators and load buses within a coherent area. In contrast to the previous works that focused on partitioning the network in dynamic models, this paper will focus on network partitioning applications in the optimal power flow (OPF). We solve the optimal power flows for each region and coordinate the multiple OPFs through an iterative update on constraint Lagrange multipliers. Naturally, the OPFs solved in each region can be implemented with the fastest

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Figure 1: Graph Partitioning of the original model of a network into three reduced sub-networks.

Laplacian matrix L that this matrix L should satisfy: L = M.Mt =D–A

(1)

where M is the incidence matrix of graph G, A is the adjacency matrix of G, and D is the diagonal matrix whose ith entry is the degree of the vertex vi (1