Implementation of a Voltage Sweep Power Flow Method and Comparison with Other Power Flow Techniques

Transcription

1 power systems eehlaboratory Feifei Teng Implementation of a Voltage Sweep Power Flow Method and Comparison with Other Power Flow Techniques Semester Thesis PSL 1432 EEH Power Systems Laboratory Swiss Federal Institute of Technology (ETH) Zurich Examiner: Prof. Dr. Göran Andersson Supervisor: Dr. Stephan Koch Zurich, November 20, 2014

2 Abstract Traditional power flow analysis techniques such as the Newton-Raphson and the Gauss-Seidel methods are widely used in analyzing power transmission systems. However, they are inefficient and may diverge due to the different network characteristics of power distribution systems such as radial and high R/X ratio. Therefore, other techniques such as the voltage sweep algorithms are developed for power distribution systems. In this thesis, the forward and backward sweep algorithm is studied and validated in Java. The sweep algorithm can deal with balanced networks which are radial or weakly meshed and contain distributed generators. Networks with different topologies are implemented to assess the convergence behavior of the sweeping algorithm and comparisons with established methods in the open-source Java power flow package JPower (the Newton-Raphson and Gauss-Seidel methods) are made. Results show that the sweeping algorithm is more efficient in analyzing power distribution systems: it has the least runtime among the three methods and the time-saving advantage is more obvious for networks with less loops and PV nodes. i

3 Acknowledgments This semester project was conducted at the ETH spin-off Adaptricity since 09/2014. The full time project made me concentrate on my work and time runs really fast for a 7-week project. First of all, I would like to express my gratitude to my supervisor Dr. Stephan Koch for his support and guidance during this work. Thanks for the opportunity he gave me to learn hands-on knowledge about Java programming and power flow analysis. I like especially the atmosphere in the Adaptricity office where everyone is so nice. Thanks my parents for their consistent encouragement. How lucky I am to have such good parents! I also would like to thank all the friends around me. Thanks for being someone in my life. ii

4 Contents List of Figures List of Tables v vi 1 Introduction Background of the project Power flow analysis Newton-Raphson method Gauss-Seidel Iteration Differences between power transmission and distribution systems Goal of the project Structure of the thesis Forward/Backward Sweep Approaches Element ordering Load flow equations Power summation method Kirchhoff s Law based method Weakly meshed networks Distributed generators modeling Transformers modeling Convergence criteria Algorithm Implementation Element ordering Loop impedance matrix Voltage sweep calculation Breakpoint power injection Convergence criteria Implementation results Assessment on the sweeping algorithm Voltage mismatches after each iteration iii

5 CONTENTS iv Influence of convergence criteria Influence of loops and PV nodes numbers Influence of PV node position Comparison with the Gauss-Seidel and the Newton-Raphson methods Runtime comparison Calculation results comparison Conclusions 25 A Benchmark networks 26 Bibliography 29

6 List of Figures 1.1 The Newton-Raphson method iteration scheme Ordered network example π circuit model of a branch Network illustrating adding breakpoints in loops and PV nodes Breakpoint current injection The breakpoint power injection method PV bus breakpoint Single-phase transformer model Flowchart of the proposed algorithm Successive voltage difference for all buses Voltage mismatches in breakpoints Runtime and iteration number under different convergence criteria Runtime and iteration number with increasing loops Runtime and iteration number with increasing PV nodes Iteration number for different PV node position Runtime comparison Voltage magnitude differences between the sweeping algorithm and the N-R method Voltage magnitude differences between the sweeping algorithm and the G-S method A.1 14-bus network A.2 29-bus network A bus network v

7 List of Tables 4.1 Characteristics of the benchmark networks Runtime comparison of different methods vi

8 Chapter 1 Introduction 1.1 Background of the project Traditional power flow analysis techniques [1] such as the Newton-Raphson (N-R) and the Gauss-Seidel (G-S) methods are frequently used in power transmission systems. However, for special cases which are usually called ill-conditioned power systems, these methods might be unstable and diverge [2]. Therefore, for power distribution systems which have high R/X ratio and are often regarded as ill-conditioned, other methods such as the forward and backward sweep algorithm are used and developed. 1.2 Power flow analysis There are basically four variables related to bus b, bus voltage magnitude U b, voltage angle θ b, active power P b and reactive power Q b. Three basic types of buses are defined depending on which of the four variables are given [3]: PQ bus: P b and Q b are given; U b and θ b are to be calculated, it represents load buses without voltage control. PV bus: P b and U b are given; Q b and θ b are to be calculated, it represents generation buses with voltage control. Uθ bus: U b and θ b are given; P b and Q b are to be calculated, it is also called slack bus and is used to balance load or generation and serves as voltage angle reference. After defining all the buses in the network, the unknown variables of every bus are calculated using equality and inequality constraints of the network. Therefore, power flow analysis is about solving a series of equations with system parameters and get the result of the system states. A general power flow equation can be written as Eq. (1.1). f(x, u, p) = 0, (1.1) 1

9 CHAPTER 1. INTRODUCTION 2 where x is a vector of unknown system states, which refers to the unknown variables of each bus. u is a vector with control output such as the active power and voltage magnitude of a PV bus. p is system parameters such as series impedance and shunt admittance of branches. Therefore, the key to analyze a power system is to find efficient ways to solve power flow Eq. (1.1). Many algorithms have been adapted and developed in the past few years. The Newton-Raphson method (N-R) and the Gauss-Seidel (G-S) method are among the most widely used methods which have an excellent convergence characteristic and calculation efficiency Newton-Raphson method The Newton-Raphson method is the most widely used iterative technique [4]. Its basic iteration scheme is illustrated in Figure 1.1. f(x) f(x 0 ) f(x 1 ) x 2 x 1 x 0 x Figure 1.1: The Newton-Raphson method iteration scheme Start with initial value of x (0), the state x is updated using Eq. (1.2) and (1.3): where J(x (v) ) is the Jacobian matrix with J(x (v) ) x (v) = f(x (v) ), (1.2) x (v+1) = x (v) + x (v), (1.3) J(x (v) ) = f(x) x. (1.4) The N-R algorithm converges to the true value of x after several iterations. For power system analysis, the state x which needs to be solved and the

10 CHAPTER 1. INTRODUCTION 3 function f are formulated as [3]: ( ) θ x =, (1.5) U ( ) P(x) f = = 0, (1.6) Q(x) where P(x) and Q(x) are the differences between active and reactive power with their specified value. The solution of x is updated using Eq. (1.7): where ( ) θ J(x (v) (v) ) U (v) + P J = θ Q θ ( P(x (v) ) ) Q(x (v) = 0, (1.7) ) P U Q. (1.8) U Reference [1] gives detailed explanation of the N-R method application in power system analysis Gauss-Seidel Iteration Given the same system equation as Eq. (1.1), the Gauss-Seidel iteration (G-S) is based on the iteration scheme of Eq. (1.9): x (v+1) = f(x (v) ). (1.9) Given an initial estimate of x (0), a series of state values could be obtained using Eq. (1.9): x (1), x (2), x (3) If the solution converges, i.e., x (v) f(x (v) < ɛ when x v x, it can be written as x = f(x ). (1.10) Differences between power transmission and distribution systems According to the respective functions of power transmission and distribution systems, they differ in multiple aspects. Voltage level: the transmission system is used to carry electricity over long distances. It is at a higher voltage level in order to reduce losses. However, the distribution system is the final step of the delivery to end consumers and is operated at a lower voltage level.

11 CHAPTER 1. INTRODUCTION 4 Topology: a transmission system is usually meshed whereas a distribution system typically has a radial or weakly meshed network. Branch type: the transmission system is mainly composed of overhead lines, which have a lower R/X ratio. For a distribution system, the structure of both overhead lines and cables makes the R/X ratio higher. The above mentioned algorithms N-R and G-S are particularly useful in power transmission systems. That is because they are especially efficient and have a good convergence characteristic in dealing with meshed networks with low R/X ratios. However, in the case of a power distribution system, convergence problems make it necessary to find new possibilities to solve the power flow equations. There are two main categories of distribution power systems analysis algorithms. One is based on modified traditional methods N-R and G-S as in [5], [6], [7], [8]. The other way is the forward and backward sweep algorithm using Kirchhoff s laws or bi-quadratic equations [9], [10]. The forward and backward sweep method makes use of the ladder structure of a radial network and solves the power flow equations step by step. It avoids complex calculation of matrices and thus gets rid of convergence problems caused by high R/X ratio but needs extra effort when dealing with meshed networks. A detailed explanation of the sweeping algorithm is presented in Chapter Goal of the project In this semester thesis, a power flow computation method based on the voltage sweeping algorithm shall be implemented in the distribution system simulator. It is realized in Java. The computations shall be validated by comparing the results to an established power flow computation tool JPower [11](Java port of MATPOWER [12]). The performance (solution time for a power flow on a large distribution grid) shall be assessed and compared with the Newton-Raphson and the Gauss-Seidel techniques available in JPower. 1.4 Structure of the thesis In Chapter 2, the principle of the sweeping algorithm is explained, including different power flow equations and the way to deal with meshed networks. The model of distributed generators is also given. In Chapter 3, the sweeping algorithm we implemented in the project is introduced. In Chapter 4, the implementation results in Java are presented including the individual assessment on the sweeping algorithm and comparisons with JPower.

12 Chapter 2 Forward/Backward Sweep Approaches During the last few decades, a variety of different approaches based on the forward/backward sweep algorithm are used by researchers to get good convergence and fast calculation speed of power distribution systems. By avoiding solving simultaneous equations and large dimension matrices, the sweeping algorithm has the advantages of less computation effort and calculation time compared to the N-R and G-S methods. During the last several years, multiple sweep based approaches are developed to adapt to different power system network structures and load types. This chapter explains the principle of the forward and backward sweep algorithms. 2.1 Element ordering An element ordering process must be carried out to determine the calculation sequence in the forward and backward sweep. Here the commonly used branch and node oriented method is introduced [13]. Figure 2.1 is an example of an ordered network. As it shows, the network is divided into different layers starting from the root bus. Each branch is ordered only after all the branches in its upper layers have been ordered. For the forward sweep, the calculation starts from the root bus to the last ordered branch. For the backward sweep, the calculation is from the last ordered bus back to the root bus. 2.2 Load flow equations According to the different power flow equations used in the sweeping process, there are power summation method, current summation method and 5

13 CHAPTER 2. FORWARD/BACKWARD SWEEP APPROACHES 6 1 root bus Figure 2.1: Ordered network example admittance summation method[10], [14]. Here the first two approaches are introduced Power summation method Step 1: Backward sweep Consider the π model of a branch i in a distribution network in Figure 2.2, the series impedance is R i + jx i and the shunt admittance is y i. The power in the sending and receiving end are P s + jq s and P r + jq r respectively. P i +jq i and P i +jq i are the power flow after the shunt element in sending end and before the shunt element in receiving end respectively. Therefore, the relation between the powers in both sides can be written as [15]: P I s + jq I s P i + jq i R i jx i P i + jq i P I r + jq I r P F r + jq F r P L s P i + jq i y i y i + jq L 2 2 s P L r + jq L r Figure 2.2: π circuit model of a branch P i = P L r + P F r P I r, (2.1) Q i = Q L r + Q F r Q I r V 2 my i /2, (2.2) where the superscripts L, F and I in P and Q stand for load, flow and injection respectively. Load power P L is given before calculation. The active

14 CHAPTER 2. FORWARD/BACKWARD SWEEP APPROACHES 7 and reactive power flow through branch i can be formulated as, P i = P i P i 2 + Q 2 i + R i V 2 r Q i = Q 2 P i 2 + Q i i + X 2 i V 2 m, (2.3) V 2 k y i/2. (2.4) If the network is radial, the power injections are assumed to be zero in every iteration. For meshed networks, the calculation of power injection P I will be explained in the meshed network part. Eq. (2.1) (2.4) are calculated backward to get active and reactive power flow for every branch. Step 2: Forward sweep With active and reactive power values available in the backward sweep, the complex voltage can then be calculated in a forward sequence using Eq. (2.5). V r = V s I i Z i = V k S i V (R i + jx i ) k = V k P i jq i (R i + jx i ) V k = (V k P i R i + Q i X i ) j( P i X i Q i R i ), (2.5) V k V k where S i = P i + jq i, P i = P i, Q i = Q i + Vk 2y i/2. Eq. (2.5) is an application of Kirchhoff s Voltage Law. Since the complex voltage of the slack bus is specified, voltage of other buses can be attained in a forward sweep calculation. The backward and forward sweep equations are calculated iteratively until it converges Kirchhoff s Law based method This approach [10] is a direct application of Kirchhoff s Current Law (KCL) and Kirchhoff s Voltage Law (KVL). Given the voltage of the root bus and an initial voltage guess of other buses, the algorithm takes three steps for each iteration: Step 1: Nodal current calculation: The current injection at each node i is calculated using Eq. (2.6): I (k) i = (S i /V (k) i ) y i V (k 1) i, i = 1, 2,..., n (2.6) where S i is the power injection at node i, V (k) i is the voltage of node i calculated from iteration k, y i is the shunt element of node i. Step 2: Backward sweep:

15 CHAPTER 2. FORWARD/BACKWARD SWEEP APPROACHES 8 Starting from the last ordered branch, current flow J l in branch l is calculated using Eq. (2.7): J (k) l = I lr + J lr, l = b, b 1,..., 1 (2.7) where I lr is the current injection of node lr calculated from step 1, J lr is the currents in branches emanating from node lr. Step 3: Forward Sweep: Starting from the root bus, the node voltages are updated using Eq. (2.8). V (k) lr = V (k) ls Z l J (k) l, l = 1, 2,..., b (2.8) where ls and lr denote the sending and receiving end of branch l, Z l is the series impedance of branch l. The element ordering process is the same as in the power summation method. There are also other forward and backward sweep methods such as [14], [16], [17]. Reference [14] proposed an admittance summation method which is quite efficient for constant admittance loads. In [16], a simplified forward and backward method is presented which has better performance than the basic sweeping algorithm. Reference [17] gives a comprehensive review and comparison of these sweep based approaches. 2.3 Weakly meshed networks Power distribution systems are radial or weakly meshed in reality. Radial networks can be solved directly from the above mentioned sweeping equations. For meshed networks, fictitious nodes are added to each loop thus the loops are broken and the network is converted to a radial one. For loops in the network, breakpoints are chosen in the ordering process. The last ordered branch in a loop is treated as a link and the breakpoints are added in every link. The PV bus is treated as a special kind of loop [15]: a breakpoint is added at the PV bus and it is assumed that the PV bus is connected to a fictitious bus of which the active power and voltage magnitude are specified. Figure 2.3 illustrates the process of adding breakpoints in the loop and PV bus. l denotes the link of the loop and l 1 and l 2 are the two ends of the breakpoint. After the conversion of the network, the interrupted current or power flow in the breakpoints can be replaced by current or power injection in the two ends of the breakpoints without affecting the network operating condition. Luo et al. proposed a compensation based power flow method [13]. It

16 CHAPTER 2. FORWARD/BACKWARD SWEEP APPROACHES 9 root bus PV bus l l 2 l 1 Figure 2.3: Network illustrating adding breakpoints in loops and PV nodes uses nodal current injections to compensate for the power flows in the breakpoints. The current injection method is illustrated in Figure 2.4. This approach is suitable for Kirchhoff s Law based sweeping algorithm in which nodal currents are calculated in each iteration. The current injection can be added directly to the calculated nodal [13]. Reference [10] improves the former method in that it uses active and reactive power as flow variables rather than complex currents thus simplifies the treatment of PV buses. The power injection method is illustrated in Figure 2.5. This approach is suitable for power summation method which active and reactive power are updated in each iteration. J j Breakpoint j J j J j Figure 2.4: Breakpoint current injection 2.4 Distributed generators modeling With the development of renewable energy and smartgrids, distributed generators become rather common and should not be omitted in distribution system analysis. They are classified as PQ or PV nodes thus different ways of modeling distributed generators have been used. If modeled as a PQ node, it is identical to a negative constant power load.

17 CHAPTER 2. FORWARD/BACKWARD SWEEP APPROACHES 10 Rest of the network Rest of the network k P, Q k k P, Q P, Q Figure 2.5: The breakpoint power injection method If modeled as a PV node in the network, the treatment is similar to that of a loop. Figure 2.6 shows the breakpoint of a PV bus. Rest of the network j Rest of the network P s, Q j P s, Q V j s Figure 2.6: PV bus breakpoint 2.5 Transformers modeling The single-phase transformer model is shown in Figure 2.7. The series impedance is Z t and the shunt element is Y m. The subscripts p and s denote primary and secondary side respectively. V p V s I Z t p I s Y m Figure 2.7: Single-phase transformer model

18 CHAPTER 2. FORWARD/BACKWARD SWEEP APPROACHES Convergence criteria Different criteria are used to check the convergence of the sweeping algorithm [13], [18], [19]. Reference [13] compares maximum active and reactive power mismatch between successive iterations to convergence criterion. Others such as [18] uses voltage differences in successive iterations to check convergence, which is explained in Eq. (2.9): V k+1 V k < ɛ. (2.9) For meshed networks or networks with PV nodes, the voltage mismatches in breakpoints can be used to check convergence: V j = V j1 V j2 < ɛ, (2.10) where j denotes a breakpoint and j 1 and j 2 are the two ends of the breakpoint. For the breakpoint of a PV node, the difference between the node voltage and its specified voltage magnitude is checked: V j = V j1 V s < ɛ. (2.11) If the successive power or voltage differences or voltage mismatches in breakpoints are below the pre-determined convergence criterion, the algorithm has converged and the calculation stops. For networks with loops and PV nodes, it is not enough to use successive voltage differences to check convergence because the voltages in the two ends of breakpoints might still mismatch even if the voltages in all buses has converged. Therefore, the breakpoints voltage mismatches are crucial to check convergence of a meshed network. Moreover, the two convergence criteria can be combined to obtain a faster and more robust sweeping algorithm. Reference [19] gives an adaptive power flow method where the successive voltage differences are checked to converge before breakpoints voltage mismatches are examined. Both the two convergence criteria are used to get faster convergence.

19 Chapter 3 Algorithm Implementation In this project, the implemented algorithm is the compensation based voltage sweep approach described in [10], [15], [20]. It deals with balanced distribution networks which are weakly meshed and contain distributed generators. Only one slack bus is assumed in the network. The branches are modeled as π model, thus shunt capacitance is considered. Constant power loads are assumed in the calculation. The parameters that need to be known before calculation are active and reactive power loads in PQ buses, slack bus voltage magnitude and angle as well as specified voltage magnitude and active power in the distributed generators. The flow-chart of the proposed method is shown in Figure 3.1. The procedure of this algorithm is explained in several parts in this chapter according to their implementation order. 3.1 Element ordering The formation of a suitable model of the electrical network is the first step of network analysis. Therefore, the element ordering process described in Chapter 2 needs to be conducted before implementing the forward and backward sweep algorithm. Before introducing the detailed process of the ordering process, some basic terminologies must be explained. In the network, line segments are called elements and their ends are called nodes. A connected subgraph containing all nodes but without closed path is called a tree. The elements of a tree are called branches. Other connecting elements which are not included in the tree are called links. Basic loop is the loop path which contains only one link. As in Reference [21], the network elements are separated into two subsets: branches and links. In the ordering process, every element is checked in the following three cases, a) none of the element nodes are in the list of ordered nodes; 12

20 CHAPTER 3. ALGORITHM IMPLEMENTATION 13 Read in data Element ordering Loop impedance matrix construction Breakpoint power injection calculations Power summation (backward sweep); Node voltage calculation (forward sweep); Breakpoint voltage calculation Incremental changes of breakpoint current calculation NO Converged? YES Print results Figure 3.1: Flowchart of the proposed algorithm

21 CHAPTER 3. ALGORITHM IMPLEMENTATION 14 b) only one of the element nodes is in the list of ordered nodes; c) both element nodes are in the list of ordered nodes. Case a changes nothing, case b denotes a branch and the element takes the next ordering element position with the other node takes the next ordering nodes position. Case c suggests a link and it takes the next element ordering position. After checking all the elements in the network, fictitious links are added if there are PV nodes in the network. Two arrays are obtained after the element ordering process, which are denoted as NS and NR. They contain the corresponding sending and receiving nodes of each branch and link. The NS and NR arrays are preparations for the sweeping process and determine the calculation sequence. 3.2 Loop impedance matrix This step is only necessary when there are loops or PV nodes in the network. Since breakpoints are added in loops, current or power injections are needed to compensate for the current flow in the breakpoints. Power injection is calculated using the incremental changes of breakpoint current I l and breakpoint voltage V l. Therefore, in order to get the power injections P inj, loop impedance matrix Z must be constructed for every breakpoint. The relation of breakpoint power injection, breakpoint current and voltage are explained in Eq. (3.1) (3.3). P inj = I l V l, (3.1) Z I l = V l, (3.2) V l = V lr V ls, (3.3) where lr and ls denotes the receiving and sending end of breakpoint l. Before introducing the way to get loop impedance matrix, basic loop incidence matrix C is constructed and the loop impedance matrix is deducted accordingly. The elements of basic loop incidence matrix are: c ij = 1 if the ith element is incident to and oriented in the same direction as the jth basic loop; c ij = 1 if the ith element is incident to and oriented in the opposite direction as the jth basic loop; c ij = 0 if the ith element is not incident to the jth basic loop. The loop impedance matrix Z is constructed based on the loop incidence matrix. The elements of matrix Z are: z ii, which is the diagonal element, is the sum of all the branches impedance which form the loop; z ij, which is the off-diagonal element, is the sum of all the mutual branches impedance of basic loop i and j, the impedance takes the opposite sign if loop i and j have opposite orientation at the mutual branches.

22 CHAPTER 3. ALGORITHM IMPLEMENTATION Voltage sweep calculation The power summation method is used in this step. The sending real and reactive power is added to the receiving power using a backward sweep calculation. Based on the introduction of power summation method in Chapter 2, Eq. (2.1) (2.4) are used to do the backward calculation and Eq. (2.5) is used in forward sweep. More precisely, Eq. (3.4) (3.9) are used to get the real and imaginary part of complex voltage. V r = V s (R s r + jx s r ) P s jq s V s V r = V s (1 P sr s r + Q s X s r V 2 s Define a and b as, a = 1 P sr s r + Q s X s r V 2 s b = Q sr s r P s X s r V 2 s, (3.4) + j Q sr s r P s X s r Vs 2 ). (3.5), (3.6). (3.7) Then the real and imaginary part of receiving voltage can be calculated using Eq. (3.8) (3.9), Re{V r } = are{v s } bim{v s }, (3.8) Im{V r } = br{v s } + aim{v s }. (3.9) 3.4 Breakpoint power injection After the sweeping process, complex voltages of each bus are known and the breakpoints power injections can be calculated using the following equations, S mr = V ms + V mr (C m jd m ), (3.10) 2 S ms = S mr. (3.11) Since the active power of a PV bus is specified, only reactive power injection is updated: V 2 ms Q ms = D m Re{V ms }. (3.12) To get the breakpoint current C m + jd m, loop impedance matrix Z and breakpoint voltage difference are used in Eq. (3.13). [ ] [ ] [ ] X R D E =, (3.13) R X C F

23 CHAPTER 3. ALGORITHM IMPLEMENTATION 16 where R and X are the real and imaginary part of loop impedance matrix Z, C and D are the real and imaginary part of incremental change of complex breakpoint current, E and F are the real and imaginary part of complex breakpoint voltage difference. At breakpoint l, the breakpoint complex voltage difference is: For the PV nodes, E l + j F = V lr V ls. (3.14) E l = ( V sp ls 1)Re{V V ls }, (3.15) ls where V sp ls is the specified value of PV nodes voltage magnitude. Thus, the power injection of every breakpoint can be updated using the above equations. 3.5 Convergence criteria In each iteration, the above mentioned steps are cycled until the results get converged. For radial networks, the difference of successive voltage is used to check convergence. For meshed networks as well as networks with PV Nodes, the breakpoints voltage mismatches are used to check convergence. The maximum successive voltage difference among all buses or maximum voltage mismatch in breakpoints are compared to the pre-determined convergence criterion. Once it is below the convergence criteria, the calculation stops.

24 Chapter 4 Implementation results The algorithm introduced in Chapter 3 was implemented in Java. In this chapter, the convergence behavior of the sweeping algorithm is assessed using different network topologies. Besides, implementation results are compared with the G-S and N-R methods in JPower. First of all, an individual assessment on the sweeping algorithm is made using a 29-bus network. The topology is given in Appendix A. As shown in Figure A.2, there are in total 29 buses with 2 loops and 3 PV buses in the network. 4.1 Assessment on the sweeping algorithm Voltage mismatches after each iteration For radial networks, the successive voltage differences for each bus are usually used to check convergence of the sweeping algorithm. By removing the loops and PV nodes in the 29-bus network, the convergence behavior of the sweeping algorithm is assessed for a radial network. The successive voltage differences for all buses are calculated and plotted in Figure 4.1. As in Figure 4.1, the successive voltage differences in all buses are decreasing with each iteration. When the convergence criterion is set to 10 7, the sweeping algorithm converges in 2 iterations. Assessment of the sweeping algorithm with meshed networks is made using the original 29-bus network. The voltage mismatches in the five breakpoints are calculated for each iteration and plotted in Figure 4.2. As in Figure 4.2, the voltage mismatches in the breakpoints are decreasing with each iteration. Once all the mismatches are within the convergence criteria, the calculation stops. When the convergence criterion is set to 10 7, the sweeping algorithm converges in 11 iterations. 17

25 CHAPTER 4. IMPLEMENTATION RESULTS Successive voltage difference (p.u.) Iteration number Figure 4.1: Successive voltage difference for all buses Voltage mismatch (p.u.) 3.0x x10-3 loop 1 loop 2 2.0x10-3 PV 1 1.5x10-3 PV 2 PV 3 1.0x x x x x x x x Iteration number Figure 4.2: Voltage mismatches in breakpoints

26 CHAPTER 4. IMPLEMENTATION RESULTS Influence of convergence criteria The convergence criteria determine how far the calculations are from the true value. Therefore, the effect of different convergence criterion is analyzed using its influence on iteration number and runtime of the sweeping algorithm. Figure 4.3 shows results of increasing runtime and iteration number of the sweeping algorithm under decreasing convergence criteria. Runtime (s) Iteration number 2.1x x x x x x x x x x E-5 1E-6 1E-7 1E-8 Convergence criteria Figure 4.3: Runtime and iteration number under different convergence criteria Influence of loops and PV nodes numbers Known from the principle of the sweeping algorithm in Chapter 2, loops and PV nodes are treated as breakpoints and need extra treatment. Therefore, the sweeping algorithm needs more effort dealing with more loops and PV nodes. This is illustrated by the following process: by adding or subtracting loops and PV nodes in the 29-bus network, the runtime and iteration number of the sweeping algorithm are calculated. Figure 4.4 shows increasing runtime and iteration number with more PV nodes in the network. Figure 4.5 shows increasing runtime and iteration number with more loops in the network. Generally speaking, the iteration number and runtime of the sweeping algorithm are increasing with more loops and PV node added since they increase the complexity of the network. However, the relationship is not necessarily a strictly positive correlation because the position of the loops or PV nodes also count. This results from the element ordering process of the sweeping algorithm. The network topology determines the calculation

27 CHAPTER 4. IMPLEMENTATION RESULTS 20 Runtime (s) 4.0x x x x x x10-4 Iteration number Loop number Figure 4.4: Runtime and iteration number with increasing loops Runtime (s) 2.8x x x x x x x x10-4 Iteration number PV node number Figure 4.5: Runtime and iteration number with increasing PV nodes

28 CHAPTER 4. IMPLEMENTATION RESULTS 21 sequence in the backward and forward sweeping process thus has an influence on the final iteration number and runtime Influence of PV node position To illustrate the influence of PV node position, only one PV bus is put in the 29-bus network and calculations are made with it in different positions. The iteration number of each calculation is recorded. As in Figure 4.6, the influence of PV node position can be shown by the different iteration number the sweeping algorithm used to do the calculation. (No result is there for bus 28 because it is the slack bus.) Iteration Number PV node position Figure 4.6: Iteration number for different PV node position 4.2 Comparison with the Gauss-Seidel and the Newton- Raphson methods In this part, the performance comparison among the sweeping algorithm, the G-S method and the N-R method is presented. Three systems are used to assess the convergence behaviors of the algorithms. The characteristics of the three grid topologies are listed in Table 4.1 and the network topologies are given in Appendix A.

29 CHAPTER 4. IMPLEMENTATION RESULTS Runtime comparison Each algorithm is run for 1000 times and Table 4.2 gives average runtime for the three networks respectively. Figure 4.7 compares the runtime of the three algorithms. Table 4.1: Characteristics of the benchmark networks Network Bus number Loops PV nodes Transformers Table 4.2: Runtime comparison of different methods Method\Network 14-bus 29-bus 106-bus Sweeping Algorithm 1.58e-4 s 6.86e-5 s 8.76e-5 s Newton-Raphson 1.82e-4 s 1.32e-4 s 2.37e-3 s Gauss-Seidel 7.92e-4 s 3.89e-3 s 3.26e-1 s 3.50x x10-1 Sweeping algorithm N-R G-S 2.5x10-4 Runtime (s) 2.0x x x x bus 29-bus 106-bus Different bus system Figure 4.7: Runtime comparison As shown in Figure 4.7, the sweeping algorithm uses the least runtime for all networks. More concrete comparison is given in Table 4.2. Comparing average runtime with the network characteristics, conclusion can be made that the time-saving advantage of the sweeping algorithm is more obvious

30 CHAPTER 4. IMPLEMENTATION RESULTS 23 for networks with less loops and PV nodes. The reason of that is easy to explain: for networks with more loops or PV nodes, more breakpoints are added thus increasing effort is needed, which results in more runtime in the calculation Calculation results comparison The voltage magnitude differences between the sweeping algorithm and the G-S and the N-R methods are calculated under different convergence criteria. For the following results, the convergence criteria of the three algorithms are equally set. Figure 4.8 shows voltage magnitude differences between the sweeping algorithm and the N-R method. Figure 4.9 shows voltage magnitude differences between the sweeping algorithm and the G-S method. The results show that the solutions of the three algorithms get closer with smaller convergence criteria and the solution differences are mostly under the convergence criteria Voltage magnitude difference (p.u.) Bus number Figure 4.8: Voltage magnitude differences between the sweeping algorithm and the N-R method

31 CHAPTER 4. IMPLEMENTATION RESULTS Voltage magnitude difference (p.u.) Bus number Figure 4.9: Voltage magnitude differences between the sweeping algorithm and the G-S method

32 Chapter 5 Conclusions In the thesis, a backward and forward voltage sweep algorithm is implemented in Java. The algorithm is validated using different network examples and comparisons with established power flow techniques (N-R and G-S) in JPower were made. The assessment of the sweeping algorithm using a 29-bus network shows that the voltage differences between successive iterations and voltage mismatches in breakpoints are decreasing with the iteration process. It needs more effort when dealing with an increasing number of loops and PV nodes which can be demonstrated by its increasing iteration number and runtime. Different convergence criteria and PV node position also have an influence on the solution. The comparison results show that the sweeping algorithm has faster convergence speed than the N-R and the G-S methods. The time-saving advantage is more obvious especially for networks with less loops or PV nodes. Since power distribution systems are often radial or weakly meshed, the sweeping algorithm shows its convergence efficiency in analyzing power distribution systems compared to N-R and G-S. 25

33 Appendix A Benchmark networks Figure A.1: 14-bus network 26

34 APPENDIX A. BENCHMARK NETWORKS 27 Figure A.2: 29-bus network

35 APPENDIX A. BENCHMARK NETWORKS 28 Figure A.3: 106-bus network

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