Identification of Source-Sink Connections for Transmission Cost Calculations by. Wan King Him D. Final Report

Project ID: FYP-99 Identification of Source-Sink Connections for Transmission Cost Calculations by Wan King Him 14074719D Final Report Bachelor of Engineering (Honours) in Electrical Engineering Of The Hong Kong Polytechnic University Supervisor: Dr. C.W. YU Date: 31 March 2018

Abstract The transmission network plays a very important role in competitive electricity markets. Utilization of a proper transmission pricing scheme deliver appropriate economic signals to the market and investors, encourage investment of transmission lines and generation along the lines, help efficient market operation and reasonable cost recovery for the investor. For this, it is important to know the contribution of each transmission service user to the network usage, as a basis of the impartial allocation of transmission fee to the users in the new restructured open access transmission system. The existing transmission pricing methods including various of incremental transmission pricing methods and embedded transmission pricing methods are being introduced. This paper focuses on an embedded transmission pricing method named Bialek Tracing Method. Bialek Tracing Method is a flow-based topological analysis technique for embedded transmission pricing. By applying Bialek Tracing Method, the information of which generators are supplying a particular load, how much use each generator is making of a transmission line and what is each generator s contribution to the system losses in the transmission system can be determinated. A detailed analysis with case studies on testing the effectiveness of Bialek Tracing Method had been conducted in this project. Case studies are adopted on IEEE 9 Bus System and IEEE 39 Bus System and simulated using MATPOWER. Keywords: Power flow, transmission cost charging, source-sink connections, open access transmission network, Bialek Tracing Method

Acknowledgements THE HONG KONG POLYTECHNIC UNIVERSITY I would like to express my gratitude to my supervisor Dr. C.W. Yu for his guidance, support and valuable advice in throughout the progress of this project.

Table of content THE HONG KONG POLYTECHNIC UNIVERSITY Abstract... 2 Acknowledgements... 3 Table of content... 4 1) Introduction... 1 2) Objectives... 4 3) Literature Review... 5 3.1 Transmission Pricing Principle... 5 3.2 Review of existing methods for transmission cost allocation... 6 3.2.1 Incremental transmission pricing methods... 7 3.2.2 Embedded Transmission Pricing... 7 4) Methodology... 11 4.1 Theories... 11 4.2 Brief Description of the project... 12 4.3 Proposed Approach (Bialek Tracing Method)... 13 4.4 Case Studies... 22 4.4.1 System 1 IEEE 9 Bus System... 23 4.4.2 System 2 IEEE 39 Bus System... 25 5) Results and Findings... 28 5.1 Power flow result of IEEE 9 Bus System... 28 5.2 Case 1 IEEE 9 Bus System... 28 5.3 Case 2 IEEE 9 Bus System... 31 5.4 Power flow result of IEEE 39 Bus System... 33 5.5 Case 3 IEEE 39 Bus System... 34 5.6 Case 4 IEEE 39 Bus System... 39 5.7 Comparison between the results of two algorithms... 44 5.8 Analysis on Bialek Tracing Method... 47 6) Conclusion and future development... 49 References... 50 Appix... A1

1) Introduction In order to attain efficiency competition in a fair environment, the tency towards deregulation and reformation of the services provided by utilities was developed in the past decades [1]. Allocating the transmission cost among the users is one of the main hindrances in reforming the service [1, 2]. Since the capital cost of transmission system development is extremely high, it is not cost-effective to build a transmission system for each power station. Number of power stations and loads share the same transmission system. All the producers and consumers are connected together by an interconnected transmission system for electricity delivery, we can call this as an electrical grid. Fig. 1 Electrical Grid Besides the benefit of capital saving, interconnect the generating units and loads enhancing the power supply stability and reliability. Neighboring units share the critical loads, different units also act as backup power remains for each another in the event of one unit s 1

failure. In addition, it allows consumer to have access to cheaper bulk energy by receiving power from different sources even in another region [3]. However, transmission cost allocation problem has to be solved before taking the advantages of electrical grid. Since the transmission system is a single, highly integrated electrical business in a mesh structure, it is complicated to figure out the transmission loss to individual loads or generators [1]. A comprehensive method to trace and measure the corresponding transmission loss in each particular transaction is necessary for fair transmission cost allocation in the restructured power market. A proper pricing scheme helps reasonable investment recovery and support efficient operation of the power market, delivering correct and useful market information to all users, ensures the reliability of the network system [4]. There are various approaches for transmission pricing, which can be divided into three types: Incremental transmission pricing methods, embedded transmission methods and composite (incremental & embedded) transmission pricing methods [5]. This project mainly focused on one of the embedded transmission methods - Bialek tracing method which is a flow-based method introduced by J. Bialek in 1996. The detail working principle of Bialek tracing method is investigated in this paper and case studies are performed to analyze the effectiveness of the tracing algorithms introduced in the proposed method. This report consists of seven sections. Objectives in point form are listed in Chapter 2 Objectives. Chapter 3 Literature Review discusses some recent studies on various approaches on tracing and measuring the system losses for each transaction, and hence the pricing methods. Chapter 4 Methodology introduces how and why this project conducted 2

including the research design, the procedures and the software I used for simulations and calculations. Chapter 5 Results and Findings show the results of project simulations and calculations discussed in the methodology, the results are then analyzed thoroughly. Conclusion of the investigation and future planning of the project are shown in Chapter 6 Conclusion and Future Development. 3

2) Objectives THE HONG KONG POLYTECHNIC UNIVERSITY (a) Analysis of the significance of fair transmission pricing in the modern power system and merits/demerits of existing transmission pricing methods. (b) Investigation of the detail principle of Bialek Tracing Method, apply the topological technique to identify the system losses in each individual source-sink connection (c) Implementation and software simulation of Bialek Tracing Method on large interconnected multi-machine power system for testing on accuracy and reliability. 4

3) Literature Review THE HONG KONG POLYTECHNIC UNIVERSITY From an economic point of view on the optimal charging strategy, efficient charging of service and property makes necessary on marginal-cost pricing [6]. However, the high capital cost of transmission network might not be recovered successfully by only marginal pricing. Marginal-based pricing induces highly volatile prices. Thus, the fast-changing marginal pricing scheme may produce incorrect messages to the company [1]. In addition to the customer side, high volatile pricing may cause a negative marginal cost of losses when the losses increase and reach a critical level [2]. Users gain benefits while increasing the load, resulting in profligacy of energy. 3.1 Transmission Pricing Principle A Switzerland report [7] published in 2015 indicated that, transmission pricing should obey the following principles, including: (a) Reasonable recovery of the transmission cost; (b) Reflect authentic operation cost of the transmission service to the market (c) Fair transmission cost allocation, no cross subsidization; (d) Unbiased treatment to all service users, disregarding their nature, wealth and other factors; (e) Comply with laws and regulations of the electricity pricing standards and power market; (f) Deliver appropriate economic message to investors and users, encourage effective allocation and reasonable use of the transmission service; 5

(g) Pricing scheme should be easily operated and managed; (h) Calculation methods of the pricing scheme should be high transparency and easy to understand. To conclude, transmission pricing scheme should be customized according to the corresponding market conditions, considering the various effects on the network operations and network users in order to charge the network users in an efficient and fair way. 3.2 Review of existing methods for transmission cost allocation The main purpose of any transmission pricing scheme is to recompense the cost of transmission system. Transmission pricing methods are the overall processes of converting transmission costs into total transmission charges [8]. These methods are shown in Figure 2. Transmission pricing methods Incremental transmission pricing methods Embedded transmission pricing methods Composite transmission pricing methods Fig 2. Types of transmission pricing methods 6

3.2.1 Incremental transmission pricing methods Incremental transmission pricing methods allocate the variable cost of the transmission transaction [5]. Although incremental transmission pricing methods reflects the actual status of the market and releasing correct signals for efficient uses of the transmission service, it only recovers the operation cost of the transmission network. The total investment cannot be fully recovered. Moreover, the price is volatility, it may lead to overload of the system when the price is relative low, affecting the transmission quality. There are four different types which are shown in Figure 3. Incremental transmission pricing methods Short-run incremental cost pricing (SRIC) Long-run incremental cost pricing (LRIC) Short-run marginal cost pricing (SRMC) Long-run marginal cost pricing(lrmc) Fig 3. Types of Incremental transmission pricing methods 3.2.2 Embedded Transmission Pricing Marginal Charge Supplement Charge Overall Charge Fig 4. Embedded transmission pricing To allocate the transmission cost to all users using the mesh network in a fair way, a popular method is adding a supplement charge to the overall charge. Reasonable recovery of transmission cost can be achieved by allocating an embedded system cost (fixed cost) to the network users. Types of Embedded Cost Transmission Pricing are shown in Figure 5. 7

Postage Stamp Method Network-based Methods Contact Path Method MW-Mile Method Embedded Cost Transmission Pricing Methods MVA-Mile Method Distribution Factors Method Flow-based Methods Bialek Tracing Method Network-based Methods Fig. 5 Types of Embedded Cost Transmission Pricing Methods All the cost incurred by the network including construction cost, operational cost, maintenance cost and other embedded cost are rolled in together. Network-based methods mainly dep on the network structure [5]. Postage Stamp Method: Postage stamp methodology is the simplest and easy for transmission pricing [9]. It is assumed that the whole system is used, excluding the equipment used for transmission. The charges are apportioned to users based on the amount of power consumed and average fixed cost. Although it is easy to be implemented, the main demerit of using postage stamp method is cross-subsidization among network users. Since Power flow calculations is not required in this method, the charges paid by the user do not reflect actual use of the network [5] [9]. Kirschen Tracing Method Besides, postage stamp method is indepent of the transmission distance and network configuration [5]. It is obviously unfair to the near generators when compare to distant 8

generators as the electricity produced by distant power stations have to transmit via the long and costly transmission network to reach their consumers. It would give incorrect economic incentives to the users of transmission network when this method is utilized to long-distance power delivery. Therefore, postage stamp method is usually used in local transmission network [9]. Contract Path Method: Contract path is the shortest, single continuous route of transmission lines which can carry the contract power between the delivery point and the receipt point [10]. The power transaction cost is charged based on this specific pre-defined path [11]. Besides the merit of simplicity, this methodology encourages efficient investment which benefits the network investors. As both the existing and new properties of the transmission network are taking into consideration for pricing, it is favorable for full cost recovery [12]. However, the contract path is not required to be a physical transaction entity and hence this methodology incurs inaccurate economic signals to the market [18]. Also, inefficient and uneconomic transactions may take place [12]. MW-MILE method: The MW-Mile method reflects the actual use of the networks by a product of the magnitude of power due to a transaction and the geographical distance this power travels in the transmission process [13, 14]. This method is DC power flow based method [10]. Limitations of contract path method are overcame since MW-Mile methodology is based on a computed set of parallel paths for a particular deal instead of a pre-defined single path [15]. Although it is an intuitive method, it is substantial supported since it is stable and maximize the utilization of the existing system [8]. 9

MVA-Mile Method: The use of transmission resources is measured by monitoring both real and reactive power in MVA-Mile Method [4]. Consider the reactive power changes in the transmission network caused by a particular user make the embedded cost pricing more reasonable and valid than MW-Mile Method. Flow-based Methods Flow-based methodologies assign the charges to transmission service users based on the network capacity used for each transaction. Magnitude of the transmitted power, the path, and the distance travelled are considered in each transaction [4]. Distribution factors method, Kirschen tracing method are introduced below. While the proposed method, Bialek tracing method would be introduced thoroughly in the chapter 4 Methodology. Distribution Factors Methods: It is based on dc linearized model of the system and superposition theorem [1]. The factors are used to determine the impact of generator and load on power flows [10]. Although it is an accurate pricing method taking the actual usage of facilities into account, this method produces negative flow and results in negative charges [1]. It is difficult for the network provider to calculate how much the user should pay when negative flow occurs. Kirschen Tracing Method: A flow-based method based on a set of definitions for domains and commons that representing the buses, and for links representing the branches [16]. According to the domains, commons and links defined, the graphical system is formed with detailed state data. The main principle used by this method is proportional sharing principle, both active and reactive power is considered [16]. The contributions from individual generators and loads to line flows can be computed. 10

4) Methodology THE HONG KONG POLYTECHNIC UNIVERSITY The procedures and details of methods implemented will be discussed in this section. Some basic concepts below should be recalled in order to have a better understanding of the project. 4.1 Theories (a) Kirchhoff s current law (KCL): the statement that the sum of the currents into a given node equals the sum of the currents out of that node [17]. For the simple circuit diagram shown in Figure 6, line 1 and 2 are inflows and line 3, 4 and 5 are outflows. In this example, the line currents are related by I1+I2 = I3+I4+I5. The proposed method is based on an assumption of lossless condition, so the whole system must obey KCL. Otherwise, the linear-based algorithms cannot be applied on load flow tracing. Fig. 6 Kirchhoff s current law (b) Counterflow: the component flows that go in the opposite direction to the total net power flow [2]. It incurs complicated calculation, results in negative charge. (c) It is also very important to know the differences between gross and net flow for understanding of two linear equation-based algorithms because these algorithms work only on lossless flows [1]. i.e. the flows at the beginning equals the flows at the. Different ways can be adopted to achieve lossless flows. 11

(i) Gross flow: It is the total power leaving a point and enter a line. When gross flows are used in calculation, a lossless transmission network is formed where the generation is the same as the actual network [18]. The lines transmit the same power as the sing of lines in the real network. (ii) Net flow: It is the total flow leaving a line and entering a point. When net flows are used in calculation, a lossless transmission network is formed where the load is the same as the actual network [18]. The lines transmit the same power as the receiving of lines in the real network. 4.2 Brief Description of the project a) Software used in the Project MATLAB (matrix laboratory) a proprietary programming language for numerical computing developed by MathWorks. Allowing algorithm implementation, matrix operation, graph plotting for data and functions, etc. [19]. MABPOWER 6.0 a free toolbox of MATLAB for solving power flow and optimal power flow problems. No conversion of program is needed when it is performed in MATLAB. Besides, there is a package of M-files consists of various of built-in cases ranging from a few buses to thousands of buses [20]. b) Aims and Flow of the Project In this project, two tracing algorithms of Bialek tracing method are applied separately to trace the source-sink connections of power systems. The bus system data (i.e. the branches states and power flow of each lines) of the studied cases are obtained by MABPOWER first, then are implemented by original MATLAB scripts for testing the performance of the 12

upstream-looking algorithm and downstream-looking algorithm of Bialek Tracing Method. The MATLAB scripts written for testing are shown in Appix A1 to A4. The proposed approach will also be analyzed under two different cases, IEEE 9 bus system and IEEE 39 bus system. so as to evaluate the performance of the proposed approach when adopted on practical power systems in various size. 4.3 Proposed Approach (Bialek Tracing Method) Bialek tracing method was introduced by J. Bialek in 1996 [2]. It is applied to find out the source-sink relationship of any pair of generation units and load in the network [21]. It means the amount of power received by a particular load from a particular generation unit can be determined. This method is suitable for tracing both active and reactive power flows. In Bialek tracing method, incoming power flows of the node are assumed to be distributed proportionally to the outgoing power flows [21]. It is topological in nature, the contribution of individual generators or loads to power flow in each single line are computed contingent on topological distribution factors [2]. The counter flow problem occurs in the conventional flow-based method (i.e. distribution factors method) is eliminated since the topological distribution factors used in the calculation are always positive. The principle of constantly positive factors would be explained in the sections below. Bialek tracing method consists of upstream- and downstream-looking algorithm, either one of the algorithms can be used for tracing the power flows of the system [2]. 13

The supplement charge for the transmission cost and the system losses are allocated to different individual units in two tracing algorithms, which are described in the below Table 1 [2]: Algorithm The supplement charge is allocated to The losses are allocated to Upstream-looking algorithm Individual generators Individual loads Downstream-looking algorithm Individual loads Individual generators Table 1 Difference between upstream and downstream looking algorithms Proportional sharing principle Proportion sharing is the main principle used to trace the power flow, which is based on Kirchhoff s current law (KCL). As the network flow is assumed to be lossless, all nodes in the transmission network must satisfy KCL, hence the power flow of each lines is assumed to be constant in this methodology [22]. Fig. 7 Bus Bm, Illustration diagram of proportional sharing principle [23] 14

According to Figure 7, four branches (i, j, k, l) are connected with node m. Line i-m and line j-m are inflows, while line m-k and line m-l are outflows. The total power flow through the system can be obtained by summing up all the nodal inflows or outflows (i.e. P in = P out, P in = P im + P jm = 90 + 10 = 100MW, P out = P mk + P ml = 80 + 20 = 100MW). The relationship between the inflows and outflows can be determined by applying proportionality principle [23]. Each of the outflows leaving the nodes is depent only on the impedance and voltage gradient of the line [2]. Let outflowing P mk in figure 7 as an example, it consists of P im P mk = 90 80 = 72MW supplied by line i-m, and P jmp mk = 10 80 = 8MW supplied P mk +P ml 80+20 P mk +P ml 80+20 by line j-m. The mathematical formulations of upstream-looking algorithms and downstream-looking algorithm are described in below section. Tracing algorithms As the tracing method only works on lossless flows, the flow at the beginning of a line must be the same with the flow at the. An assumption is made for the tracing algorithms, that is the total transmission loss can be divided perfectly and then assigned to individual generators or loads. By this assumption, the algorithms are possible for tracing since the lossy flow problem is eliminated. 15

Upstream-looking algorithm THE HONG KONG POLYTECHNIC UNIVERSITY (1) Determination of the relationship between gross demand P gross and actual power generation P G The allocated transmission loss adding with the sum of the actual demand of a particular load is referred as gross demand to the particular load. The total gross demand is equal to the total actual power generated. Let P g m be the gross nodal power flow through node m and P g mn be the gross line power flow through line m-n. When looking at the sing of the line, the gross power balance equation at node m can be defined as P g m = n αu P g m mn + P Gm for m = 1, 2, 3,, k (1) Where α m u is the set of nodes sing power directly to node m and P Gm is the power generation at node m. In normal condition, the transmissions losses are small. Therefore, it is assumed that P g mn g P m = P mn, where P P mn and P m refer to the actual line flow from node m m to node n and the actual nodal flow at node m respectively. Also, by the assumption of the equivalence of the actual generation and gross demand, equation (1) can be rewritten as P mn P Gm = P g m n αu m P g n or A u P gross = P G (2) P n (2) Formation of upstream distribution matrix, Au Au is the upstream distribution matrix, Pgross is called the vector of gross nodal flows, while PG is the vector of nodal generations. 16

Au is sparse and non-symmetric, THE HONG KONG POLYTECHNIC UNIVERSITY [A u ] mn ={ P mn P m 1 for m = n for n α m u 0 otherwise. (3) (3) Determination of the contribution of generators to lines If the inverse matrix of Au exists, then Pgross = Au -1 PG, the gross nodal power flows can be determined by equation (4). j P g m = [A 1 i=1 u ] mi P Gi for m = 1, 2, 3,, k (4) The gross line power flows can then be determined by equation (5). This equation shows how the m-th gross nodal power is supplied from all the generators in the system. By using proportionality sharing, the gross outflow in the line m-n can be calculated as, P g mn = P g mn g P m P g m = P g mn j g [A 1 j g P i=1 u ] m mi P Gi = d i=1 D mn,i P Gi for n α m (5) Where α d g m is the the set of nodes receiving power directly from node m and D mn,i = P g mn [A u ] mi /P g g m. This equation defines D mn,i as the topological generation distribution factor that is part of generation due to i-th generator that flows in line m-n. These factors represent the share of a particular generation in the total line flow. Therefore, the topological generation distribution factors are always positive. (4) Determination of the contribution of generators to loads The nodal load demand P Lm is an outflow and it can be calculated also by proportional sharing principle, P Lm g = P g Lm g P m P g m = P g mn j g [A 1 j g P i=1 u ] m mi P Gi = d i=1 D mn,i P Gi for n α m (6) Equation (6) show the contribution of the m-th generation unit to the i-th load demand. 17

Thus, the amount of power flows through a particular pair of source-sink connections can be determined. Downstream-looking algorithm (1) Determination of the relationship between actual demand P L and net power generation P net The allocated transmission loss is subtracted by the actual generation of a particular generator is referred as net generation. The actual demand is the same with the net power net generated. Let P m be the net nodal power flow through node m and P net mn be the net line power flow through line m-n. When looking at the receiving of the line, the gross power balance equation at node m can be defined as P net m = n αu P net m mn + P Lm for m = 1, 2, 3,, k (7) Where α m d is the set of nodes receiving power directly from node m and P Lm is the load demand at node m. In normal condition, the transmissions losses are small. Therefore, it is assumed that P mn net Pnet m = P mn, where P P mn and P m refer to the actual line flow from node m m to node n and the actual nodal flow at node m respectively. Also, by the assumption of the equivalence of the net generation and actual demand, equation (1) can be rewritten as P mn P Lm = P net m net n αu m P n or A d P net = P L (8) P n (2) Formation of Downstream Distribution Matrix, Ad Ad is the downstream distribution matrix, Pnet is called the vector of net nodal flows, while PL is the vector of nodal load demands. Ad is non-symmetric and sparse, 18

[A d ] mn ={ P nm P m 1 for m = n for n α m d 0 otherwise. (9) (3) Determination of the contribution of loads impact the line flows If the inverse matrix of Ad exists, then Pnet = Ad -1 PL, j P net m = [A 1 i=1 d ] mi P Li for m = 1, 2, 3,, k (10) This equation shows how the m-th net nodal power is transmitted to all the loads in the system. By using proportionality sharing, the net inflow from the line m-n can be calculated as, P net mn = P net mn net j Pnet m P m net P m net = P mn j net [A d 1 i=1 ] mi P Li = u i=1 D mn,i P Li for n α m (11) Where α u net m is the the set of nodes sing power directly to node m and D mn,i = P net mn [A d ] mi /P net m. This equation defines D net mn,i as the topological generation distribution factor that is part of load demand that flows in line m-n. These factors represent the share of a particular generation in the total line flow. Therefore, the topological generation distribution factors are always positive. (4) Determination of the contribution of loads impact to generation The net nodal generation P Gm is an inflow and it can be calculated also by proportional sharing principle, P net Gm = P Gm net net j Pnet m P m net P m net = P Gm j net [A d 1 i=1 ] mi P Li = u i=1 D mn,i P Li for n α m (12) Equation (12) show the share of output among the generation units used to supply load demand. Thus, determining the amount of power flows through a particular pair of sourcesink connections. 19

Diagram of the key steps of Script Written for tracing calculations Upstream-looking algorithm 1) Determination of the relationship between gross demand and actual power generation, A u P gross = P G 2) Formation of upstream distribution Downstream-looking algorithm 1) Determination of the relationship between actual demand and net power generation, A d P net = P L 2) Formation of downstream distribution matrix, A u 3) Inverse to get topological distribution factors, A u 1 4) Determination of the contribution of generators to line flows, P g mn applying equation (5) Fig. 8 Summary of the key steps of tracing algorithms by 5) Determination of the contribution of generators to loads, P Lm g equation (6) by applying matrix, A d 3) Inverse A d to get topological distribution factors, A d 1 4) Determination of the contribution of net loads impact line flows, P mn by applying equation (11). 5) Determination of the contribution of net loads impact to generation, P Gm by applying equation (12). 20

Flow Diagram of identification of source-sink connections The following is a block diagram describing the overall rundown of source-sink identification. Fig. 9 A flow diagram showing the overall operation process of source-sink identification 21

4.4 Case Studies As mentioned in the previous section, the proposed method would be examined on practical cases. Two IEEE standard systems are introduced for method implementation, including IEEE 9 Bus System and IEEE 39 Bus System to investigate the performance of the Bialek tracing method for determining the source-sink relationships in various system size. There are 4 cases to be studied. A brief description of all the cases is listed in the below table. Case System adopted Algorithm used 1. IEEE 9 Bus System Upstream-looking algorithm 2. IEEE 9 Bus System Downstream-looking algorithm 3. IEEE 39 Bus System Upstream-looking algorithm 4. IEEE 39 Bus System Downstream-looking algorithm Table 2. A brief description of the cases studied The following Figure 10 & 12 shows the systems to be examined in this project. 22

4.4.1 System 1 IEEE 9 Bus System Fig. 10 IEEE 9 Bus System [24] IEEE 9 Bus System are introduced as an illustration sample showing how the tracing algorithms work out the results. It is chosen in current case study since there are 6 lines, 9 buses, 3 two-winding power transformers, 3 generators and 3 loads in this standardized bus system, which is also favorable for investigating the performance of the tracing algorithms on small bus system [24]. Detail system parameters are listed below. 23

System Parameters THE HONG KONG POLYTECHNIC UNIVERSITY %% system MVA base mpc.basemva = 100; %% bus data % bus_i type Pd Qd Gs Bs area Vm Va basekv zone Vmax Vmin mpc.bus = [ 1 3 0 0 0 0 1 1 0 345 1 1.1 0.9; 2 2 0 0 0 0 1 1 0 345 1 1.1 0.9; 3 2 0 0 0 0 1 1 0 345 1 1.1 0.9; 4 1 0 0 0 0 1 1 0 345 1 1.1 0.9; 5 1 90 30 0 0 1 1 0 345 1 1.1 0.9; 6 1 0 0 0 0 1 1 0 345 1 1.1 0.9; 7 1 100 35 0 0 1 1 0 345 1 1.1 0.9; 8 1 0 0 0 0 1 1 0 345 1 1.1 0.9; 9 1 125 50 0 0 1 1 0 345 1 1.1 0.9; ]; %% generator data % bus Pg Qg Qmax Qmin Vg mbase status Pmax Pmin Pc1 Pc2 Qc1min Qc1max Qc2min Qc2max ramp_agc ramp_10 ramp_30 ramp_q apf mpc.gen = [ 1 0 0 300-300 1 100 1 250 10 0 0 0 0 0 0 0 0 0 0 0; 2 163 0 300-300 1 100 1 300 10 0 0 0 0 0 0 0 0 0 0 0; 3 85 0 300-300 1 100 1 270 10 0 0 0 0 0 0 0 0 0 0 0; ]; %% branch data %fbus tbus r x b ratea rateb ratec ratio angle status angmin angmax mpc.branch = [ 1 4 0 0.0576 0 250 250 250 0 0 1-360 360; 4 5 0.017 0.092 0.158 250 250 250 0 0 1-360 360; 5 6 0.039 0.17 0.358 150 150 150 0 0 1-360 360; 3 6 0 0.0586 0 300 300 300 0 0 1-360 360; 6 7 0.0119 0.1008 0.209 150 150 150 0 0 1-360 360; 7 8 0.0085 0.072 0.149 250 250 250 0 0 1-360 360; 8 2 0 0.0625 0 250 250 250 0 0 1-360 360; 8 9 0.032 0.161 0.306 250 250 250 0 0 1-360 360; 9 4 0.01 0.085 0.176 250 250 250 0 0 1-360 360; ]; Fig. 11 Matrix showing parameters for buses, branches and generators of IEEE 9 Bus System adopted in Case 1 and 2 24

4.4.2 System 2 IEEE 39 Bus System Fig. 12 IEEE 39 Bus System [25] Another standardized case to be studied in the current project is IEEE 39 Bus System shown above. It consists of 46 transmission lines, 39 buses, 10 generators and 21 loads, which is a favorable size to examine the performance of the method when implemented into a real power system in practical [25]. Detail parameters of the system are listed below. 25

System Parameters THE HONG KONG POLYTECHNIC UNIVERSITY mpc.basemva = 100; %% system MVA base %% bus data %bus_i type Pd Qd Gs Bs area Vm Va basekv zone Vmax Vmin mpc.bus = [ 1 1 97.6 44.2 0 0 2 1.0393836-13.536602 345 1 1.06 0.94; 2 1 0 0 0 0 2 1.0484941-9.7852666 345 1 1.06 0.94; 3 1 322 2.4 0 0 2 1.0307077-12.276384 345 1 1.06 0.94; 4 1 500 184 0 0 1 1.00446-12.626734 345 1 1.06 0.94; 5 1 0 0 0 0 1 1.0060063-11.192339 345 1 1.06 0.94; 6 1 0 0 0 0 1 1.0082256-10.40833 345 1 1.06 0.94; 7 1 233.8 84 0 0 1 0.99839728-12.755626 345 1 1.06 0.94; 8 1 522 176.6 0 0 1 0.99787232-13.335844 345 1 1.06 0.94; 9 1 6.5-66.6 0 0 1 1.038332-14.178442 345 1 1.06 0.94; 10 1 0 0 0 0 1 1.0178431-8.170875 345 1 1.06 0.94; 11 1 0 0 0 0 1 1.0133858-8.9369663 345 1 1.06 0.94; 12 1 8.53 88 0 0 1 1.000815-8.9988236 345 1 1.06 0.94; 13 1 0 0 0 0 1 1.014923-8.9299272 345 1 1.06 0.94; 14 1 0 0 0 0 1 1.012319-10.715295 345 1 1.06 0.94; 15 1 320 153 0 0 3 1.0161854-11.345399 345 1 1.06 0.94; 16 1 329 32.3 0 0 3 1.0325203-10.033348 345 1 1.06 0.94; 17 1 0 0 0 0 2 1.0342365-11.116436 345 1 1.06 0.94; 18 1 158 30 0 0 2 1.0315726-11.986168 345 1 1.06 0.94; 19 1 0 0 0 0 3 1.0501068-5.4100729 345 1 1.06 0.94; 20 1 680 103 0 0 3 0.99101054-6.8211783 345 1 1.06 0.94; 21 1 274 115 0 0 3 1.0323192-7.6287461 345 1 1.06 0.94; 22 1 0 0 0 0 3 1.0501427-3.1831199 345 1 1.06 0.94; 23 1 247.5 84.6 0 0 3 1.0451451-3.3812763 345 1 1.06 0.94; 24 1 308.6-92.2 0 0 3 1.038001-9.9137585 345 1 1.06 0.94; 25 1 224 47.2 0 0 2 1.0576827-8.3692354 345 1 1.06 0.94; 26 1 139 17 0 0 2 1.0525613-9.4387696 345 1 1.06 0.94; 27 1 281 75.5 0 0 2 1.0383449-11.362152 345 1 1.06 0.94; 28 1 206 27.6 0 0 3 1.0503737-5.9283592 345 1 1.06 0.94; 29 1 283.5 26.9 0 0 3 1.0501149-3.1698741 345 1 1.06 0.94; 30 2 0 0 0 0 2 1.0499-7.3704746 345 1 1.06 0.94; 31 3 9.2 4.6 0 0 1 0.982 0 345 1 1.06 0.94; 32 2 0 0 0 0 1 0.9841-0.1884374 345 1 1.06 0.94; 33 2 0 0 0 0 3 0.9972-0.19317445 345 1 1.06 0.94; 34 2 0 0 0 0 3 1.0123-1.631119 345 1 1.06 0.94; 35 2 0 0 0 0 3 1.0494 1.7765069 345 1 1.06 0.94; 36 2 0 0 0 0 3 1.0636 4.4684374 345 1 1.06 0.94; 37 2 0 0 0 0 2 1.0275-1.5828988 345 1 1.06 0.94; 38 2 0 0 0 0 3 1.0265 3.8928177 345 1 1.06 0.94; 39 2 1104 250 0 0 1 1.03-14.535256 345 1 1.06 0.94; ]; %% generator data % bus Pg Qg Qmax Qmin Vg mbase status Pmax Pmin Pc1 Pc2 Qc1min Qc1max Qc2min Qc2max ramp_agc ramp_10 ramp_30 ramp_q apf mpc.gen = [ 30 250 161.762 400 140 1.049 100 1 1040 0 0 0 0 0 0 0 0 0 0 0 0; 31 677.9 221.574 300-100 0.982 100 1 646 0 0 0 0 0 0 0 0 0 0 0 0; 32 650 206.965 300 150 0.9841 100 1 725 0 0 0 0 0 0 0 0 0 0 0 0; 33 632 108.293 250 0 0.9972 100 1 652 0 0 0 0 0 0 0 0 0 0 0 0; 34 508 166.688 167 0 1.0123 100 1 508 0 0 0 0 0 0 0 0 0 0 0 0; 35 650 210.661 300-100 1.0494 100 1 687 0 0 0 0 0 0 0 0 0 0 0 0; 26

36 560 100.165 240 0 1.0636 100 1 580 0 0 0 0 0 0 0 0 0 0 0 0; 37 540-1.3695 250 0 1.0275 100 1 564 0 0 0 0 0 0 0 0 0 0 0 0; 38 830 21.7327 300-150 1.0265 100 1 865 0 0 0 0 0 0 0 0 0 0 0 0; 39 1000 78.4674 300-100 1.03 100 1 1100 0 0 0 0 0 0 0 0 0 0 0 0; ];%% branch data % fbus tbus r x b ratea rateb ratec ratio angle status angmin angmax mpc.branch = [ 1 2 0.0035 0.0411 0.6987 600 600 600 0 0 1-360 360; 1 39 0.001 0.025 0.75 1000 1000 1000 0 0 1-360 360; 2 3 0.0013 0.0151 0.2572 500 500 500 0 0 1-360 360; 2 25 0.007 0.0086 0.146 500 500 500 0 0 1-360 360; 2 30 0 0.0181 0 900 900 2500 1.025 0 1-360 360; 3 4 0.0013 0.0213 0.2214 500 500 500 0 0 1-360 360; 3 18 0.0011 0.0133 0.2138 500 500 500 0 0 1-360 360; 4 5 0.0008 0.0128 0.1342 600 600 600 0 0 1-360 360; 4 14 0.0008 0.0129 0.1382 500 500 500 0 0 1-360 360; 5 6 0.0002 0.0026 0.0434 1200 1200 1200 0 0 1-360 360; 5 8 0.0008 0.0112 0.1476 900 900 900 0 0 1-360 360; 6 7 0.0006 0.0092 0.113 900 900 900 0 0 1-360 360; 6 11 0.0007 0.0082 0.1389 480 480 480 0 0 1-360 360; 6 31 0 0.025 0 1800 1800 1800 1.07 0 1-360 360; 7 8 0.0004 0.0046 0.078 900 900 900 0 0 1-360 360; 8 9 0.0023 0.0363 0.3804 900 900 900 0 0 1-360 360; 9 39 0.001 0.025 1.2 900 900 900 0 0 1-360 360; 10 11 0.0004 0.0043 0.0729 600 600 600 0 0 1-360 360; 10 13 0.0004 0.0043 0.0729 600 600 600 0 0 1-360 360; 10 32 0 0.02 0 900 900 2500 1.07 0 1-360 360; 12 11 0.0016 0.0435 0 500 500 500 1.006 0 1-360 360; 12 13 0.0016 0.0435 0 500 500 500 1.006 0 1-360 360; 13 14 0.0009 0.0101 0.1723 600 600 600 0 0 1-360 360; 14 15 0.0018 0.0217 0.366 600 600 600 0 0 1-360 360; 15 16 0.0009 0.0094 0.171 600 600 600 0 0 1-360 360; 16 17 0.0007 0.0089 0.1342 600 600 600 0 0 1-360 360; 16 19 0.0016 0.0195 0.304 600 600 2500 0 0 1-360 360; 16 21 0.0008 0.0135 0.2548 600 600 600 0 0 1-360 360; 16 24 0.0003 0.0059 0.068 600 600 600 0 0 1-360 360; 17 18 0.0007 0.0082 0.1319 600 600 600 0 0 1-360 360; 17 27 0.0013 0.0173 0.3216 600 600 600 0 0 1-360 360; 19 20 0.0007 0.0138 0 900 900 2500 1.06 0 1-360 360; 19 33 0.0007 0.0142 0 900 900 2500 1.07 0 1-360 360; 20 34 0.0009 0.018 0 900 900 2500 1.009 0 1-360 360; 21 22 0.0008 0.014 0.2565 900 900 900 0 0 1-360 360; 22 23 0.0006 0.0096 0.1846 600 600 600 0 0 1-360 360; 22 35 0 0.0143 0 900 900 2500 1.025 0 1-360 360; 23 24 0.0022 0.035 0.361 600 600 600 0 0 1-360 360; 23 36 0.0005 0.0272 0 900 900 2500 1 0 1-360 360; 25 26 0.0032 0.0323 0.531 600 600 600 0 0 1-360 360; 25 37 0.0006 0.0232 0 900 900 2500 1.025 0 1-360 360; 26 27 0.0014 0.0147 0.2396 600 600 600 0 0 1-360 360; 26 28 0.0043 0.0474 0.7802 600 600 600 0 0 1-360 360; 26 29 0.0057 0.0625 1.029 600 600 600 0 0 1-360 360; 28 29 0.0014 0.0151 0.249 600 600 600 0 0 1-360 360; 29 38 0.0008 0.0156 0 1200 1200 2500 1.025 0 1-360 360; Fig. 13 Matrix showing parameters for buses, branches and generators of IEEE 39 Bus System adopted in Case 3 and 4 27

5) Results and Findings THE HONG KONG POLYTECHNIC UNIVERSITY 5.1 Power flow result of IEEE 9 Bus System The following tables shown the testing results of the studied cases. Bus Line Flow (MW) Bus Generation Load (MW) (MW) Line no. From To Sing Receiving Loss 1 71.95 0 1 1 4 71.95 71.95 0 2 163 0 2 4 5 30.73 30.55 0.174 3 85 0 3 6 5 60.89 59.45 1.449 4 0 0 4 3 6 85 85 0 5 0 90 5 6 7 24.11 24.01 0.095 6 0 0 6 8 7 76.5 75.99 0.506 7 0 100 7 2 8 163 163 0 8 0 0 8 8 9 86.5 84.04 2.465 9 0 125 9 4 9 41.23 40.96 0.266 Table 3 IEEE 9 Bus System, Power flow result from MATPOWER 5.2 Case 1 IEEE 9 Bus System Topological distribution factors for IEEE 9 Bus System, A u 1 = 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0.4270 0 0.7164 0.4270 1 0.7164 0 0 0 0 0 1 0 0 1 0 0 0 0 0.4693 0.2836 0 0 0.2836 1 0.4693 0 0 1 0 0 0 0 0 1 0 0.5730 0.5307 0 0.5730 0 0 0 0.5307 1 1 Fig. 14 Matrix of topological generation distribution factors A u by upstream-looking algorithm 28

Gross Power (MW) THE HONG KONG POLYTECHNIC UNIVERSITY 180.0 160.0 140.0 120.0 100.0 80.0 60.0 40.0 20.0 0.0 Gross Nodal Power of IEEE 9 Bus System 163.0 163.0 127.7 100.6 91.6 85.0 85.0 72.0 72.0 1 2 3 4 5 6 7 8 9 Bus Fig. 15 Gross nodal power of the 9 buses of IEEE 9 Bus System Load Bus 5 7 9 Coefficients (PL/Pi) 1 1 1 Table 4 Coefficients of load buses Line Generation (MW) Sum (MW) Line loss (MW) No. From bus To bus G1 G2 G3 1 1 4 71.950 - - 71.950 0.0000 2 4 5 30.726 - - 30.726 0.0043 3 6 5 - - 60.890 60.890 0.0000 4 3 6 - - 85.000 85.000 0.0000 5 6 7 - - 24.110 24.110 0.0000 6 8 7-76.500-76.500 0.0000 7 2 8-163.000-163.000 0.0000 8 8 9-86.500-86.500 0.0000 9 4 9 41.224 - - 41.224 0.0057 Table 5 Contribution of particular generators to line flow and line loss 29

Line flow (MW) THE HONG KONG POLYTECHNIC UNIVERSITY 200.000 Contribution of particular generators to line flows 150.000 100.000 50.000 G1 G2 G3 0.000 1 2 3 4 5 6 7 8 9 Line no. Fig. 16 Graph showing the contribution of particular generators to line flow and line loss G1 G2 G3 Total Gross Loss demand Allocated L5 30.73-60.89 91.62 1.62 L7 0.00 76.50 24.11 100.61 0.61 L9 41.22 86.50-127.72 2.72 Total Actual Gen. 71.95 163.00 85.00 319.95 4.95 Table 6 Contribution to power flows to each source-sink pair in IEEE 39 bus system by upstream-looking algorithm 30

Net Power(MW) 5.3 Case 2 IEEE 9 Bus System THE HONG KONG POLYTECHNIC UNIVERSITY Topological distribution factors for IEEE 9 Bus System, A d 1 = 1 0 0 0.9999 0.3394 0 0 0 0.3276 0 1 0 0 0 0 0.7599 1 0.6723 0 0 1 0 0.6606 1 0.2401 0 0 0 0 0 1 0.3394 0 0 0 0.3277 0 0 0 0 1 0 0 0 0 0 0 0 0 0.6606 1 0.2401 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0.7599 1 0.6723 0 0 0 0 0 0 0 0 1 1 Fig. 17. Matrix of topological distribution factors, A d by downstream-looking algorithm 180.0 160.0 140.0 120.0 100.0 80.0 60.0 40.0 20.0 0.0 Net Nodal Power of IEEE 9 Bus System 160.0 160.0 125.0 100.0 90.0 83.5 83.5 71.5 71.5 1 2 3 4 5 6 7 8 9 Bus Fig. 18 Net nodal power of the 9 buses of IEEE 9 Bus System Gen. Bus 1 2 3 Coefficients (PG/Pi) 1 1 1 Table 7 Coefficients of generator buses 31

Line flow (MW) THE HONG KONG POLYTECHNIC UNIVERSITY Line Load (MW) Sum (MW) Line loss (MW) No. From bus To bus L5 L7 L9 1 1 4 30.546-40.954 71.500 0.4499 2 4 5 12.970-17.389 30.359 0.1910 3 6 5 41.580 16.793-58.373 1.0771 4 3 6 59.450 24.010-83.460 1.5400 5 6 7 16.793 6.782-23.575 0.4350 6 8 7-35.426 39.179 74.605 1.3846 7 2 8-75.990 84.040 160.030 2.9700 8 8 9-39.179 43.330 82.509 1.5313 9 4 9 17.389-23.315 40.704 0.2561 Table 8 Contribution of particular loads to line flow and line loss 200.000 Contribution of particular loads to line flows 150.000 100.000 50.000 L5 L7 L9 0.000 1 2 3 4 5 6 7 8 9 Line no. Fig. 19 Graph showing the contribution of particular loads to line flow and line loss L5 L7 L9 Total Net Gen. Loss Allocated G1 30.55-40.95 71.50 0.45 G2-75.99 84.04 160.03 2.97 G3 59.45 24.01-83.46 1.54 Total Actual demand 90.00 100.00 124.99 314.99 4.96 Table 9 Contribution to power flows to each source-sink pair in IEEE 39 bus system by downstream-looking algorithm 32

5.4 Power flow result of IEEE 39 Bus System Bus Generation (MW) Load (MW) Bus Line no. From To Line Flow (MW) Table 10 IEEE 39 Bus System, Power flow result from MATPOWER Sing Receving Loss 1 0 97.6 1 2 1 174.68 173.7 0.978 2 0 0 2 1 39 76.1 76.03 0.066 3 0 322 3 2 3 319.91 318.58 1.335 4 0 500 4 25 2 248.93 244.59 4.337 5 0 0 5 30 2 250 250 0 6 0 0 6 3 4 37.34 37.13 0.208 7 0 233.8 7 18 3 40.78 40.76 0.017 8 0 522 8 5 4 197.76 197.45 0.309 9 0 6.5 9 14 4 265.99 265.42 0.571 10 0 0 10 6 5 537.51 536.94 0.573 11 0 0 11 5 8 339.18 338.24 0.933 12 0 8.53 12 6 7 453.82 452.56 1.261 13 0 0 13 11 6 323.38 322.65 0.724 14 0 0 14 31 6 668.67 668.67 0 15 0 320 15 7 8 218.76 218.56 0.192 16 0 329 16 8 9 34.81 34.48 0.324 17 0 0 17 9 39 27.98 27.97 0.018 18 0 158 18 10 11 327.9 327.46 0.438 19 0 0 19 10 13 322.1 321.69 0.407 20 0 680 20 32 10 650 650 0 21 0 274 21 11 12 4.09 4.06 0.029 22 0 0 22 13 12 4.51 4.47 0.034 23 0 247.5 23 13 14 317.18 316.3 0.879 24 0 308.6 24 14 15 50.31 50.26 0.053 25 0 224 25 16 15 270.56 269.74 0.825 26 0 139 26 16 17 224.02 223.68 0.338 27 0 281 27 19 16 454.38 451.3 3.078 28 0 206 28 21 16 330.42 329.6 0.821 29 0 283.5 29 24 16 42.71 42.68 0.03 30 250 0 30 17 18 199.04 198.78 0.261 31 677.87 9.2 31 17 27 24.64 24.62 0.016 32 650 0 32 19 20 174.73 174.51 0.218 33 632 0 33 33 19 632 629.11 2.894 34 508 0 34 34 20 508 505.49 2.511 35 650 0 35 22 21 607.21 604.42 2.783 36 560 0 36 22 23 42.79 42.77 0.025 37 540 0 37 35 22 650 650 0 38 830 0 38 23 24 353.84 351.31 2.529 39 1000 1104 39 36 23 560 558.57 1.43 40 25 26 65.41 65.29 0.126 41 37 25 540 538.34 1.657 42 26 27 257.3 256.38 0.92 43 28 26 141.61 140.82 0.788 44 29 26 192.1 190.19 1.914 45 29 28 349.16 347.61 1.556 46 38 29 830 824.77 5.234 33

Gross power (MW) 5.5 Case 3 IEEE 39 Bus System THE HONG KONG POLYTECHNIC UNIVERSITY The detailed information of topological distribution factor matrix can be found in the Appix A5. 1200.0 1000.0 800.0 600.0 400.0 200.0 Gross Nodal Power of IEEE 39 Bus System 0.0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 Bus Fig. 20 Gross nodal power of the 39 buses of IEEE 39 Bus System Bus Power Power Bus Power (MW) Bus Power (MW) Bus (MW) (MW) 1 176.5 11 327.9 21 607.2 31 668.7 2 499.7 12 8.6 22 650.0 32 650.0 3 364.5 13 322.1 23 602.8 33 632.0 4 503.1 14 317.6 24 354.7 34 508.0 5 538.1 15 323.7 25 540.0 35 650.0 6 992.5 16 831.5 26 402.1 36 560.0 7 454.3 17 226.2 27 286.0 37 540.0 8 559.6 18 201.3 28 351.4 38 830.0 9 35.0 19 632.0 29 830.0 39 105.7 10 650.0 20 683.5 30 250.0 Table 11 Gross nodal power of the 39 buses of IEEE 39 Bus System in tabular form 34

Line flow (MW) THE HONG KONG POLYTECHNIC UNIVERSITY Load Bus 1 3 4 7 8 9 12 15 Coefficients (PL/Pi) 0.5619 0.8961 1.0000 0.5166 0.9375 0.1885 1.0000 1.0000 Load Bus 16 18 20 21 23 24 25 26 Coefficients (PL/Pi) 0.3995 0.7948 1.0000 0.4533 0.4116 0.8784 0.4161 0.3507 Load Bus 27 28 29 31 39 Coefficients (PL/Pi) 1.0000 0.5926 0.3437 0.0000 1.0000 Table 12 Coefficients of load buses 900 800 700 Contribution of particular generators to line flow G30 G31 G32 G33 G34 G35 G36 G37 G38 600 500 400 300 200 100 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 Line no. Fig. 21 Graph showing the contribution of particular generators to line flow Absence of generator at bus 39 From the power flow result obtained from MATPOWER in Table 10, we can know that actual there is a generation of 1000MW at bus 39. However, from the result of upstreamlooking algorithm shown in figure 21 and table 13, the 1000MW generator is absence in the contribution to any transmission line. It is because there is also 1104MW local load at bus 39. It is assumed that the local generation would serve the entire local load first in linear equation-based tracing algorithms. Therefore, the load at bus 39 become 1104-1000 = 104 MW. Note that bus 31 is also assumed to be a pure generation bus with 677.87-9.2 = 668.67 MW generation due to the same assumption. 35

Line Generation (MW) Sum Line Loss No. From To G30 G31 G32 G33 G34 G35 G36 G37 G38 (MW) (MW) 1 2 1 88.30 - - - - - - 88.19-176.48-1.804 2 1 39 38.68 - - - - - - 38.64-77.32-1.220 3 2 3 161.70 - - - - - - 161.51-323.21-3.304 4 25 2 - - - - - - - 249.70-249.70-0.768 5 30 2 250.00 - - - - - - - - 250.00 0 6 3 4 16.80 - - 2.36-1.73 0.21 16.78-37.88-0.537 7 18 3 - - - 22.67-16.63 1.99 - - 41.29-0.510 8 5 4-133.53 64.66 - - - - - - 198.20-0.439 9 14 4 - - 267.07 - - - - - - 267.07-1.080 10 6 5-362.56 175.57 - - - - - - 538.13-0.621 11 5 8-229.03 110.91 - - - - - - 339.93-0.752 12 6 7-306.11 148.23 - - - - - - 454.34-0.524 13 11 6 - - 323.80 - - - - - - 323.80-0.425 14 31 6-668.67 - - - - - - - 668.67 0 15 7 8-147.97 71.65 - - - - - - 219.62-0.862 16 8 9-23.57 11.41 - - - - - - 34.98-0.172 17 9 39-19.13 9.26 - - - - - - 28.39-0.407 18 10 11 - - 327.90 - - - - - - 327.90 0 19 10 13 - - 322.10 - - - - - - 322.10 0 20 32 10 - - 650.00 - - - - - - 650.00 0 21 11 12 - - 4.10 - - - - - - 4.10-0.005 22 13 12 - - 4.52 - - - - - - 4.52-0.006 23 13 14 - - 317.58 - - - - - - 317.58-0.404 24 14 15 - - 50.51 - - - - - - 50.51-0.204 25 16 15 - - - 149.96-110.06 13.16 - - 273.17-2.613 26 16 17 - - - 124.16-91.12 10.90 - - 226.18-2.164 27 19 16 - - - 456.47 - - - - - 456.47-2.087 28 21 16 - - - - - 331.95 - - - 331.95-1.525 29 24 16 - - - - - 3.06 40.06 - - 43.12-0.411 30 17 18 - - - 110.49-81.09 9.70 - - 201.27-2.228 31 17 27 - - - 13.68-10.04 1.20 - - 24.92-0.276 32 19 20 - - - 175.53 - - - - - 175.53-0.803 33 33 19 - - - 632.00 - - - - - 632.00 0 34 34 20 - - - - 508.00 - - - - 508.00 0 35 22 21 - - - - - 607.21 - - - 607.21 0 36 22 23 - - - - - 42.79 - - - 42.79 0 37 35 22 - - - - - 650.00 - - - 650.00 0 38 23 24 - - - - - 25.18 329.51 - - 354.69-0.853 39 36 23 - - - - - - 560.00 - - 560.00 0 40 25 26 - - - - - - - 65.61-65.61-0.202 41 37 25 - - - - - - - 540.00-540.00 0 42 26 27 - - - - - - - 42.60 218.45 261.05-3.751 43 28 26 - - - - - - - - 143.15 143.15-1.535 44 29 26 - - - - - - - - 193.32 193.32-1.221 45 29 28 - - - - - - - - 351.38 351.38-2.218 46 38 29 - - - - - - - - 830.00 830.00 0 Table 13 Contribution of particular generators to line flow 36

Power flow corresponding to each source-sink pair G30 G31 G32 G33 G34 G35 L1 49.61 - - - - - L3 144.90 - - 20.31-14.91 L4 16.80 133.53 331.73 2.36-1.73 L7-158.14 76.58 - - - L8-353.43 171.15 - - - L9-4.44 2.15 - - - L12 - - 8.61 - - - L15 - - 50.51 149.96-110.06 L16 - - - 182.35-133.83 L18 - - - 87.82-64.45 L20 - - - 175.53 508.00 - L21 - - - - - 275.26 L23 - - - - - 17.61 L24 - - - - - 22.12 L25 - - - - - - L26 - - - - - - L27 - - - 13.68-10.04 L28 - - - - - - L29 - - - - - - L31 - - - - - - L39 38.68 19.13 9.26 - - - Total Actual 250.00 668.67 650.00 632.00 508.00 650.00 Gen. Table 14a Contribution to power flows to each source-sink pair in IEEE 39 bus system by upstream-looking algorithm 37

G36 G37 G38 G39 Total Gross Demand Loss Allocated L1-49.55 - - 99.16 1.56 L3 1.78 144.73 - - 326.63 4.63 L4 0.21 16.78 - - 503.15 3.15 L7 - - - - 234.72 0.92 L8 - - - - 524.57 2.57 L9 - - - - 6.59 0.09 L12 - - - - 8.61 0.08 L15 13.16 - - - 323.69 3.69 L16 16.00 - - - 332.18 3.18 L18 7.71 - - - 159.98 1.98 L20 - - - - 683.53 3.53 L21 - - - - 275.26 1.26 L23 230.49 - - - 248.10 0.60 L24 289.45 - - - 311.57 2.97 L25-224.69 - - 224.69 0.69 L26-23.01 118.01-141.03 2.03 L27 1.20 42.60 218.45-285.97 4.97 L28 - - 208.23-208.23 2.23 L29 - - 285.30-285.30 1.80 L31 - - - - 0.00 0.00 L39-38.64 - - 105.71 1.71 Total Actual Gen. 560.00 540.00 830.00 0.00 5288.67 43.64 Table 14b Contribution to power flows to each source-sink pair in IEEE 39 bus system by upstream-looking algorithm 38

Net power (MW) 5.6 Case 4 IEEE 39 Bus System THE HONG KONG POLYTECHNIC UNIVERSITY The detailed information of topological distribution factor matrix can be found in the Appix A6. 1200.0 Net Nodal Power of IEEE 39 Bus System 1000.0 800.0 600.0 400.0 200.0 0.0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 Bus Fig. 22 Net Nodal Power of the 39 buses of IEEE 39 Bus System by upstream-looking algorithm Bus Power Power Bus Power (MW) Bus Power (MW) Bus (MW) (MW) 1 173.6 11 325.5 21 603.0 31 666.2 2 492.0 12 8.5 22 645.6 32 645.7 3 359.1 13 320.2 23 598.7 33 625.0 4 500.0 14 315.7 24 351.2 34 505.5 5 535.5 15 320.0 25 532.5 35 645.6 6 987.7 16 822.1 26 395.4 36 556.1 7 452.2 17 223.4 27 281.0 37 532.5 8 556.5 18 198.7 28 346.5 38 819.8 9 34.5 19 625.0 29 819.7 39 104.0 10 645.7 20 680.0 30 248.7 Table 15 Net nodal power of the 39 buses of IEEE 39 Bus System by upstream-looking algorithm in tabular form Gen. Bus 30 31 32 33 34 35 36 37 38 39 Coefficients 1 1 1 1 1 1 1 1 1 0 (PG/Pi) Table 16 Coefficients of generation bus 39

Line Load (MW) No. From To L1 L3 L4 L7 L8 L9 L12 L15 L16 L18 1 2 1 34.28 100.26 11.56 - - - - - - - 2 1 39 42.72 - - - - - - - - - 3 2 3 62.87 183.88 21.20 - - - - - - - 4 25 2 21.93 64.14 7.40 - - - - - - - 5 30 2 49.33 144.30 16.64 - - - - - - - 6 3 4-33.27 3.84 - - - - - - - 7 18 3-7.49 0.86 - - - - - - 32.40 8 5 4 - - 72.61-116.61 1.45 - - - - 9 14 4 - - 222.72 - - - - 42.18 - - 10 6 5 - - 106.95 126.63 282.73 3.52 - - - - 11 5 8 - - 124.38 0.00 199.75 2.49 - - - - 12 6 7 - - 90.14 106.73 238.30 2.97 - - - - 13 11 6 - - 63.32 74.98 167.39 2.08 4.00 - - - 14 31 6 - - 133.18 157.70 352.09 4.38 - - - - 15 7 8 - - - 112.91 98.95 1.23 - - - - 16 8 9 - - - - 32.32 0.40 - - - - 17 9 39 - - - - - 5.27 - - - - 18 10 11 - - 166.09 38.33 85.59 1.07 4.30 25.32 - - 19 10 13 - - 163.16 37.66 84.08 1.05 4.22 24.87 - - 20 32 10 - - 329.68 76.09 169.89 2.12 8.53 50.26 - - 21 11 12 - - 0.80 0.94 2.11 0.03 0.05 - - - 22 13 12 - - 3.69 - - - 0.06 0.70 - - 23 13 14 - - 260.97 - - - 4.40 49.42 - - 24 14 15 - - 42.18 - - - - 7.99 - - 25 16 15-11.96 1.38 - - - - 88.35 107.75 51.75 26 16 17-9.92 1.14 - - - - 73.26 89.35 42.91 27 19 16-14.36 1.66 - - - - 106.03 129.33 62.11 28 21 16-7.97 0.92 - - - - 58.87 71.80 34.48 29 24 16-0.23 0.03 - - - - 1.70 2.07 0.99 30 17 18-32.46 3.74 - - - - - - 140.41 31 17 27-4.02 0.46 - - - - - - 17.39 32 19 20-5.55 0.64 - - - - 41.00 50.01 24.02 33 33 19-19.92 2.30 - - - - 147.13 179.46 86.18 34 34 20-0.00 0.00 - - - - - - - 35 22 21-13.72 1.58 - - - - 101.31 123.56 59.34 36 22 23-0.97 0.11 - - - - 7.17 8.74 4.20 37 35 22-14.75 1.70 - - - - 108.95 132.88 63.81 38 23 24-1.11 0.13 - - - - 8.17 9.96 4.78 39 36 23-1.75 0.20 - - - - 12.95 15.80 7.59 40 25 26 5.85 17.12 1.97 - - - - - - - 41 37 25 48.12 140.74 16.23 - - - - - - - 42 26 27 - - - - - - - - - - 43 28 26 - - - - - - - - - - 44 29 26 - - - - - - - - - - 45 29 28 - - - - - - - - - - 46 38 29 - - - - - - - - - - Table 17a Contribution of particular loads to line flow 40

Load (MW) Line No. From To L21 L23 L24 L25 L26 L27 L28 L29 L39 Sum (MW) Line loss (MW) 1 2 1 - - - - - - - - 26.70 172.80 0.901 2 1 39 - - - - - - - - 33.28 76.00 0.031 3 2 3 - - - - - - - - 48.97 316.93 1.653 4 25 2 - - - 101.77 10.40 19.19 - - 17.08 241.92 2.672 5 30 2 - - - - - - - - 38.43 248.70 1.297 6 3 4 - - - - - - - - - 37.11 0.022 7 18 3 - - - - - - - - - 40.75 0.009 8 5 4 - - - - - - - - 6.25 196.91 0.536 9 14 4 - - - - - - - - - 264.90 0.520 10 6 5 - - - - - - - - 15.15 534.98 1.961 11 5 8 - - - - - - - - 10.70 337.32 0.918 12 6 7 - - - - - - - - 12.77 450.91 1.653 13 11 6 - - - - - - - - 8.97 320.74 1.910 14 31 6 - - - - - - - - 18.87 666.23 2.442 15 7 8 - - - - - - - - 5.30 218.40 0.161 16 8 9 - - - - - - - - 1.73 34.46 0.021 17 9 39 - - - - - - - - 22.69 27.96 0.008 18 10 11 - - - - - - - - 4.59 325.28 2.180 19 10 13 - - - - - - - - 4.51 319.55 2.142 20 32 10 - - - - - - - - 9.10 645.67 4.328 21 11 12 - - - - - - - - 0.11 4.04 0.024 22 13 12 - - - - - - - - - 4.45 0.021 23 13 14 - - - - - - - - - 314.79 1.514 24 14 15 - - - - - - - - - 50.16 0.099 25 16 15 - - - - - 8.06 - - - 269.25 0.486 26 16 17 - - - - - 6.69 - - - 223.28 0.403 27 19 16 - - - - - 9.68 - - - 448.35 2.951 28 21 16 149.42 - - - - 5.37 - - - 328.83 0.771 29 24 16 - - 37.49 - - 0.16 - - - 42.67 0.013 30 17 18 - - - - - 21.88 - - - 198.49 0.288 31 17 27 - - - - - 2.71 - - - 24.58 0.036 32 19 20 - - - - - 3.74 - - - 173.37 1.141 33 33 19 - - - - - 13.43 - - - 622.14 6.971 34 34 20 - - - - - - - - - 502.99 2.498 35 22 21 254.79 16.37 20.41 - - 9.25 - - - 600.32 4.102 36 22 23 18.03 1.16 1.44 - - 0.65 - - - 42.48 0.290 37 35 22 274.00 17.60 21.95 - - 9.94 - - - 645.59 4.411 38 23 24-144.59 180.29 - - 0.75 - - - 349.77 1.541 39 36 23-229.31 285.92 - - 1.18 - - - 554.70 3.869 40 25 26 - - - 27.17 2.78 5.12 - - 4.56 64.58 0.713 41 37 25 - - - 223.31 22.83 42.11 - - 37.48 530.82 7.517 42 26 27 - - - - 89.92 165.86 - - - 255.78 0.595 43 28 26 - - - - 20.01 36.91 83.45 - - 140.37 0.452 44 29 26 - - - - 26.77 49.38 47.50 65.38-189.03 1.157 45 29 28 - - - - 48.93 90.25 86.82 119.49-345.49 2.115 46 38 29 - - - - 115.37 212.79 204.70 281.72-814.59 10.184 Table 17b Contribution of particular loads to line flow 41

Line flow (MW) THE HONG KONG POLYTECHNIC UNIVERSITY 900 800 700 Contribution of particular loads to line flow L1 L3 L4 L7 L8 L9 L12 L15 L16 L18 L20 L21 L23 L24 L25 L26 L27 L28 L29 L39 600 500 400 300 200 100 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 Line no. Fig. 23 Graph showing the contribution of particular loads to line flow Power flow corresponding to each source-sink pair L1 L3 L4 L7 L8 L9 L12 L15 G30 49.33 144.30 16.64 - - - - - G31 - - 133.18 157.70 352.09 4.38 - - G32 - - 329.68 76.09 169.89 2.12 8.53 50.26 G33-20.01 2.31 - - - - 147.81 G34 - - - - - - - - G35-14.75 1.70 - - - - 108.95 G36-1.76 0.20 - - - - 12.98 G37 48.27 141.18 16.28 - - - - - G38 - - - - - - - - G39 - - - - - - - - Total Actual Demand 97.60 322.00 500.00 233.80 521.98 6.50 8.53 320.00 Table 18a Contribution to power flows to each source-sink pair in IEEE 39 bus system by downstream-looking algorithm 42

L16 L18 L20 L21 L23 L24 L25 L26 G30 - - - - - - - - G31 - - - - - - - - G32 - - - - - - - - G33 180.28 86.58 174.51 - - - - - G34 - - 505.49 - - - - - G35 132.88 63.81-274.00 17.60 21.95 - - G36 15.84 7.61 - - 229.90 286.65 - - G37 - - - - - - 224.00 22.90 G38 - - - - - - - 116.10 G39 - - - - - - - - Total Actual Demand 329.00 158.00 680.00 274.00 247.50 308.60 224.00 139.00 Table 18b Contribution to power flows to each source-sink pair in IEEE 39 bus system by downstream-looking algorithm L27 L28 L29 L31 L39 Table 18c Contribution to power flows to each source-sink pair in IEEE 39 bus system by downstream-looking algorithm 43 Total Net Generatio n Loss Allocated G30 - - - - 38.43 248.70 1.30 G31 - - - - 18.87 666.23 2.44 G32 - - - - 9.10 645.67 4.33 G33 13.49 - - - - 625.00 7.00 G34 - - - - - 505.49 2.51 G35 9.94 - - - - 645.59 4.41 G36 1.19 - - - - 556.12 3.88 G37 42.24 - - - 37.60 532.46 7.54 G38 214.14 206.00 283.50 - - 819.75 10.25 G39 - - - - - - - Total Actual Demand 281.00 206.00 283.50 0.00 104.00 5187.71 43.66

5.7 Comparison between the results of two algorithms Comparison between Case 1 and Case 2 on the source-sink identification Sourcesink pair Case 1 (upstream-looking algorithm) Power flow (MW) Case 2(downstreamlooking algorithm) Relative error (%) 100%] [ case1 case2 case1 G1-L5 30.73 30.55 0.586% G1-L9 41.22 40.95 0.655% G2-L5 0.00 0.00 0.000% G2-L7 76.50 75.99 0.667% G2-L9 86.50 84.04 2.844% G3-L5 60.89 59.45 2.365% G3-L7 24.11 24.01 0.415% Table 19 Relative error between the power flows for source-sink pairs of Case 1 and 2 Bus Case 1 (upstream-looking algorithm) Power flow (MW) Case 2(downstreamlooking algorithm) Relative error (%) 100%] [ case1 case2 case1 G1 71.95 71.50 0.625% G2 163.00 160.03 1.822% G3 85.00 83.46 1.812% L5 91.62 90.00 1.768% L7 100.61 100.00 0.606% L9 127.72 124.99 2.137% Total 319.95 314.99 1.550% Total loss 4.95 4.96-0.201% Table 20 Relative error between the nodal power flow s for generator buses and load buses of Case 1 and Case 2 44

Comparison between Case 3 and 4 on the source-sink identification Buses Case 3 (upstreamlooking algorithm) Power flow (MW) Case 4(downstreamlooking algorithm) Relative error (%) 100%] [ case1 case2 case1 G30 250.00 248.70 0.52% G31 668.67 647.36 0.37% G32 650.00 645.67 0.67% G33 632.00 625.00 1.11% G34 508.00 505.49 0.49% G35 650.00 645.59 0.68% G36 560.00 556.12 0.69% G37 540.00 532.46 1.40% G38 830.00 819.75 1.23% L1 99.16 97.60 1.58% L3 326.63 322.00 1.42% L4 503.15 500.00 0.63% L7 234.72 233.80 0.39% L8 524.57 521.98 0.49% L9 6.59 6.50 1.44% L12 8.61 8.53 0.94% L15 323.69 320.00 1.14% L16 332.18 329.00 0.96% L18 159.98 158.00 1.24% L20 683.53 680.00 0.52% L21 275.26 274.00 0.46% L23 248.10 247.50 0.24% L24 311.57 308.60 0.95% L25 224.69 224.00 0.31% L26 141.03 139.00 1.44% L27 285.97 281.00 1.74% L28 208.23 206.00 1.07% L29 285.30 283.50 0.63% L39 105.71 104.00 1.62% Total 5288.67 5187.71 1.91% Loss 43.64 43.66-0.05% Table 21 Relative error between the power flow contribution to buses in Case 3 and 4 45

Differences in power flows result of 2 algorithms As identical system (i.e. IEEE 9 Bus System) is adopted to both Case 1 and Case 2, some may expect the results from Case 2 should be same with Case 1. However, we can find that in table 19 to table 20, there are relative errors about 0% to 2% between the power flows computed from upstream-looking algorithm and downstream-looking algorithms. This also appears between Case 3 and Case 4 in table 21. The differences in power flow when using upstream- and downstream- looking algorithms is due to the kinds of power flow measured in two algorithms. In the upstream-looking algorithm, gross power flow is used. The generation and incoming lines of the network is the same with the actual network. Since we have to fulfill KCL for linear calculation, the load is increased. While in the downstream-looking algorithm, net power flow is used. The load and outgoing lines of the actual network is taken for calculation. To fulfill KCL, the generation is decreased in the network. Thus, the power flow of generation bus in cases using downstream-looking algorithm is slightly lower than that using upstream-looking algorithm. As a result, the power flow of both the generation bus and load bus in cases using upstreamlooking algorithm is slightly higher than that using downstream-looking algorithm. 46

5.8 Analysis on Bialek Tracing Method From the introduction and test results of the proposed method in the previous sections, we have a brief understanding of the principles and calculation processes of the tracing method. In this section, the merits and limitations of Bialek Tracing Method are analyzed. Advantages of Bialek Tracing Method: (a) An uncomplicated methodology since it is a topological method based on Kirchhoff s current law and proportional sharing principle, which are linear equation-based, can be explained to the customers with simple diagrams. (b) Based on share of power, instead of impact of a particular load or generator to the network, which eliminate the counter flow problem. Hence, the calculation operation simplified. The service providers can manage and operate the pricing scheme easily. (c) Users can understand the data and tables of the transaction states even that they do not have engineering background [2]. Hence, high reliability and transparency of the pricing method are enhanced. (d) Actual usage of the transmission facilities is considered, which is fair to all the network users, each of them pay for their own usage, no cross-subsidization. (e) Enhances reasonable capital and operation cost recovery. (f) Complex networks can be handled well by linear equation-based algorithms, and linear equations can be programmed easily. 47

Disadvantages/limitations of Bialek Tracing Method: (a) The two algorithms in the proposed method generate distinct results which may confuse the users. (b) Inversion of matrix is involved in the tracing algorithms, it may be time-consuming for large systems. (c) Transmission loss due to reactive power is considered indepently with that due to active power. 48

6) Conclusion and future development In this paper, the existing transmission pricing methods including incremental transmission pricing methods and embedded transmission pricing methods are being introduced. A detailed analysis on Bialek tracing technique had been conducted in this project. Bialek Tracing Method is a flow-based topological analysis technique for embedded transmission pricing in the new restructured power supply industry, calculating how a particular user contributing to the transmission usage. The information of which generators are supplying a particular load, what is each generator s contribution to the transmission line usage and losses in the transmission system can be determinated by using this flowbased tracing method. Fair service charges can be allocated according to the actual transmission capacity used, enhancing proper economic message to the power supply providers and their customers. Case studies of the Bialek Tracing Method have be carried out on IEEE 9 Bus System and IEEE 39 Bus System and simulated using MATPOWER. Two MATLAB scripts are written for solving two linear-based algorithms. Different power flows in the system are mearsured in two algorithms, gross power flows are traced in upstream-looking algorithms, while net power flows are traced in downstream-looking algorithms. The contributions of generator and load to eah other and to lines can be computed by applying either one of the algorithms. The scripts are shown in Appix A1 to A4. To conclude, we have a comprehensive evaluation of the operation process and functions of the proposed tracing method. The effectiveness of Bialek Tracing Method is tested and verified in this project, which having a high feasibility and accuracy in power flow tracing of various system. 49

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Appix The followings are the MATLAB scripts for the calculations. (*Variable bus and branch in matrix format should be imported from excel file to MABLAB before run the script.) Script written for upstream-looking algorithm: bus_data= [bus(:,4)-bus(:,6) ];%sort the useful data branch_data=[branch(:,2) branch(:,3) branch(:,4) branch(:,6) branch(:,8)];%sort the useful data a=0; for n=1:length(branch_data) %shifting the position for convenience operation if branch_data(n,3)<0 a=branch_data(n,1); branch_data(n,1)=branch_data(n,2); branch_data(n,2)=a; a=branch_data(n,3); branch_data(n,3)=branch_data(n,4); branch_data(n,4)=a; n=length(bus_data) ; T=zeros(n,n);% create a empty matrix in right size for n=1:length(branch_data) %formation of upstream matrix T(branch_data(n,2),branch_data(n,1))=-branch_data(n,3); actual_gen=bus_data; %forming actual input matrix actual_gen(actual_gen<0)=0; bus_outflow=bus_data; %prepare for bus outflow matrix bus_outflow(bus_outflow>0)=0; bus_outflow(bus_outflow<0)=-bus_outflow(bus_outflow<0); bus_demand=bus_outflow; for n=1:length(bus_outflow) %calculate total outflow of each bus for m=1:length(branch_data) if branch_data(m,1)==n bus_outflow(n)= bus_outflow(n)+branch_data(m,3); for n=1:length(t) T(:,n)=T(:,n)/bus_outflow(n);% divide each coloum by the corresponding total power outflow of each bus for n=1:length(t) T(n,n)=1;%formation of upstream matrix t = inv(t) % get inverse matrix, topological factors gross_power = t*actual_gen % get gross power for each node A1

c=[]; for n = 1:length(bus_demand) c(n)= abs(bus_demand(n))/bus_outflow(n);% find (PL/Pi)coefficient c % show coefficient value fprintf('\nresults') for n = 1:length(bus_demand) % find load node load_loss(n) = c(n)*gross_power(n)-abs(bus_demand(n)); % find loss correponding to load node load_loss %show result total_load_loss=sum(load_loss)% show value %fprintf('%.2f\n',totalloss) for n= 1:length(bus_data) % load in row for m = 1:length(bus_data)% generator in column power_sspair(n,m)= c(n)'*t(n,m)*actual_gen(m); %find the loss corresponding to each source-sink pair power_sspair %show result for n = 1: length(bus_data) total_demand_of_each_load(n)= sum(power_sspair(n,:)); total_generation_of_each_gen(n)= sum(power_sspair(:,n)); TD=total_demand_of_each_load'; %transpose the matric for convenience TG=total_generation_of_each_gen'; total_powerflow=sum(sum(power_sspair)); %total power flow througth the system for m = 1:length(actual_gen) %contribution of gen to line flow for n= 1:length(branch_data) usline_flow(m,n)=[abs(branch_data(n,3))/bus_outflow(branch_data(n,1))]*t(bra nch_data(n,1),m)*actual_gen(m); usline_flow=usline_flow' for n= 1:length(branch_data) total_usline_flow(n)=sum(usline_flow(n,:)); total_usline_flow filename = 'utestdata.xlsx'; %export an excel file for essential data xlswrite(filename,t,1,'a1') xlswrite(filename,gross_power,2,'a1') xlswrite(filename,c,3,'a1') xlswrite(filename,power_sspair,4,'a1') xlswrite(filename,tg,5,'a1') xlswrite(filename,td,6,'b1') xlswrite(filename,load_loss,7,'a2') xlswrite(filename,branch_data,8,'a2') xlswrite(filename,usline_flow,9,'a2') xlswrite(filename,total_usline_flow,10,'a2') A2

Script written for downstream-looking algorithm: bus_data= [bus(:,4)-bus(:,6) ];%sort the useful data branch_data=[branch(:,2) branch(:,3) branch(:,4) branch(:,6),branch(:,8)]; %sort the useful data a=0; for n=1:length(branch_data) %shifting the position for convinence operation if branch_data(n,3)<0 a=branch_data(n,1); branch_data(n,1)=branch_data(n,2); branch_data(n,2)=a; a=branch_data(n,3); branch_data(n,3)=branch_data(n,4); branch_data(n,4)=a; n=length(bus_data); T=zeros(n,n);% create a empty matrix in right size for n=1:length(branch_data) %formation of downstream matrix T(branch_data(n,1),branch_data(n,2))=branch_data(n,4); net_gen=bus_data; %forming actual input matrix net_gen(net_gen<0)=0; bus_inflow=bus_data; %prepare for bus inflow matrix bus_inflow(bus_inflow>0)=0; bus_inflow(bus_inflow<0)=-bus_inflow(bus_inflow<0); bus_output=bus_inflow; for n=1:length(bus_inflow) %calculate for total inflow power of each bus for m=1:length(branch_data) if branch_data(m,1)==n bus_inflow(n)= bus_inflow(n)+branch_data(m,3); for n=1:length(t) T(:,n)=T(:,n)/bus_inflow(n);% divide each coloum by the corresponding total power inflow of each bus for n=1:length(t) T(n,n)=1;%formation of downstream matrix t = inv(t) % get inverse matrix, topological factors net_power = t*bus_output % get net power for each node c=[]; for n = 1:length(net_gen) c(n)= net_gen(n)/bus_inflow(n);% find (PG/Pi)coefficient c % show coefficient value fprintf('\nresults\n') A3

for n = 1:length(net_gen) % find gen node gen_loss(n) = -c(n)*net_power(n)+abs(net_gen(n)); % find loss correponding to gen node gen_loss %show result total_gen_loss=sum(gen_loss) % find total loss of the transmission system for n= 1:length(bus_data) % load in row for m = 1:length(bus_data)% generator in column power_sspair(n,m)= c(n)'*t(n,m)*bus_output(m); %find the loss corresponding to each source-sink pair power_sspair %show result for n = 1: length(bus_data) total_demand_of_each_load(n)= sum(power_sspair(:,n)); total_generation_of_each_gen(n)= sum(power_sspair(n,:)); TD=total_demand_of_each_load';%transpose the matric for convenience TG=total_generation_of_each_gen'; total_powerflow=sum(sum(power_sspair)); %total power flow througth the system for m = 1:length(bus_output) %contribution of load to line flow for n= 1:length(branch_data) dsline_flow(m,n)=[abs(branch_data(n,4))/bus_inflow(branch_data(n,1))]*t(bran ch_data(n,1),m)*bus_output(m); dsline_flow=dsline_flow' for n= 1:length(branch_data) total_dsline_flow(n)=sum(dsline_flow(n,:)); total_dsline_flow filename = 'dtestdata.xlsx';%export an excel file for essential data xlswrite(filename,t,1,'a1') xlswrite(filename,net_power,2,'a1') xlswrite(filename,c,3,'a1') xlswrite(filename,power_sspair,4,'a1') xlswrite(filename,tg,5,'a1') xlswrite(filename,td,6,'a2') xlswrite(filename,gen_loss,7,'a1') xlswrite(filename,branch_data,8,'a2') xlswrite(filename,dsline_flow,9,'a2') xlswrite(filename,total_dsline_flow,10,'a2') A4

Case 3 Topological distribution factor A5

Case 4 Topological distribution factor A6