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Distributed Algorithms with Theoretic Scalability Analysis of Radial and Looped Load flows for Power Distribution Systems

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Distributed Algorithms with Theoretic Scalability Analysis of Radial and Looped Load flows for Power Distribution Systems Electric Power Systems Research 65 (2003) 169Á/177 www.elsevier.com/locate/epsr Distributed algorithms with theoretic scalability analysis of radial and looped load flows for power distribution systems Fangxing Li *, Robert P. Broadwater ECE Department Virginia Tech, Blacksburg, VA 24060, USA Received 15 April 2002; received in revised form 14 December 2002; accepted 16 December 2002 Abstract This paper presents distributed algorithms for both radial and looped load flows for unbalanced, multi-phase power distribution systems. The distributed algorithms are developed from tree-based sequential algorithms. Formulas of scalability for the distributed algorithms are presented. It is shown that computation time dominates communication time in the distributed computing model. This provides benefits to real-time load flow calculations, network reconfigurations, and optimization studies that rely on load flow calculations. Also, test results match the predictions of derived formulas. This shows the formulas can be used to predict the computation time when additional processors are involved. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Distributed computing; Scalability analysis; Radial load flow; Looped load flow; Power distribution systems 1. Introduction Also, the method presented in Ref. [10] was tested in radial distribution systems with no more than 528 buses. Parallel and distributed computing has been applied More recent works [11Á/14] presented distributed to many scientific and engineering computations such as implementations for power flows or power flow based weather forecasting and nuclear simulations [1,2]. It also algorithms like optimizations and contingency analysis. has been applied to power system analysis calculations These works also targeted power transmission systems. [3Á/14]. Parallel computing is usually referred to as Instead of using shared memory, a scheme of message computing carried out on a single machine with multiple passing was used. CPUs and shared memory. Distributed computing is This paper discusses distributed algorithms to calcu- usually thought of as a less ‘coupled’ computing carried late load flows for radial and looped power distribution out among multiple, separate machines without shared systems based on tree traverses. The algorithms are memory. implemented in up to 8 workstations distributed in an Parallel schemes for power flow based on sparse Ethernet LAN. Ways in which this work differs from previous works are now considered: matrices have been investigated in previous works [3Á/ 9] for transmission systems. These works employed (1) Previous works presented scalability based on test results. This work not only presents test results of factorization to utilize sparse matrices in parallel scalability, but also presents theoretic scalability formulas computers. Power transmission systems were the target from the point of view of algorithm analysis (AA). AA of these works. The previous work [10] presented a [1,2] is an approach to explore the performance of parallel power flow method for radial distribution algorithms from the point of view of mathematics. systems. Like [3 /9], a Jacobin matrix was employed. Á Scalability, including speedup and efficiency, is the most important measurement of performance for par- allel and distributed algorithms. It is a function of n and p, where n is the number of elements in a power * Corresponding author. Tel.: /1-919-807-5707; fax: /1-919-807- 5060. distribution system, and p is the number of processors E-mail addresses: [email protected], [email protected] (F. Li). involved in parallel or distributed schemes. This paper 0378-7796/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0378-7796(03)00021-X 170 F. Li, R.P. Broadwater / Electric Power Systems Research 65 (2003) 169Á/177 explores theoretic scalability analysis and estimates this work are based on the (sequential) algorithms in the scalability for the proposed distributed algorithms. previous work [15Á/19], the advantages of the tree-based Estimations will be checked with test results. approach flow naturally to distributed algorithms. (2) Previous works were based on dedicated parallel/ (4) Previous works focused on power transmission distributed architecture, while this work is based on systems or small distribution systems, while this work Ethernet LANs that are widely available. Some previous focuses on power distribution systems having a much parallel solutions of load flows were implemented in larger system scale. The number of elements (including dedicated and/or expensive high-performance parallel customer loads) in a large-scale power distribution computers [4Á/10], such as the Sequent Symmetry S81, system model may well exceed the number of elements iPSC/2, Transputer. Other previous work utilized spe- in a transmission system model. The previous works [3Á/ cific software like parallel virtual machine [11Á/13] or 9,12Á/14] were tested on transmission systems with at specific virtual architecture [14] like ring and hyper- most two thousand elements. The previous work [10] cube. Here we use existing, generic computer networks handled small radial distribution systems with less than like Ethernet LANs, without any extra hardware or 600 elements. This work is tested with radial and looped software, to fulfill distributed solutions for load flows. distribution systems consisting of approximately 40 000 This makes the proposed algorithms very practical since elements. Ethernet LANs are widely available. The communica- Section 2 presents a general approach for distributed tion delay in a LAN is generally longer than in a specific algorithms of tree-based systems. Section 3 presents a architecture, but there is no additional cost. Fig. 1 shows distributed algorithm for radial load flow. Section 4 the Ethernet LAN architecture. Since there is no shared presents a distributed algorithm for looped load flow. memory or storage, message passing through the Section 5 employs techniques in AA to derive formulas standard TCP/IP protocol is used to exchange informa- of the scalability (speedup and efficiency) of the tion among different machines. Here multi-instruction distributed algorithms discussed in Sections 3 and 4 and multi-data is used in the distributed algorithm. and Section 6 presents test results of a distribution Also, a multithreading approach is employed to fully system with approximately 40 000 elements. utilize the concurrency of hardware. (3) Previous works used matrix-based load flows, while this work uses a tree-based approach for unbalanced, 2. General approach for distributed algorithms of tree- multi-phase distribution systems, including both radial and based analysis looped systems. Previous works employed the sparse/ Jacobin matrix to solve load flow or load flow based 2.1. Controller and workers algorithms for transmission systems [3Á/9,11Á/14] and radial distribution systems [10]. This work utilizes a tree- The distributed algorithm defines two types of based approach to solve the unbalanced, multi-phase processes, controllers and workers, which play different load flow for radial and looped distribution systems, roles in the message passing scheme. A controller is a based on the previous works [15Á/19], especially the process with the following responsibilities: previous work [19]. Compared with the matrix approach, the tree-base . To accept user input approach may reduce the complexity of software . To assign initial data (job) to workers . To invoke workers to execute tasks implementation for distribution systems [20Á/22]. The reason is that the matrix approach needs to model nodes . To send/receive intermediate data to/from workers and edges for the system topology, while the tree-based . To do system-wide, non-intensive calculations approach only needs to model edges. In addition, with . To terminate the algorithm at the appropriate time the tree-based approach, the looped load flow can be and notify workers to stop. viewed as inherited from the radial load flow. Hence, A worker is a process with the following respon- development and implementation works can be reduced sibilities: in practice. Since the distributed algorithms presented in . To receive initial data from a controller . To do intensive calculations upon a controller’s request . To send/receive intermediate data to/from a control- ler . To stop when notified. In the approach here there is only one controller, while there are multiple workers. The controller is Fig. 1. An Ethernet LAN with multiple workstations. mainly responsible for coordinating all workers and F. Li, R.P. Broadwater / Electric Power Systems Research 65 (2003) 169Á/177 171 maybe some non-intensive computations, while workers . Assign (p/1)th feeder to pth processor, (p/2)th are responsible for intensive computations. The con- feeder to (p/1)th processor,...,(2p)th feeder to 1st troller has system-wide knowledge, while a worker has processor. knowledge of only a part of the system, i.e., the job . Assign (2p/1)th feeder to 1st processor, (2p/2)th assigned by the controller. In this sense, a controller feeder to 2nd processor, and so on. behaves like a manager in a company, while a worker . Stop when all k feeders are assigned. behaves like a salesman or an engineer. 2.2. General approach 3. Distributed algorithm for radial load flow Fig. 2 illustrates the above steps using unified model- ing language (UML) notation [23], a standard for software design. In this figure, CController is the controller class and CWorker is the worker class. 3.1. Sequential algorithm based on tree approach Also, c is an object of class CController,andw is an object of class CWorker. Each directed line represents a The term sequential is used here to indicate one message or an activity, ordered by its sequence in time. processor. The radial load flow algorithm used here is The name of each message or activity is self-explana- based on tree traverses [15Á/20]. Compared with tradi- tory. Also, an asterisk (*) indicates a repeated activity. tional load flow algorithms like GaussÁ/Seidel and NewtonÁ/Raphson, this approach suits power distribu- tion systems. First, it is designed for unbalanced, multi- 2.3.
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