2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia Allocation of Loss Cost by Optimal and Proportional Tracing Methods Nahid Aslani Amoli* and Shahram Jadid** Center of Excellence for Power System Automation and Operation Electrical Engineering Department, Iran University of Science and Technology (IUST), Tehran, Iran * Email: [email protected] ** Email: [email protected] software. Nevertheless, the allocation of loss cost between system loads was not investigated in [5]. Since optimal real power tracing is a nonlinear programming problem and with regard to high ability of GAMS software in solving such problems, in this paper, after problem linearization, it is solved as linear programming problem by GAMS. Also, allocation of loss cost between loads with presence of congestion in transmission lines by optimal and proportional tracing methods is investigated. Due to lines congestion, optimal power flow results instead of AC power flow results are employed by tracing methods. This paper is organized as follows. In section II, the concepts of proportional tracing method for loss allocation are presented. Formulation of optimal real power tracing and description of related constraints are stated in section III. In next section, the obtained results of solving optimal tracing problem by GAMS are compared with proportional tracing results in the modified IEEE 14bus system. Section V concludes the paper. Abstract—Allocation of transmission loss between network users is a challenging and contentious issue in a fully deregulated system. Also, the cost of loss must be compensated in a fair manner by users which use the transmission network. Power flow tracing can find the extent of network usage by the users that can be used for loss allocation. In this paper, proportional tracing and optimal real power tracing methods are used for allocation of loss and its cost with presence of congestion in transmission lines. GAMS software is employed for solving optimal real power tracing as linear programming problem. The results obtained by proportional and optimal tracing are compared in the modified IEEE 14-bus system and the results show that optimal tracing is more fair approach than proportional tracing. Keywords ü Power flow tracing; Loss allocation; Proportional sharing principle; Linear programming; I. INTRODUCTION In recent years, electricity systems worldwide were restructured in order to introduce market concepts. One of the problems to be faced is the allocation of transmission loss between network users. Also, the cost of the losses has to be paid by the market participants which have access to the network and the allocation must be transparent and non discriminatory. For this purpose, it is necessary to assess the extent of network usage by the participants. The advent of the tracing methods solves the problem of finding the extent of use of a network. The proportional sharing principle has been used to develop different methods for loss allocation. References [1-3] are examples of these methods, where the results of a converged power flow are used along with a linear proportional sharing rule to allocate transmission losses between network users. In these methods, the power flow of generators and loads is traced to determine the transmission system usage by each generator and load. Then, transmission losses caused by each user are specified. In [4], loss is attributed to a particular generator or load in proportion to its share on the line power flows. By this method, loss allocation can be done in exact manner. Abhyankar et al. [5] proposed that with regard to multiplicity of solution space in real power tracing, this problem can be formulated as a linear constrained optimization problem. Obtained results of solving optimal real power tracing can be used for loss allocation between system loads in a fair manner. The authors implemented the proposed method for power flow tracing in MATLAB 1-4244-2405-4/08/$20.00 ©2008 IEEE II. THE CONCEPTS OF PROPORTIONAL TRACING A. Proportional Sharing Principle The main principle used for power flow tracing in [1-3], is that of proportional sharing. This principle is explained with the help Fig. 1. The figure shows bus A, where there are two inflows of real power and two outflows. The lines i and j carry 20 and 80 MW of power to bus A respectively. Hence total power inflow to bus A is 100 MW. The lines p and q carry 60 and 40 MW of power away from bus A. As electricity is indistinguishable and the power flowing through lines is dependent only upon voltage gradient and impedance of these lines, it may be assumed that each of the MW leaving bus A contains the same proportion of the inflows. Hence, 60 MW of outflow 20 on line p contains (60 ) MW of power flowing through 100 80 ) MW of power flowing through line j. line i and (60 100 Figure 1. Proportional sharing principle 994 2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia Loss P = ¦ Loss Plm . Similarly, 40 MW of outflow on line q contains 20 (40 ) MW of power flowing through line i and 100 80 (40 ) MW of power flowing through line j [4]. 100 Li III. 1) Upstream Tracing Algorithm Upstream tracing begins at a pure sink. Pure sink node is defined as a node which has no outflows associated with it. It corresponds to node with the lowest node angle (δ ) . If one traverses the gragh in the direction of power flow, it will not be possible to revisit the same point (no circular flow). In the upstream algorithm, nodes are eliminated in the ascending order of the node angles. A node elimination operation involves: 1) Deletion of the node and its associated components like lines, generators, loads, etc. 2) Insertion of tagged fictitious load at sending end to maintain flow equivalence with the rest of the network. For this purpose, proportional sharing rule is employed. Step 1 and 2 are carried out repeatedly, till all the nodes left to be deleted are pure sources. 2) Loss Modeling The loss allocation is based on a simple proposition that, as the individual entities’ flows over a transmission line are shared in proportion to the nodal outflows, the losses can also be shared on the same basis. Let Plms and Problem OPT(x,y): min f (x , y ) . {x , y }∈S lm respectively. Let PLvirt be the virtual load in MW of i load i, on node m, which has come out of deletion of the earlier node. Now, the loss in the line lm, ( Plms − Plmr ) MW can be allocated to load i as follows: Li i ¦ nL P virt i =1 L . (1) i Where, n L is total number of loads in the system. An algorithm for upstream tracing after incorporating losses can be given as below: Step 1: Start with a pure sink node. Delete pure sink node. Step 2: Insert tagged virtual load PLi at sending end l as follows: PLvirt = l i Plms Plmr ¦ Plmr Pr ∀lm lm PLvirt . m i OPTIMAL TRACING A. Defining an Optimal Tracing Problem The optimal tracing problem can be summarized as follows: Plmr be sending and receiving end powers in MW on line PLvirt (3) In this method, the total system loss is allocated to loads. In the next section, loss allocation to various loads by optimal real power tracing will be discussed. B. Loss Allocation Loss Plm = (Plms − Plmr ) × Li ∀lm (4) The set S represents the set of all possible tracing solutions and specific set of x and y vectors represents a solution to generation and load tracing problem. The set S can be characterized by a set of linear equality and inequality constraints. These constraints are grouped into the following categories: • flow specification constraints for series branches, i.e., transmission lines and transformers; • source and sink specification constraints pertaining to shunts, e.g., generators and loads; • conservation of commodity flow constraints. Inequality constraints are associated with flow bounds. In the following subsection, mathematical representations for these constraints are given. 1) Flow Specification Constraints Traditionally, two types of tracing problems, viz., generation tracing and load tracing, are discussed. Generation tracing traces generator flows to loads, while load tracing traces load flows to generators. First, modeling of the flow specification constraints for generation tracing is discussed. a) Generation tracing Let Plm (MW) be the flow on a line lm. Flow Plm supplied from system generators, is presented as follows: Plm = PlmG1 + PlmG 2 + ... + PlmG nG . (5) The component of generator G k on line lm can be k of the total injection by expressed as fraction x lm generator G k , i.e., PG k . Therefore (2) k PlmG = x lm .PG . (6) k Plm = ¦ x lm .PG k , ∀ set of lines . (7) k k Step 3: After each deletion of a pure sink, there is at least one pure sink node left in the system. Delete this pure sink node and repeat step 2. Step 4: Repeat step 2 and 3 till all the nodes left in the system are pure sources. Equation (1) assigns losses over a line to various loads. Total losses incurred due to load PLi are given as follows [6]: nG k =1 Since the branch flows are known and x are unknown, flow equations for generation allocation can be written as follows: 995 2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia [A flow d ][x flow ] = [bflow d ] . (8) Matrix A flow d has nb rows and nb × nG columns, where [Ainju ][ y inj ] = [binju ] . nb is the number of branches and nG the number of generators. b) Load Tracing The power flow Plm on line lm can also be expressed as a summation of load components, i.e., L Plm = PlmL1 + PlmL2 + ... + PlmnL . (9) i of load PLi as follows: a fraction y lm i .PLi . PlmLi = y lm (10) i .PLi , ∀ set of lines . Plm = ¦ y lm (11) Thus, nL i =1 The matrix form of above equation is: (12) Matrix A flow u has nb rows and nb × n L columns, It should be noted that all x-fractions are restricted from 0 to 1. These limits correspond to flow bound constraints. The lower limit ensures that the flow component should have the same direction as the arc flow, while the upper limit ensures that no flow component exceeds the corresponding generation. It is worth mentioning that in traditional tracing methods [1-3], the fractions k i x lm , y lm , x ik and y ki are frozen by application of the proportional sharing principle. In the optimal tracing, these fractions are decision variables and are set as a result of the optimization problem. 3) Conservation of Commodity Flow Constraints The conservation of flow constraints can be neatly expressed by using arc or bus incidence matrix M of the underlying graph. In the matrix M, rows correspond to nodes and columns to arcs. The entry M (i , j ) is set to 1 if arc j is outgoing at node i; it is -1 if the arc is incoming at node i; else , it is set to zero. a) Generation Tracing Let ϑ (ν , ε ) represent the graph of network, where ν represents a set of all nodes and ε the set of arcs. Let ε be partitioned as follows: ε = {e n } * {e L } * {eG } . b where n L is the number of loads. 2) Source and Sink Specification Constraints a) Generation Tracing In a generation tracing problem, it is necessary to write sink (load) constraints. They express contribution of generators in loads. This statement is presented by following constraint: (17) Where subset {enb } represents the set of series branches, subset {eG } indicates the set of shunt branches due to generators, and subset {e L } represents set of shunt branches due to loads. Then, partitioning of ε induces the following column partitions on M : nG PLi = ¦ x ik PG k . (16) Matrix A inju has nG rows and nG × n L columns. The component of load (PLi ) on line lm is expressed as [A flow u ][ y flow ] = [bflow u ] . In the matrix form, generator equations for generation allocation can be written as follows: (13) M = [ M nb , M L , M G ] . (18) k =1 Gk Li Where P Further, let M d represent submatrix of M formed by considering series branches and shunt loads, is the component of load PLi met by generator G k . In the matrix form, the load equations for generation allocation can be written as follows: [Ainj d ][x inj ] = [binj d ] . M d = [M nb , M L ] . (14) Matrix A inj d has n L rows and n L × nG columns. nL PG k = ¦ y ki PLi . Finally, conservation of commodity flow constraints for generation tracing are stated as follows: [M d ] ª¬ x k º¼ = ª¬e k º¼ , k = 1...nG . b) Load Tracing In the load tracing problem, it is necessary to model the share of loads in a generator, i.e., (15) i =1 996 (19) (20) Where x k represents the set of x-variables for lines and loads associated with the kth generator. k is the node at which the kth generator is connected, and e k is the kth column of identity matrix. The set of the continuity equations (20) can be rearranged and written in block 2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia min f ( x, y ) . matrix notations with x flow and xinj variable partition, in the same way as shown in (8) and (14), as follows: ªA ¬« cont flow d Acont inj d x º ⋅ «ª flow »º = ªb º . (21) ¼» ¬ x inj ¼ ¬ contd ¼ [M u ] ª y º = − ¬ªe i ¼º , i = 1...n L . ¬ ¼ The set of above equations can be rearranged into the following form: flow u ª y flow Acont inj º ⋅ « u ¼ ¬ y inj º » = ª¬bcont º¼ . ¼ u (23) B. Loss Allocation The objective function of optimal tracing problem is equitable distribution of losses between loads. On the other hand, loss incurred in supplying demand PLi is given by: § nG loss PL = ¨¨ ¦ y ki PLi i © k =1 · ¸¸ − PL . ¹ i · ¸¸ − 1 . ¹ . (29) (30) [ y ] ≥ [0,0...0]T . (31) As can be seen, the optimal tracing is a nonlinear programming problem that can be solved by GAMS. More detailed explanations about above equations have been presented in [5]. Due to congestion in the transmission lines, OPF dispatch provides locational marginal price (LMP) at any bus. Also, every load must be responsible for the system losses he causes. Therefore, after assigning losses between system loads by optimal and proportional tracing methods, every load pays the charge according to loss P × LMPi Li (24) (25) Therefore, the objective function f ( x, y ) in (4) is defined as follows: nL i − k L∗ . f {x , y } = ¦ loss pu ∀k ∈ {1,2,..., nG } ∀i ∈ {1,2,..., n L } for its allocated loss. By dividing (24) by PLi ,the per unit loss for load i is obtained as follows: § nG i = ¨¨ ¦ y ki loss pu © k =1 (28) [0,0...0]T ≤ [ x] ≤ [1,1...1]T . (22) Where i is the node at which the ith load is connected and M u = [M nb , M G ] . ª Acont ¬ 0 º ª x º ªbd º = . Au »¼ «¬ y »¼ «¬bu »¼ y ki PLi − xik PGk ≥ 0 , b) Load Tracing For the load tracing problem, the continuity equations are given as follows: i ª Ad «0 ¬ (27) (26) i =1 Where k L∗ is the ratio of total system loss to total system load. The aim of optimal tracing method is to compute the closest traceable solution to the proportionate distribution of transmission system losses. In other words, the loss allocated to each load should be equal to k L∗ . In practical power system, this condition may not get satisfied, and hence, it is tried to adjust it as near as possible to k L∗ , within the tracing framework. Now, the OPT problem that was defined in (4) can be explicitly formulated as follows: 997 IV. SIMULATION RESULTS The methods discussed for allocation of loss and its cost are applied to modified IEEE 14-bus system in which bus 8 and line 7-8 (due to zero power flow) have been deleted. The system is illustrated in Fig. 2. To obtain the locational marginal prices (LMPs) as influenced by congestion, it is assumed that the power flow limits have been decreased to 50% of the original value for both the 2-4 and 6-12 lines and to 80% for the line 4-5. In order to remove the lines congestion, OPF is run in the MATPOWER environment [7]. The OPF results have been shown in the table I. Now, for tracing power flows in the network, OPF results are used. The k L∗ value for this system is 0.0367. To assign losses by proportional tracing method, first, the pure sink nodes in system are identified. Then, they are eliminated in the ascending order of the node angles. The elimination sequence for this system is as follows: 13-12-11-9-10-8-6-7-3-4-5-2 Based on this sequence, steps 1 to 4 are carried out. The obtained results by this method have been presented in table II. As mentioned before, the optimal real power tracing for the aim of loss allocation is formulated as a nonlinear programming problem that is solved using the GAMS/CPLEX solver [8]. This problem essentially points to the first norm that is converted to a linear programming problem. In order to allocate losses between loads by optimal tracing, OPF results are given to GAMS as input data. After executing program, the fractions i k x lm , y lm , x ik and y ki are obtained. By using (24), the loss allocated to load i is specified. The table II shows the obtained results. In this table, all loss allocations are greater than zero which means that no cross subsidies on the system unlike incremental loss allocation methods. Indeed, the aim of optimal tracing method is to minimize the sum of absolute deviations of per unit loss of 2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia this result is helpful for system loads from a fair viewpoint, but it can be ignored due to little difference in load payments. By changing amount or number of lines involved in congestion, the obtained result may become better. Overall, these results verify the effectiveness of optimal tracing method in terms of equitable loss allocation and also a bit less charge paid by system loads. all loads from the k L∗ value. In other words, it aims to achieve overall nearness of all loads to k L∗ . The objective function value for the results obtained by optimal tracing method is 0.048, while the sum of absolute deviations calculated for proportional tracing results is 0.098. The comparison of these figures shows that by optimal tracing method, system losses are distributed between loads in a more equitable manner rather than by proportional tracing method. Fig. 3 depicts per unit loss allocated to various loads, calculated by optimal as well as proportional tracing methods. It obviously shows that the majority of system loads near to k L∗ value by optimal tracing method that it implies equitable distribution of losses between system loads. This fact is not derived from the results of proportional tracing method. The table III shows the load payments for loss. In optimal tracing, the total load payments is 383.63 ($ hr ) , while in proportional tracing, it is 386.47 ($ hr ) . TABLE II. MW LOSS ALLOCATION TO VARIOUS LOADS Load Bus No. 2 3 4 5 6 8 9 10 11 12 13 Proportional Tracing 0.3205 3.0291 1.9758 0.2582 0.3806 1.2193 0.422 0.158 0.2695 0.6438 0.8392 Optimal Tracing 0.4991 2.826 2.39 0.2812 0.4144 1.2095 0.351 0.126 0.2257 0.4995 0.6407 Figure 2. The Modified IEEE 14-bus system TABLE I. OPF RESULTS FOR THE MODIFIED IEEE-14 BUS SYSTEM Bus No. PL (MW) LMP ($/MWh) 196.48 - 36.909 37.14 21.7 38.568 34.9 94.2 40.698 Angle (deg) PG (MW) 1 0 2 -4.056 3 -9.741 4 -8.811 - 47.8 40.523 5 -7.576 - 7.6 39.958 6 -13.134 0 11.2 39.993 7 -11.94 - - 40.554 8 -13.586 - 29.5 40.576 9 -13.795 - 9 40.713 10 -13.59 - 3.5 40.488 11 -13.995 - 6.1 40.65 12 -14.046 - 13.5 40.87 13 -14.82 - 14.9 41.586 Figure 3. Comparing per unit loss allocation by two methods TABLE III. LOAD PAYMENTS FOR ALLOCATED LOSS Load Bus No. Proportional Tracing Optimal Tracing 2 3 4 5 6 8 9 10 11 12 13 12.36 123.28 80.07 10.32 15.22 49.47 17.18 6.4 10.96 26.31 34.9 19.25 115.01 96.85 11.24 16.57 49.08 14.29 5.1 9.18 20.42 26.64 V. CONCLUSIONS In this paper, allocation of loss and its cost between system loads by proportional tracing and optimal tracing methods has been investigated. Since optimal tracing is a nonlinear programming problem, GAMS software has been used. Because of congestion in transmission lines, It is clear that a total load payment in optimal tracing is a little less than that in proportional tracing. Of course, 998 2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia tracing power flows in system was achieved through OPF results. The simulation results have shown that optimal tracing method is more competent than proportional tracing method from the viewpoint of equitable distribution of system losses between loads which is crucial for transparency in market operation. Also, it is proved that optimal tracing is a little fair rather than proportional tracing from allocation of loss cost viewpoint. Hence, it is recommended that the optimal tracing method be used for the aim of allocation of loss and its cost between loads. [3] [4] [5] [6] REFERENCES [1] [2] J. W. Bialek, “Tracing the flow of electricity,” IEE Proc.-Gener. Transm. 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