Development of simplified power grid models in EU project Spine

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IN

DEGREE PROJECT ELECTRICAL ENGINEERING, SECOND CYCLE, 30 CREDITS

,

STOCKHOLM SWEDEN 2020

Development of simplified power

grid models in EU project Spine

MOHAMMAD ALHARBI

KTH ROYAL INSTITUTE OF TECHNOLOGY

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Author

Mohammad Alharbi <alharbi@kth.se> Electrical Engineering and Computer Science KTH Royal Institute of Technology

Examiner

Dr. Mikael Amelin Associate Professor

KTH Royal Institute of Technology

Supervisor

Dr. Iasonas Kouveliotis-Lysikatos Postdoctoral researcher

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Abstract

The electric power system is among the biggest and most complex man-made physical network worldwide. The increase of electricity demand, the integration of ICT technologies for the modernization of the electric grid and the introduction of intermittent renewable generation has resulted in further increasing the complexity of operating and planning the grid optimally. For this reason the analysis of large-scale power systems considering all state variables is a very complicated procedure. Thus, it is necessary to explore methods that represent the original network with smaller equivalent networks in order to simplify power system studies. The equivalent network should provide an accurate and efficient estimation of the behavior of the original power system network without considering the full analytical modelling of the grid infrastructure.

This thesis investigates partitioning methods and reduction methodologies in order to develop a proper reduction model. The K-means and K-medoids clustering algorithms are employed to partition the network into numerous clusters of buses. In this thesis the Radial, Equivalent, and Independent (REI) method is further developed, implemented, and evaluated for obtaining a reduced, equivalent circuit of each cluster of the original power system. The basic idea of REI method is to aggregate the power injections of the eliminated buses to two fictitious buses through the zero power balance network.

The method is implemented using Julia language and the PowerModels.jl package. The reduction methodology is evaluated using the IEEE 5-bus, 30-bus, and 118-bus systems, by comparing a series of accuracy and performance indices. Factors examined in the comparison include the chosen number of clusters, different assumptions for the slack bus as well as the effect of the imposed voltage limits on the fictitious REI buses.

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Keywords

Reduction Techniques, Aggregated Power Flow Models, REI, Optimal Power Flow, Julia.

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Abstract

Elsystemet är ett av de största och mest komplexa människotillverkade fysiska nätverken i världen. Ökad elförbrukning, integration av informationsteknik för att modernisera elnäten samt införandet av varierande förnybar elproduktion har resulterat i ytterligare ökad komplexitet för att driva nätet optimalt. Därför är det mycket komplicerat att analysera storskaliga elsystem samtidigt som man tar hänsyn till alla tillståndsvariabler. Det är således nödvändigt att utforska metoder för att modellera det ursprungliga nätverket med ett mindre ekvivalent nätverk för att underlätta studier av elsystem. Det ekvivalenta nätverket ska ge en noggrann och effektiv uppskattning av det ursprungliga systemets egenskaper utan att inkludera en kompletta analytisk modell av nätverkets stuktur.

Den här rapporten undersöker metoder för att dela upp och reducera ett nätverk för att få fram en lämplig ekvivalent modell. Klusteranalysalgotmerna K-means och K-medoids används för att dela in nätverket i ett antal kluster av noder. I rapporten vidareutvecklas, implementeras och utvärderas REI-metoden för att ta fram reducerade ekvivalenta nätverk för varje kluster i det ursprungliga systemet. Den grundläggande idén med REI-metoden är att den aggregerar elproduktionen i de elminerade noderna i två fiktiva noder genom ett nolleffektbalansnätverk.

Metoden är implementerad i programspråket Julia och programpaketet PowerModels.jl. Reduceringsmetoderna utvärderas på IEEE:s system med 5 noder, 30 noder respektive 118 noder, genom att jämföra ett antal index för noggrannhet och prestanda. De faktorer som undersäks i jämförelsen inkluderar det valda antalet kluster, olika antagande om slacknoden samt följderna av spänningsgränserna för de fiktiva REI-noderna.

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Acknowledgements

I would like to express my gratitude and appreciation to my supervisor, Iasonas Kouveliotis-Lysikatos, for his guidance and enthusiastic support throughout this thesis project to make my thesis a success.

I would also like to thank my examiner, Mikael Amelin, for his valuable feedbacks and support in preparing this thesis

My appreciation also goes to ABB Saudi Arabia for supporting my education at KTH.

My deepest thanks goes to my family for their endless love and support during all these years.

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List of Figures

2.1 K-means clustering. . . 6

2.2 K-medoids clustering. . . 7

2.3 The difference between K-means and K-medoids [17]. . . 8

2.4 Network reduction technique. . . 11

3.1 K-means algorithm flow chart. . . 14

3.2 Construction of the zero power balance network. . . 15

3.3 Network reduction flow chart. . . 20

4.1 The 5-bus system. . . 22

4.2 Single line diagram of the IEEE bus-30 system. . . 24

4.3 Selected clustering solution for the 30-bus system. . . 26

4.4 Comparison of active power flow in the original network and its equivalent. 28 4.5 Active power flow between clusters for the selected solution. The power flow in red color represents the original power flow . . . 28

4.6 Single-line diagram of the IEEE 118-bus system, by University of Washington [36]. . . 29

4.7 Comparison of active power flow for the K-means clustering solution. . 33

4.8 Comparison of active power flow for the K-medoids clustering solution. 33 4.9 Comparison of the voltage angle error. . . 36

4.10 Voltage magnitude of the retained buses for different voltage tolerance limit at REI buses. . . 37

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List of Tables

4.1 Comparison of the original and equivalent networks of the 5-bus system. 23 4.2 Optimal power flow of case 5 equivalent network. . . 23 4.3 Power injection at case 5 equivalent network buses. . . 23 4.4 Comparison between case 5 original and its equivalent. . . 24 4.5 Comparison of the original and equivalent networks of the 30-bus system. 25 4.6 Summary of the network equivalent results for a selected clustering

solution. . . 26 4.7 Comparison of the original and equivalent networks of the 30-bus system. 27 4.8 Comparison of the original and equivalent networks of the 118-bus system. 30 4.9 OPF result of 118-bus equivalent network by applying K-means. . . 30 4.10 OPF result of 118-bus equivalent network by applying K-medoids. . . . 31 4.11 Comparison of original and equivalent 118-bus system networks. . . 32 4.12 Comparison of the effects of number of clusters. . . 34 4.13 Comparison of the equivalences obtained by retaining and eliminating

the slack bus. . . 35 4.14 Comparison of OPF results for retaining the original slack bus. . . 36

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Chapter 1 Introduction

1.1 Background

Power systems are built to safely transport and distribute electricity produced in power plants [1]. The interconnection of different power systems and the implementation of smart grid technologies for integrating renewable resources and electric vehicles have enormous advantages, but have also led to a large and complex grid infrastructure. In recent years, power systems comprise either large or extremely large networks, and therefore conducting full analytical modelling for operation and planning of the power system network is computationally intensive and difficult to carry out. For this reason, reducing the computational burden is of vital importance for large-scale power system networks for performing a wide variety of research studies within the power systems field.

With growing industrialisation, planning the networks’ expansion has become a tedious process; consequently, numerous approaches are utilized to provide simplified power network models. Partitioning large power networks into clusters of buses is a significant aspect in facilitating the analysis, on the ground that smaller dimension network matrices are acquired. Small dimension matrices of a large-scale network cannot be attained without dividing the power network into areas such that nodes belonging to the same area are closely connected [2].

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CHAPTER 1. INTRODUCTION

In addition, due to the fact that modern power systems have become increasingly complicated as a result of growing demands and heavily integrated power systems, it is a normal practice for researchers and engineers to replace the original model with an equivalent reduced model for various reasons. Some of which include: (i) an interest in observing the behaviour of only a limited set of the system variables; (ii) it is not required to obtain a complete detailed modelling of a distant power system; (iii) lower computational burden [3].

An essential characteristic of a reduced equivalent network is that it should represent the original network with small errors. Some of the widely used network reduction techniques are REI, abbreviated for Radial, Equivalent, and Independent due to the radial structure of the network, and WARD.

1.2 Spine Project

The EU Spine Project aims at developing and evaluating a set of software tools, known as the Spine Toolbox, for modelling energy systems. The primary objective for innovating the Spine Project is to create a modeling toolbox that would facilitate the planning of the future of European energy grids by providing the ability for the clients to develop and validate energy system models input data and visualize the resulted data analysis [4, 5]. The Spine Toolbox can collect and process data from varied sources and in different formats such as tree view, tabular view and graph view. Data structure and user-friendly data display patterns are the main features of the Spine Project. It is noted that the project is funded by the European Union’s Horizon 2020 research and innovation program [6].

Simply put, the Spine Project targets at solving problems related to European energy grids. The system is designed to be user-friendly with data structures that are easy to interpret.

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1.3 PowerModels

The role of power transmission grids has become significant over recent years. As a result, power conservation has gained importance and different approaches are designed for solving issues related to AC power optimisation. Julia is an open-source and effective programming language that is an emerging numerical computing tool that serves as a multi-purpose programming language. An example of an open-source package in Julia is JuMP where a modeling layer is available for optimisation [7].

PowerModels, which is a julia and JuMP package, is an open-source power network optimising toolkit. It involves the application of Optimal Transmission Switching (OTS), Optimal Power Flow (OPF), and Transmission Network Expansion Planning (TNEP). PowerModels is a mathematical programming network developed for optimising power networks. In this framework, power system mathematical programmes can be divided into two different components: problem specifications and mathematical formulations [8]. The main purpose of PowerModels is supplying system abstractions that would assist the designing and contrasting of different power network optimisation problems. The initial type of PowerModels was developed based on MATPOWER which is also an open-source framework for optimizations of electric power systems. Moreover, PowerModels provides support to specific types of buses, generators and transformers. It assists in solving AC power transmission problems, detects failures and promotes planning of sustainable energy future [8].

1.4 Problem Definition

As presented earlier, it is not convenient to compute all variables of large bulk power systems. Therefore, it becomes necessary to reduce the size and complexity of the network and develop a methodology that provide an accurate and efficient estimate simulation of the power system.

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1.5 Objective

This thesis project aims to develop a reduction model which can be applied on large-scale power system networks in order to reduce the complexity of the network to simplify, for instance, power system operational planning studies. The main objectives of this thesis are identified as follows:

• To conduct a literature review on various methods of power system partitioning • To conduct a literature review on REI implementation

• To develop both partitioning and REI equivalent

• To evaluate the developed models (efficiency and accuracy)

1.6 Report Structure

This thesis project is organized in five chapters as follows: Introduction in chapter 1, Literature Study in chapter 2, Methodology in chapter 3, Results and Evaluation in chapter 4, and Conclusion in chapter 5. Chapter 1 contains the background, problem definition, and goals of the project. It also provides the reader with a brief introduction of spine project and PowerModels. Chapter 2 presents a brief review of different partitioning methods and equivalent networks. Chapter 3 describes the methods used for the development of the model in details. Chapter 4 provides a detailed analysis of the equivalent networks produced by the developed model. Chapter 5 summarizes the findings of this project and provides suggestions for future work.

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Chapter 2 Literature Study

2.1 Network partitioning

The planning and operation of a large power system is a very complicated task because the demand of electricity and interconnections between various electricity markets keep growing, resulting in new planning decisions for transmission lines and generators. Furthermore, it can be cumbersome to efficiently obtain necessary information from large power system networks, and therefore data clustering using partitioning algorithm becomes very useful in retrieving information easily by identifying natural groupings of a set of nodes [9, 10]. A reduced equivalent of the power system is essential to reduce the enormous amount of variables in the network resulting in a lower computational burden and better estimation of the behavior of the network [11]. The first step of network reduction is to partition the entire power systems network into smaller areas. Some of the partitioning methods are illustrated in the following sections.

2.2 K-means method

J. McQueen first introduced the K-means algorithm which was later improved by J.A. Hartigan and M.A. Wong [10]. The main purpose of the K-means method is to

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CHAPTER 2. LITERATURE STUDY

categorise a set of data into K number of groups on the basis of similar characteristics. The method is considered to be one of the oldest and most commonly used clustering method. It is a simple partitioning clustering method that seeks to divide the data sets in a given network to achieve K clusters without overlapping such that the distance among the points within each cluster is minimized, while it is maximized for points in different clusters [12, 13].

The clusters are classified by their centroids, which are represented by a reference point in each cluster. The input is a set of data points which we will refer to as buses or nodes. The parameter K defines the desired number of clusters that need to be generated. It starts by placing K centroids in random locations in the power systems network that need to be partitioned and iteratively do the following: the method first goes throughout all the nodes in the power system, and for each node, it will find the nearest centroid point to that node. The distance between the node and cluster centroid is compared for every cluster. The node is then assigned to the cluster, which has the minimum distance (nearest centroid). Once all buses of the power network are assigned to clusters, the K-means algorithm runs through each cluster centroid to recompute its position. This is achieved by averaging all nodes positions located within the cluster to determine the new location of the centroid. Figure 2.1 shows a step by step illustration process of the K-means method.

1 2 3

4

Initial network Determine random

centroid Assign clusters Recompute centroid

Figure 2.1: K-means clustering.

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obtained by the K-means method. For instance, reference [14] uses a hybrid approach for dividing the network into clusters. It first utilizes the k-means method to obtain an initial solution for partitioning the network and then enhances this solution by using an evolutionary algorithm, which is a population-based optimization algorithm, to yield a solution that maintains a balance of multiple measures of attributes.

2.3 K-mediods method

The K-medoids method, as the name suggests, utilises medoids, which are real object points, to represent clusters instead of centroids. This algorithm is similar to the K-means clustering method, and its goal is to enhance the result derived from K-K-means algorithm [9]. When all data assignment is completed, a new medoid is computed and selected to represent clusters instead of the previous medoid. The process is continued until medoids do not undergo any change. An illustrative example of the k-medoids style is shown in Figure 2.2.

1 2 3 4

Figure 2.2: K-medoids clustering.

Overall, it has been found that partition-based algorithms like K-means and K-medoids are suitable for locating spherical-shaped clusters in data points ranging between small to medium size. K-means takes less time in case of few data points, but it takes a long time when there is a large set of data points. In this scenario, the k-medoids algorithm converges in less iterations but with higher total costs [15]. In addition, the k-means achieves acceptable results when all data points packed together as it is very sensitive

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to outliers, which are points that are different or located far form the rest of data points. On this account, the K-medoids is normally applied since it utilizes centrally located points instead of centroids [16].

K-means K-medoids

Figure 2.3: The difference between K-means and K-medoids [17].

2.4 Optimal K-decomposition

A method specifically designed for power system partitioning is the optimal k-decomposition. It basically partitions the power systems network into k clusters by taking into consideration the admittance of the power system.

The number of clusters and minimum number of buses of each cluster are provided, and the objective function of the optimal K-decomposition is to minimize the impedance of the connected nodes within a cluster [2]. This can be also achieved by maximizing the electrical admittance between same-area nodes. Reference [18] provides a solution to the optimal k-decomposition without approximation. The objective function of the optimization problem is to ensure that nodes that are electrically strongly-connected are placed in one cluster while buses that are weakly connected are placed in different clusters.

2.5 Evolution of network equivalence

In 1929, before the development of digital computers, mathematical calculation is utilized to solve power flow problems for the analysis and development of power

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systems. In 1929, network analyzers which are large special purpose analogue computers are created by General Electric in collaboration with Massachusetts Institute of Technology to solve complex power flow equations [19, 20]. Few years later, the utilization of network analyzers was a standard practice among electrical engineers and researchers for the operation and planning of power systems. Nevertheless, the interconnections between different power systems and power system expansions had significantly increased resulting in complicated power flow equations, which therefore made the analysis of the power systems network to be very sophisticated [21]. The limitation of the network analyzers led engineers to search for methods to simplify and reduce the power systems network. As early as 1946, Jennings and Quinan indicated that power system studies can be carried out by using accounting machines of the company International Business Machines (IBM) instead of network analyzers for the reason that they can substantially facilitate the tedious calculations involving sequence of operations while maintaining full accuracy and keeping the real and imaginary terms separate [22].

In 1949, J. B. Ward proposed two methods to decrease the large size of the network. The first approach is that the power system to be examined in depth is terminated at the interconnecting points, while the neighboring system are represented by generators or loads attached to the terminals. The second approach is to replace the power system of interest with an equivalent circuit, in which the tie-lines and generator nodes are kept intact, and the transmission lines between these nodes are represented by an equivalent network [21].

2.6 Network equivalent

Methods for power system reduction are commonly employed in various studies. For instance, ref. [23] and [24] construct network equivalence techniques to reliably carry out system security monitoring and analysis. In ref. [25] equivalencing techniques are used for performing power system planning studies. Ref. [26] simulates reactive power system optimization of multi-regional power systems based on equivalent areas

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to improve the calculation speed. Moreover, ref. [27] reviews and provides a critical analysis of the characteristics of different equivalencing methods to study the power flow in large interconnected power systems. Some of the widely used reduction techniques in static power system studies are REI and Ward [28].

2.7 Ward equivalent

The Ward method was originally introduced by J. B. Ward in 1949 [21]. The method partitions the power system networks into internal and external subsystems and boundary nodes. Two versions are proposed in Ward’s paper to develop equivalencing techniques; the first version is known as Ward Injection and the other version is Ward admittance. The former converts the complex power and voltage of each node into injected current prior to reduction and then converted back to injected power after reduction. The latter version requires that all node powers are transformed to shunt admittance before reduction resulting in an passive equivalent network. It appears that Ward injection method provides a more accurate result due to its reliability and attractive properties [21, 27].

2.8 REI equivalent

The REI (abbreviated for Radial, Equivalent, and Independent) technique was initially proposed by Dimo in 1975 [29]. The prime objective of the REI equivalencing technique is to aggregate the apparent power injections for multiple nodes, which need to be eliminated, into two fictitious nodes, one of which is for aggregating the power injection at the production buses and the other is utilized to aggregate the power injections at the load buses in each cluster of the network [18]. The main procedure of REI equivalencing technique starts with identifying the essential and non-essential nodes of the power system in each area. The essential nodes are represented by the border nodes and vital nodes, which are the buses of interests. The non-essential buses are those which need to be reduced. The REI buses are thereafter created. one for

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grouping all generators and the other one is for combining all loads [18]. Additionally, the total power injections and the corresponding injected currents at these newly created nodes are calculated by summing up the power injection and injected current at the non-essential nodes in the cluster. After that, the zero power balance network is developed for each area and the injections at the non-essential nodes are aggregated to the newly formed nodes. At this time, network reduction is performed on the area via Gaussian elimination to remove the non-essential nodes of the cluster. This results in an area consisting of only essential nodes and the REI generation and load nodes. Figure 2.4 presents an example of a network reduction.

~ ~ Non-essential buses Essential buses Essential buses Equivalent Netowrk ~ ~ ~ Non-essential buses Essential buses Modified Network ~ Zero power balance network REI-L REI-G REI-L REI-G Original Network

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Chapter 3 Methodology

The aim of this project is to reduce the complexity of large-scale power systems. To achieve this goal, the k-means and K-mediods are encountered in the literature study and therefore will be used to partition large networks before applying the reduction technique by using the REI method.

In this chapter, the K-means algorithm and REI reduction technique introduced in the previous chapter are presented in detail.

3.1 Electrical Distance

There are different approaches to calculate the electrical distance of a power network [30, 31]. However, as all nodes in one cluster should have low impedance and high admittance path to each other, the absolute value of the inverse of the Ybus matrix

Ybus is utilized to compute the electrical distance Delec in this project as presented in

Equation 3.1. For instance, the electrical distance from node i to node j in the network is determined by the element in the ith _{row and j}th _{column of the electrical distance}

matrix. Delec =        |Y−1 bus| if i ̸= j 0 if i = j (3.1)

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CHAPTER 3. METHODOLOGY

3.2 Partitioning methods

The k-means and K-medoids are employed to partition the network buses into clusters such that the electrical distances between buses within any cluster are minimized while they are maximized for buses across clusters. The k-means method seeks to minimize the total distance between buses within clusters. The process for partitioning the network by using the K-means algorithm is described in detail in reference [32] and is briefly summarized in this section. The process of partitioning a power network using the K-means method is the following:

1. Select random initial set of cluster centroids cj(j = 1, 2, ..., k)in the XY plane for

the k clusters i.e., A1, A2, ..., Ak.

2. Allocate each bus in the network to the cluster with the closest centroid based on their electrical distance such that the following equation hold for all buses:

Delec(bij, cj)≤ Delec(bij, c

′

j)∀i, j, j

′

: j ̸= j′ (3.2)

where bij is the bus coordinate indicating bus number i in cluster j.

3. Calculate the new cluster centroids by taking the mean of the buses in each cluster as shown in Equation 3.3. cj = 1 nj nj ∑ i=1 bij (3.3)

where nj is the total number of buses in cluster Aj

4. Repeat step 2-3 until the cluster centroids are not revised anymore

Figure 3.1 presents a flow chart of the K-means method.

The K-medoids algorithm attempts to minimize the intra-cluster dissimilarities of all buses. The procedure for partitioning a network of buses using the K-medoids algorithm is similar to the K-means. But the K-medoids algorithm uses medoids instead of centroids to represents clusters. The updated cluster’s medoid is calculated by taking the median of the data points within the cluster.

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CHAPTER 3. METHODOLOGY Start Number of clusters (k) Select/recalculate initial K centroids Calculate electrical distance of buses to centroids Assign buses to clusters based on minimum distance No bus changes cluster? No Yes End

Figure 3.1: K-means algorithm flow chart.

3.3 REI Equivalent Technique

The procedure for calculating the REI equivalent has been described in more details in references [18, 28, 33].

One distinctive feature of the REI equivalent is that it preserves power losses in the original and equivalent network through the zero power balance network. After identifying the essential and non essential buses of the network, the REI equivalent is constructed from the solved optimal power flow by computing the admittance values between the non-essential buses and the fictitious ground buses in the zero power balance network as shown in Equation 3.4-3.5. For instance, the admittance of a non-essential bus with a generation injection is identified by YGi. Similarly, the admittance

of a non-essential bus with a load injection is specified by YLi, where SGi and SLi are

the apparent power of generation and load in the non-essential bus i, respectively, and

Vi is the phase voltage of bus i.

Some buses might have both generation and load injections which need to be split up; Consequently, two admittances are calculated for each injection. Figure 3.2 provides

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a demonstration of constructing the zero power balance network.

Figure 3.2: Construction of the zero power balance network.

YGi = −S∗ Gi |Vi|2 (3.4) YLi = −S ∗ Li |Vi|2 (3.5)

The corresponding current injection IGiand ILifor each of the non-essential buses are

determined as IGi = S_Gi∗ V_i∗ (3.6) ILi= S_Li∗ V_i∗ (3.7)

The apparent injected power at non-essential buses is calculated as follows:

SGi = PGi+ jQGi (3.8)

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where P and Q are the active and reactive power injections, respectively.

The REI equivalent introduces two buses in the network, the generator bus and the load bus. The former aggregates the total generation of the eliminated buses while the latter aggregates the total demand of the eliminated buses. SG and SL represent the

apparent power injections at the REI generation and load bus, respectively, and are calculated using Equation 3.10-3.11.

SG = Nc ∑ i=1 SGi (3.10) SL= Nc ∑ i=1 SLi (3.11)

Where Ncis the total number of non-essential buses in the cluster. The current flows

IGand ILbetween the newly formed REI buses and the common grounded buses are

identified using Kirchoff’s theorem:

IG = Nc ∑ i=1 IGi (3.12) IL= Nc ∑ i=1 ILi (3.13)

In accordance with apparent power injections and corresponding injected current, the voltage values at the REI buses must be:

VG = SG I_G∗ (3.14) VL = SL I_L∗ (3.15)

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grounded buses are calculated as:

YG= S_G∗ |vG|2 (3.16) YL= S_L∗ |vL|2 (3.17)

In addition, the total shunt conductance and shunt susceptance of all non-essential buses in the network are divided by two in order to aggregated them equally to two new created REI buses as shown in Equation 3.18-3.19.

GREI−L = GREI−G = N c ∑ i=1 Gi/2 (3.18) BREI−L= BREI−G= N c ∑ i=1 Bi/2 (3.19)

3.4 Building REI Equivalent

In order to obtain an equivalent reduced network, it is necessary to eliminate the non-essential buses and common grounded buses in the developed zero power balance network, using the algorithm explained in the previous section, by applying the Gaussian elimination method. The new admittance matrix after introducing the two REI buses has the following form:

Ybus=  YEE YEN YN E YN N   (3.20)

Where YEE and YN N are the elements of the admittance matrix corresponding to all

essential and non-essential buses of the network, respectively.

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CHAPTER 3. METHODOLOGY demonstrated below.  YEE YEN YN E YN N    VEE VN N   =  IEE IN N   (3.21)

which can be rewritten as

YEEVEE + YENVN N = IEE (3.22)

YN EVEE + YN NVN N = IN N (3.23)

Solving for VN N in equation 3.23 and substituting in equation 3.22 results in:

VEE(YEE − YENY_{N N}−1YN E) = IEE − YENY_{N N}−1IN N (3.24)

It is therefore observed that

Yreduced= YEE − YENYN N−1YN E (3.25)

Ireduced= IEE − YENYN N−1IN N (3.26)

Now, the equivalent of the original network consists of the essential nodes consisting of boundary and retained buses, and the two REI buses.

Algorithm 1 REI method for each network area

Require: Cluster network data, retained buses, boundary buses

1. Calculate the admittance matrix of the network 2. Run OPF

3. Create two REI generation and load buses

3. Aggregate eliminated buses generation and load to the REI buses 4. Aggregate eliminated buses susceptance values to the REI buses 5. Build the zero power balance network

6. Calculate the equivalent network admittance matrix 7. Compute the transmission lines of the network

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3.5 Network Reduction Algorithm

In this subsection, the whole procedure of network reduction employed in this project is explained.

First of all, the user needs to provide the number k of clusters, minimum number of buses in each cluster, and retained buses. The retained buses are those of the user’s interest which could be, for instance, buses with high load or generation in order to investigate their impact on the network. Thereafter, the electrical distance of all bus pairs in the network is computed. It is a measure of the electrical connectivity between all nodes in the network in order to be used for the partitioning method to generate electrically cohesive clusters of buses. Based on the network data and user-defined inputs, the network is partitioned into k clusters of buses. After that, boundary buses connecting different clusters via tie lines are determined and therefore retained. In the event of eliminating the slack bus, a retained bus with the closest electrical distance is selected to be the reference bus.

The next step after identifying all buses to be eliminated in each cluster is to model the clusters by their equivalent circuits. From the solved Optimal Power Flow (OPF), the REI method is applied to each of these clusters to acquire simplified models. The procedure details of REI method is illustrated in section 3.3-3.4. As the equivalent circuits contain fewer number of buses, the numbering of all buses is recalculated. Finally, the equivalent circuits are reconnected through the tie lines resulting in an equivalent power system network. The process of network reduction is described in Figure 3.3.

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CHAPTER 3. METHODOLOGY Start Import Network Data Calculate Electrical Distance Apply partitioning method Input number of clusters (k), minimum number of buses in each

cluster, and retained buses All clusters meet clustering requirements No Yes

Apply REI reduction method on cluster i

Determine boundary buses

Stop Update slack bus

if it is eliminated i <= k No Yes Calculate bus numbering Reassemble network

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Chapter 4 Case Studies

It is necessary for the reduced equivalent network to provide similar results as those of the original network. In order to assess the performance of the reduction technique employed in this project, the equivalent network is compared with the original network by comparing the magnitude voltages of the retained buses, power loss on the branches that kept intact, and total power generation. The power system test cases utilised to examine and evaluate the performance of the proposed method are shown below.

1. The PJG 5-bus system [34]. 2. IEEE 30-bus case [35]. 3. IEEE 118-bus case [35].

The 5-bus system is utilized to examine the developed REI algorithm. And the IEEE 30-bus system is divided into three areas by using the K-medoids method followed by applying the REI algorithm to every area of the network. Similarly, IEEE 118-bus system is partitioned into three clusters by using both K-means and K-medoids. A detailed comparison and analysis of the equivalent and original networks is provided for every test case in the following sections. It should be noted that the partitioning methods are based on non-deterministic algorithms, meaning that different clusters of buses might be produced for every execution. Thus, the results shown in the section are executed consecutively and the evaluation metrics are calculated by taking their

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CHAPTER 4. CASE STUDIES

mean values for a number of 10 iterations, which was empirically chosen, and for larger numbers of executions the average values are not much different.

The following metrics are defined for assessing the accuracy of the method: • Flows of transmission lines that are kept intact

• Essential buses’ active power generation • Total active power generation

• Total reactive power generation

• Computational time for running the OPF • Memory allocated by the model

4.1 Test case: 5-bus system

G 2 3 4 5 1 G G G G

Figure 4.1: The 5-bus system.

As depicted in Figure 4.1, the 5-bus system includes 6 transmission lines, 5 generators, and 3 loads. The algorithm simulation of the reduced equivalent network is implemented in Julia. The equivalent network consist of 3 buses: Load, REI-Generation, and bus 1 in the original system. A comparison of the performances of

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the original and equivalent networks are shown in Table 4.1. The magnitude voltage, phase-angle, generation, and load of the buses in the obtained equivalent network are summarized in Table 4.2. In addition, the calculated results of the net active and reactive power injections in the equivalent network are listed in Table 4.3.

Table 4.1: Comparison of the original and equivalent networks of the 5-bus system. Parameter Original Network Equivalent Network

Number of buses 5 3

Active power generation (MW) 1005.191 1005.037 Reactive power generation (MVAr) 371.658 368.969 Computational time (sec) 0.178 0.029 Memory allocation (KiB1) 1634.304 381.266 Objective function 129660.694 14961.114

Table 4.2: Optimal power flow of case 5 equivalent network.

Bus Voltage Generation Load

Mag (pu) Ang (rad) P (MW) Q (MVAr) P (MW) Q (MVAr)

REI-L 1.103 -0.053 0.000 0.000 1000.0 328.7

REI-G 1.135 -0.011 795.037 211.469 0.000 0.000

1 1.100 0.000 210.000 157.500 0.000 0.000

Table 4.3: Power injection at case 5 equivalent network buses.

From Bus To Bus From Bus LineFlow To Bus LineFlow Loss

P (MW) Q (MVAr) P (MW) Q (MVAr) P (MW) Q (MVAr)

REI-L REI-G -543.084 -129.639 552.449 156.805 9.365 27.166

REI-L 1 -456.916 -195.736 444.545 219.255 12.371 23.518

REI-G 1 242.587 58.170 -234.545 -59.034 8.0424 0.864

Total -757.412 -267.205 762.449 317.026 5.037 49.821

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Table 4.4 shows a comparison of the voltage magnitude and active power generation in the original and equivalent networks. Some variables are existent either in the original network or reduced network, and therefore a hyphen is used to indicate a non-existent variable.

Table 4.4: Comparison between case 5 original and its equivalent.

Bus Voltage Magnitude (pu) Active power Generation (MW Original Network Reduced Network Original Network Reduced Network

1 1.078 1.100 210.000 210.00 2 1.084 - 0.00 -3 1.100 - 324.498 -4 1.064 - 0.000 -5 1.069 - 470.693 -REI-L - 1.103 - 0.000 REI-G - 1.135 - 795.037 Total 1005.191 1005.037

4.2 Test case: 30-bus system

G G G G G G 29 30 27 28 26 25 23 24 18 19 15 20 21 17 22 14 16 10 9 11 12 13 1 3 4 6 8 7 5 2

Figure 4.2: Single line diagram of the IEEE bus-30 system.

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methods. The network is first divided into 3 areas and simplified by reducing each area of the network. The original network consist of 30 buses, 41 branches, 6 generators, and 20 loads. Figure 4.2 shows the original IEEE 30-bus power system.

Because the K-medoids algorithm is non-deterministic, the mean of 10 scenarios was taken into account to compare the results of the reduction technique. A comparison of the original system and its equivalent is given in Table 4.5.

Table 4.5: Comparison of the original and equivalent networks of the 30-bus system. Parameter Original Network Equivalent Network

Number of buses 30 19.3

Active power generation (MW) 192.060 191.083 Reactive power generation (MVAr) 105.079 103.980 Computational time (sec) 0.143 0.091 Memory allocation (MiB2) 8.300 4.430 Objective function 576.892 573.080

A selected clustering solution of the network is examined to look more in-depth at the equivalent reduced network. Each area in the equivalent network includes the two newly created REI buses and essential buses. In this test case, essential buses consist of only boundary buses which have at least one connection with any bus in different area in the network. The selected clustering solution is shown in Figure 4.3 .

The AC optimal power flow method is used in order to determine the voltage magnitude, voltage angle, net real power injection, and net reactive power injection at every bus of the equivalent network, as shown in Table 4.6. As seen from the table, there are 2 REI buses for each cluster resulting in a total of 6 REI buses in the entire network. The REI-Ln and REI-Gn labels indicate the REI load bus and the REI generation bus, respectively, of the nth _{cluster. The remaining buses are border buses}

which are connected together, and, for comparison purposes, they are assigned the same bus number as in the original network.

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CHAPTER 4. CASE STUDIES 29 30 27 28 26 25 23 24 18 19 15 20 21 17 22 14 16 10 9 11 12 13 1 3 4 6 8 7 5 2 G G G G G G

Figure 4.3: Selected clustering solution for the 30-bus system.

Table 4.6: Summary of the network equivalent results for a selected clustering solution.

Cluster Bus Number Voltage Generation Load

1 REI-L1 0.988 -0.049 0.000 0.000 40.300 22.100 REI-G1 1.012 -0.058 22.478 27.375 0.000 0.000 5 0.984 -0.028 0.000 0.000 0.000 0.000 6 0.982 -0.039 0.000 0.000 0.000 0.000 10 0.998 -0.061 0.000 0.000 5.800 2.000 17 0.994 -0.065 0.000 0.000 9.000 5.800 24 1.016 -0.050 0.000 0.000 8.700 6.700 2 REI-L2 1.000 -0.062 0.000 0.000 40.700 17.100 REI-G2 1.026 0.036 55.465 10.866 0.000 0.000 2 0.995 0.000 59.513 23.878 21.700 12.700 4 0.985 -0.032 0.000 0.000 7.600 1.600 16 1.000 -0.062 0.000 0.000 3.500 1.800 20 0.988 -0.076 0.000 0.000 2.200 0.700 23 1.023 -0.048 14.220 7.475 3.200 1.600 3 REI-L3 1.047 -0.029 0.000 0.000 16.500 5.100 REI-G3 1.080 0.001 39.751 34.432 0.000 0.000 8 0.971 -0.047 0.000 0.000 30.000 30.000 25 1.050 -0.021 0.000 0.000 0.000 0.000 28 0.991 -0.039 0.000 0.000 0.000 0.000

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A comparison of the voltage magnitude and active power generation at each node of the 30-bus system original and equivalent networks is presented in Table 4.7. It is worth noting out that some variables are existent either in the original network or reduced network, and therefore a hyphen “-” is used to indicate a non-existent variable.

Table 4.7: Comparison of the original and equivalent networks of the 30-bus system.

Bus Number Voltage Magnitude (pu) Active Power Generation (MW)

Original Network Equivalent Network Original Network Equivalent Network

1 0.982 - 41.542 -2 0.979 0.995 55.402 59.513 3 0.977 - 0 -4 0.976 0.985 0 0 5 0.97 0.984 0 0 6 0.972 0.982 0 0 7 0.962 - 0 -8 0.961 0.971 0 0 9 0.990 - 0 -10 0.999 0.998 0 0 11 0.990 - 0 -12 1.017 - 0 -13 1.064 - 16.2 -14 1.007 - 0 -15 1.009 - 0 -16 1.002 1.000 0 0 17 0.995 0.994 0 0 18 0.993 - 0 -19 0.987 - 0 -20 0.989 0.988 0 0 21 1.009 - 0 -22 1.016 - 22.741 -23 1.025 1.023 16.267 14.220 24 1.016 1.016 0 0 25 1.043 1.050 0 0 26 1.027 - 0 -27 1.069 - 39.909 -28 0.982 0.991 0 0 29 1.050 - 0 -30 1.039 - 0 -REI-L1 - 0.988 - -REI-G1 - 1.012 - 22.478 REI-L2 - 1.000 - 0 REI-G2 - 1.026 - 55.465 REI-L3 - 1.047 - 0 REI-G3 - 1.080 - 39.751 Total 192.061 191.427

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The power flow through the transmission lines between all clusters’ boundary buses is also compared with its corresponding branches in the original network. Figure 4.4 compares the the power flow of the branches that kept intact, and Figure 4.5 shows the power flow between the clusters.

0 5 10 15 20 25 30 2-5 2-6 4-6 6-8 16-17 10-20 23-24 25-24 28-6 Re al P ow er F lo w (M W )

Branch (From - To)

Original network Equivaent network

Figure 4.4: Comparison of active power flow in the original network and its equivalent.

G = 39.751 MW D = 46.5 MW G = 129.198 MW D = 78.900 MW G = 22.478 MW D = 63.800 MW 48.391 MW 7.495 MW 49.541 MW 7.314 MW

Figure 4.5: Active power flow between clusters for the selected solution. The power flow in red color represents the original power flow

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4.3 Test case: 118-bus system

Figure 4.6: Single-line diagram of the IEEE 118-bus system, by University of Washington [36].

The IEEE 118-bus system is utilized in this section to examine the reduction method. The original version of the IEEE 118-bus system is shown in Figure 4.6. The system network contains 118 buses, 54 generators, 99 loads, and 186 transmission lines. The partitioning methods K-means and K-medoids are applied to partition the system network into 3 clusters. Since both methods may not produce the same clusters for different runs, the mean of 10 scenarios are considered. The results of comparing the performance of the reduction method by using both k-means and K-medoids are summarized in Table 4.8.

The AC optimal power flow is calculated on the equivalent network for a selected clustering solution of both k-means and k-medoids. The OPF results of the equivalent networks are shown in Table 4.9-4.10. The selected clustering solution of the k-means

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comprises 14 buses and 6 fictitious REI buses. Similarly, the clustering solution based on the k-medoids includes 20 buses, of which 6 buses are REI generation and load buses.

Table 4.8: Comparison of the original and equivalent networks of the 118-bus system.

Parameter Original Network Eq.3(K-means) Eq. (K-medoids)

Number of buses 118 21.9 22.6

Active power generation (MW) 4319.401 4302.745 4305.242

Reactive power generation (MVAr) 388.264 360.555 372.351

Computational time (sec) 0.677 0.0.137 0.125

Memory allocation (MiB) 35.350 7.100 7.030

Objective function 129660.690 128801.890 128818.920

Table 4.9: OPF result of 118-bus equivalent network by applying K-means.

Cluster Bus Voltage Generation Load

1 REI-L1 1.050 -0.094 0.000 0.000 1397.000 558.000 REI-G1 1.087 -0.047 1304.442 405.117 0.000 0.000 69 1.060 0.000 473.661 -219.029 0.000 0.000 74 1.021 -0.108 33.415 9.000 68.000 27.000 75 1.025 -0.101 0.000 0.000 47.000 11.000 81 1.060 -0.043 0.000 0.000 0.000 0.000 2 REI-L2 1.059 -0.225 0.000 0.000 1050.000 380.000 REI-G2 1.118 -0.038 998.318 -62.830 0.000 0.000 24 1.060 -0.088 30.113 -7.423 13.000 0.000 34 1.046 -0.210 37.907 -7.466 59.000 26.000 37 1.049 -0.205 0.000 0.000 0.000 0.000 38 1.060 -0.149 0.000 0.000 0.000 0.000 39 1.031 -0.227 0.000 0.000 27.000 11.000 3 REI-L3 1.035 -0.148 0.000 0.000 1298.000 345.000 REI-G3 1.060 -0.112 729.548 58.317 0.000 0.000 40 1.031 -0.225 77.168 20.014 66.000 23.000 43 1.019 -0.225 0.000 0.000 18.000 7.000 47 1.037 -0.119 0.000 0.000 34.000 0.000 49 1.048 -0.113 216.444 117.851 87.000 30.000 65 1.060 -0.034 379.443 -1.207 0.000 0.000 68 1.052 -0.042 0.000 0.000 0.000 0.000 70 1.044 -0.093 11.918 30.311 66.000 20.000 72 1.053 -0.086 17.458 -4.904 12.000 0.000

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Table 4.10: OPF result of 118-bus equivalent network by applying K-medoids.

Cluster Bus Voltage Generation Load

1 REI-L1 1.061 -0.281 0.000 0.000 1050.000 380.000 REI-G1 1.119 -0.094 986.323 -57.842 0.000 0.000 24 1.060 -0.126 24.824 -4.283 13.000 0.000 34 1.049 -0.277 15.531 17.182 59.000 26.000 37 1.051 -0.269 0.000 0.000 0.000 0.000 38 1.058 -0.204 0.000 0.000 0.000 0.000 39 1.027 -0.299 0.000 0.000 27.000 11.000 2 REI-L2 1.046 -0.038 0.000 0.000 1702.000 596.000 REI-G2 1.082 0.019 1445.921 357.021 0.000 0.000 68 1.060 -0.052 0.000 0.000 0.000 0.000 69 1.044 0.000 476.793 -86.566 0.000 0.000 70 1.034 -0.093 29.527 32.000 66.000 20.000 72 1.050 -0.104 20.475 -4.268 12.000 0.000 3 REI-L3 1.011 -0.213 0.000 0.000 1108.000 345.000 REI-G3 1.043 -0.150 683.726 66.689 0.000 0.000 40 1.024 -0.302 52.285 24.874 66.000 23.000 43 1.015 -0.286 0.000 0.000 18.000 7.000 47 1.018 -0.155 0.000 0.000 34.000 0.000 49 1.029 -0.153 203.391 103.482 87.000 30.000 65 1.053 -0.058 367.240 -67.000 0.000 0.000 Total 4306.036 381.289 4242.000 1438.000

As seen from Table 4.9-4.10, the equivalent networks consist only of the fictitious REI buses, REI-L for load and REI-G for generation, and the essential buses, which are assigned the same bus number as in the original network. As an illustration, the equivalent network (Table 4.9) is a representation of the IEEE 118-bus system consisting of three clusters. Each cluster contains two REI buses and boundary buses, and all clusters are connected together to form one power system network.

Table 4.11 compares the voltage magnitudes and active power generation at the essential buses and REI buses. As observed in the table, the magnitude voltages at the essential buses are not identical to those of the original network as they contain small errors with a percentage error less than 5%. Figure 4.7-4.8 provide a comparison of the active power flow through the transmission lines between boundary buses of the selected K-means and K-medoids clustering solutions.

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Table 4.11: Comparison of original and equivalent 118-bus system networks.

Bus id Magnitude Voltage (pu) Active Power Generation (MW) Original k-medoids k-means Original k-medoids k-means 24 1.046 1.060 1.060 0.000 24.824 30.113 34 1.056 1.049 1.046 4.888 15.531 37.907 37 1.060 1.051 1.049 0.000 0.000 0.000 38 1.015 1.058 1.060 0.000 0.000 0.000 39 1.042 1.027 1.031 0.000 0.000 0.000 40 1.042 1.024 1.031 49.322 52.285 77.168 43 1.040 1.015 1.019 0.000 0.000 0.000 47 1.045 1.018 1.037 0.000 0.000 0.000 49 1.050 1.029 1.048 193.332 203.391 216.444 65 1.016 1.053 1.060 352.236 367.240 379.443 68 1.015 1.060 1.052 0.000 0.000 0.000 69 1.060 1.044 1.060 453.666 476.793 473.661 70 1.039 1.034 1.044 0.000 29.527 11.918 72 1.040 1.050 1.053 0.000 20.475 17.458 74 1.022 - 1.021 16.931 - 33.415 75 1.024 - 1.025 0.000 - 0.000 81 1.011 - 1.060 0.000 - 0.000 REI-L1 - 1.061 1.050 - 0.000 0.000 REI-G1 - 1.119 1.087 - 986.323 1304.442 REI-L2 - 1.046 1.059 - 0.000 0.000 REI-G2 - 1.082 1.118 - 1445.921 998.318 REI-L3 - 1.011 1.035 - 0.000 0.000 REI-G3 - 1.043 1.060 - 683.726 729.548

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CHAPTER 4. CASE STUDIES 0 20 40 60 80 100 120 140 160 37-40 39-40 34-43 65-38 69-47 69-49 69-68 70-24 69-70 24-72 70-74 70-75 81-68 Re al P ow er F lo w ( M W )

Branch (From - To)

Original network Equivalent network

Figure 4.7: Comparison of active power flow for the K-means clustering solution.

44 16 18 3.63499759 -1.1150955 -3.7369331 0.56570367 45 16 19 -12.656934 7.64107919 13.7177299 -5.6956915 46 16 20 49.7120392 21.4966869 -44.250446 -33.846875 47 17 18 -7.4705062 2.01168262 7.68969298 -1.0249514 48 17 19 -10.56676 7.53906208 11.6254357 -6.1560033 49 17 20 2.12700589 11.6722898 0.58697012 -12.296766 50 18 19 -9.9043243 -17.226518 9.97196429 17.4320998 51 18 20 1.00012503 3.19516272 -0.7084531 -3.3913144 52 19 20 -73.697873 -36.860037 71.5267591 44.7437436 53 3 12 -8.6953949 6.86601883 8.69780935 -6.4164595 54 3 13 -10.637609 8.27256709 10.7164786 -7.9557956 55 5 16 24.0084479 8.78538127 -23.657639 -7.7915206 56 7 16 6.70085229 3.59616658 -6.6907703 -3.5630167 57 4 17 9.88923443 18.4245914 -9.7249905 -17.756083 58 6 20 -161.51248 32.2427236 163.694025 -8.3692554 59 20 10 -40.737787 -40.044292 40.7783709 40.5148267 60 18 11 -55.427375 11.9605423 58.046185 -3.3408103 61 19 11 -46.443243 13.2286662 48.6133916 -6.0903103 0 20 40 60 80 100 120 140 160 180 37-40 39-40 34-43 65-38 69-47 69-49 70-24 72-24 65-68 Re al P ow er F lo w (M W )

Branch (From - To)

Original network Equivalent network

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4.4 Evaluation

In this section, a series of scenarios is executed for the evaluation of the efficiency and accuracy of the developed model. The evaluation findings helps us obtain a better understanding of the performance of the model. The scenarios considered for the evaluation of the model are the number of clusters, the effect of slack bus assumptions, and the voltage tolerance limits of the REI buses.

4.4.1 Number of Clusters

The IEEE 118-bus system is utilized to test the effect that the number of clusters has on the efficiency of the reduction technique. The K-means clustering algorithm is employed to generate the clusters for each number of clusters. The developed model was executed 10 times and the mean values of the results were obtained. Table 4.12 provides comparison for clustering solutions with different selection of the number of clusters.

Table 4.12: Comparison of the effects of number of clusters.

Clusters 2 3 4 5 6

Number of buses 14 21.9 30.3 36.2 43.7

Active power generation error 0.33% 0.38% 0.85% 0.44% 0.89%

Reactive power generation error 10.93% 4.23% 7.94% 8.12% 9.06%

Computational time (sec) 0.108 0.136 0.178 0.196 0.235

Memory allocation (MiB) 4.373 7.100 9.290 11.107 13.290

Objective function error 0.61% 0.66% 1.12% 0.59% 1.14%

As shown in the table, an equivalent circuit of a power system network with a few number of clusters can reduce the total execution time as well as the memory allocation. Additionally, it can be observed that partitioning the IEEE 118-bus network into 5 clusters gives worse quality results indicating it is not the optimal number of clusters for the IEEE 118-bus system network. Overall, dividing the network into 3 clusters significantly reduces the total number of buses of the equivalent network while delivering the best performance with low errors of active and reactive power generation.

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4.4.2 Retaining Original Slack Bus

The slack bus of the network acts as a reference point for all voltage angles of the buses in the network. The slack bus in the original network might need to be eliminated during the reduction process if it is not of interest. In this case, a new slack bus is assigned to the retained bus with the closest electrical distance. To better investigate the effect of retaining, or eliminating, the slack bus, the slack bus of the considered network should not be a boundary bus meaning that it can be eliminated if desired. For this reason, the IEEE 30-bus system is used to test the effect of the slack bus. Table 4.13 shows a comparison between a network equivalent of which a new slack bus is selected and a network equivalent with the original slack bus being retained.

Table 4.13: Comparison of the equivalences obtained by retaining and eliminating the slack bus.

Criteria Eq. with updated slack Eq. with original slack

Number of buses 19.3 20.5

Active power generation error 0.51% 0.27%

Reactive power generation error 1.05% 3.00%

Computational time (sec) 0.087 0.098

Memory allocation (MiB) 4.430 4.520

Objective function error 0.66% 0.39%

Although retaining the original slack in the corresponding equivalent network does not seem to have an impact on the computational time and the memory allocation, it minimises both the real power generation and objective function errors. In addition, the phase-angle voltage error of the retained buses decreases when the original reference bus is retained, as illustrated in Figure 4.9. The voltage magnitude profile of the equivalent circuits is presented in Table 4.14.

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CHAPTER 4. CASE STUDIES 18 -0.096 -0.077 -0.088 23 -0.066 -0.043 -0.059 24 -0.068 -0.046 -0.064 25 -0.036 -0.020 -0.039 0 0,005 0,01 0,015 0,02 0,025 0,03 2 4 5 6 7 9 10 15 16 17 18 23 24 25 Ab so lu te V ol ta ge An gl e Er ro r ( Ra d) Bus Number

Eliminated slack bus Retained slack bus

Figure 4.9: Comparison of the voltage angle error.

Table 4.14: Comparison of OPF results for retaining the original slack bus.

Bus Num. Original Eliminated slack bus Retained slack bus

VM (pu) VA (rad) VM (pu) VA (rad) VM (pu) VA (rad)

1 0.982 0.000 - - 1.050 0.000 2 0.979 -0.013 1.063 0.000 1.048 -0.013 3 0.977 -0.042 - - - -4 0.976 -0.050 1.031 -0.028 1.030 -0.043 5 0.971 -0.043 1.039 -0.025 1.032 -0.040 6 0.972 -0.056 1.021 -0.033 1.023 -0.050 7 0.962 -0.061 1.019 -0.038 1.017 -0.054 8 0.961 -0.064 - - - -9 0.990 -0.072 1.013 -0.051 1.031 -0.066 10 1.000 -0.080 1.008 -0.061 1.036 -0.074 11 0.990 -0.072 - - - -12 1.017 -0.079 - - - -13 1.064 -0.058 - - - -14 1.007 -0.088 - - - -15 1.009 -0.084 1.021 -0.066 1.044 -0.076 16 1.003 -0.084 1.020 -0.067 1.040 -0.077 17 0.995 -0.085 1.006 -0.066 1.032 -0.079 18 0.993 -0.096 1.004 -0.077 1.029 -0.088 19 0.987 -0.099 - - - -20 0.990 -0.096 - - - -21 1.009 -0.081 - - - -22 1.016 -0.079 - - - -23 1.026 -0.066 1.017 -0.043 1.056 -0.059 24 1.017 -0.068 1.003 -0.046 1.039 -0.064 25 1.044 -0.036 1.009 -0.020 1.040 -0.039 26 1.027 -0.043 - - - -27 1.069 -0.012 - - - -28 0.982 -0.056 - - - -29 1.050 -0.032 - - - -30 1.039 -0.046 - - -

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-CHAPTER 4. CASE STUDIES

4.4.3 Voltage Limits of REI Buses

In this section, the maximum and minimum voltage limit of the fictitious REI buses are modified and tested. The IEEE 30-bus system is utilized to study the impact of the voltage tolerance limit of the REI buses on the voltage magnitude of the retained buses. It is to point out that increasing the tolerance limits more than 5% makes no difference. Figure 4.10 shows the impact of the REI buses when their voltage limits vary.

0,95 0,97 0,99 1,01 1,03 1,05 1,07 1 2 4 5 6 7 9 10 15 16 17 18 23 24 25 Vo lta ge M ag ni tu de (p u) Bus Number Original 1% 2% 3% 4% 5%

Figure 4.10: Voltage magnitude of the retained buses for different voltage tolerance limit at REI buses.

The resulted magnitude voltages vary the most when the tolerance limit is set to 5% while they behave closely to those of the original system when the limit is set to 1%. Therefore, minimizing the the voltage limits of the REI buses gives the best performance. However, tightening the voltage limit constraints of the REI buses may result in a non-convergent solution.

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Chapter 5 Conclusion

A reduction model based on the REI methodology has been developed, and numerically tested, in order to reduce the complexity of the power flow modelling of large power networks. The resulted equivalent network can be utilized rather than the original system for various network planning and analysis studies. The effectiveness of the proposed model was tested on 3 different test cases, namely the PJM 5-bus, the IEEE 30-bus, and the IEEE 118-bus systems. The performance of the equivalent networks is compared with that of the original network by assessing several indices. Those indices include the power flow of the transmission lines which are kept intact, the active power generation of the essential buses, the total power generation, the computational time, and the memory allocation.

The equivalent circuit of the 5-bus system contains 3 buses with active and reactive power generation errors of less than 1.00% of its original system. The 30-bus system equivalent network contains 19 buses in average, with active and reactive power generation errors of 0.50% and 1.00 %, respectively. The computation time of the OPF is decreased about 36.36% while maintaining a small OPF objective function error of 0.66%. By applying the reduction model to the 118-bus 10 times due the non-deterministic behaviour of the clustering methods, it is observed that the number of buses is reduced to 22 buses. The resulted active and reactive power generation present an error of 0.35% and 5.87%, respectively. Also, the simulation results of the equivalent

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CHAPTER 5. CONCLUSION

network have shown a significant improvement in the computational time and memory allocation.

The simulation results indicated that the developed reduction model can be used for efficiently producing estimates of the original system’s operational aspects. Additionally, the results show that the reduction model substantially improve the computational time of the optimal power flow and the memory allocation. Furthermore, the evaluations show that the performance of the reduction model can be enhanced by selecting the suitable number of clusters and minimizing the tolerance limit of the fictitious REI buses while retaining the original slack bus in the equivalent network.

5.1 Future Work

The model presented in this thesis project is based on the K-means and the K-medoids algorithms for network partitioning, and the REI methodology for creating a reduced power flow model. The partitioning methods tend to calculate solutions based on random initializations, meaning that different clusters of buses are generated for every execution. Therefore, the influence of the random start points for the clustering can be examined to determine whether it is preferable to utilize user-defined start points or other partitioning methods of increased quality with deterministic algorithm. In addition, the developed model can be further investigated and tested by evaluating the performance of the resulted equivalent network and effort required to complete it by using very large power system networks. Moreover, other equivalencing techniques can be implemented and integrated with the reduction model of this project, and their results can be compared with those obtained by applying REI method.

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Development of simplified power grid models in EU project Spine

Development of simplified power

grid models in EU project Spine

MOHAMMAD ALHARBI

Author

Examiner

Supervisor

Abstract

Keywords

Abstract

Acknowledgements

Contents

1 Introduction

1

2 Literature Study

5

3 Methodology

12

4 Case Studies

21

5 Conclusion

38

List of Figures

List of Tables

Chapter 1

Introduction

1.1

Background

1.2

Spine Project

1.3

PowerModels

1.4

Problem Definition

1.5

Objective

1.6

Report Structure

Chapter 2

Literature Study

2.1

Network partitioning

2.2

K-means method

2.3

K-mediods method

2.4

Optimal K-decomposition

2.5

Evolution of network equivalence

2.6

Network equivalent

2.7

Ward equivalent

2.8

REI equivalent

Chapter 3

Methodology

3.1

Electrical Distance

3.2

Partitioning methods

3.3

REI Equivalent Technique

3.4

Building REI Equivalent

3.5

Network Reduction Algorithm

Chapter 4

Case Studies

4.1

Test case: 5-bus system

4.2

Test case: 30-bus system

4.3

Test case: 118-bus system

4.4

Evaluation

4.4.1

Number of Clusters