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A flow-tracing method for transmission networks
Energy 30 (2005) 1781–1792 www.elsevier.com/locate/energy
A flow-tracing method for transmission networks Milosˇ Pantosˇ*, Ferdinand Gubina Faculty of Electrical Engineering, University of Ljubljana, Trzˇasˇka 25, SI-1000 Ljubljana, Slovenia Received 3 February 2004
Abstract In this article, we present a new network flow-tracing method. This new method is designed to trace flow paths across a network from selected consumers to a specified producer. Flow tracing gives fundamental insight into the relations between consumers and producers. The new method is based on a matrix calculation that considers the transmission losses in a simple way. We tested the new flow-tracing method on an 18-bus electric-power test system. Since all the transmission systems have some common characteristics, knowledge from one technical field can be imposingly used in the other fields. The idea is to spread the knowledge and to find some basic principles of observed phenomena enabling to solve problems in a proper manner. In this way, a better insight into the system operation and control can be obtained, especially nowadays when deregulation and liberalization of transmission systems are introduced. q 2004 Elsevier Ltd. All rights reserved.
1. Introduction Throughout both the developed and developing worlds, the needs of the population are increasing in terms of the supply of utilities like gas, oil, hot water, and electricity. As a result of geography, weather conditions and the location of natural resources, high-consumption areas are usually supplied from distant production canters. When supplying these utilities, the main criteria are the preservation of quality, uninterrupted delivery and the minimizations of transmission losses combined with minimized costs. This applies to all forms of transmission networks, including pipelines, gas networks, heat networks and electric-power networks. At first sight, the question ‘which producer supplies a particular consumer?’ in a transmission network seems to be a very complicated one; however, with
* Corresponding author. Tel.: C386 1 4768 240; fax: C386 1 426 46 51. E-mail address: [email protected] (M. Pantosˇ). 0360-5442/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2004.11.009
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the requirement for open access to networks and the need to follow network flows and losses in order to remunerate producers in a transparent and fair way, this question needs to be answered. Transmission-loss management is becoming a significant part of the field of network operation and control, which are being influenced by deregulation and the liberalization of network use. In the energy market, for instance, the financial flow follows the contracts between consumers and producers, while the electrical energy flows to the consumer from the producers in compliance with Kirchoff’s laws. For instance, in the market environment, transmission-service pricing has an important role because of the need for a transparent and fair division of the network costs and transmission losses to the consumers according to their network usage. A few methods of electric-power flow tracing were already proposed. The Topological Generation Distribution Factor (TGDF) method [1–3] is based on matrix calculation of producers’ and consumers’ shares in the line flow. Since it considers transmission losses by introducing virtual nodes on every branch, the calculation becomes very complex and time consuming. The Domain Generation Distribution Factor (DGDF) method [4–6] is based on a generator’s domain, i.e. a set of buses supplied by the same set of generators. The obtained clusters are viewed as new nodes, connected together with tie branches. Such a simplification leads to a significant change in the generator’s contribution to some line flows due to a slight change in the system’s topology. The Nodal Generation Distribution Factor (NGDF) method [7] is based on search algorithm that needs additional algorithm in case of circular flows, thus it is time consuming method. To overcome these problems, the paper presents a modification of the TGDF calculation. In order to avoid the matrix expansion, it introduces four novelties: equivalent model of a line, different consideration of the transmission losses, decoupled extended distribution-matrix and finally Load Distribution Factor and Load Share Factor. Since electric networks consist of a number of producers and consumers that are connected by transmission lines, the features of the proposed method can be effectively demonstrated. Thus the proposed decoupled TGDF method was tested on an 18-bus electricpower test system. The power-flow tracing methods assume all the line- and nodal-power flows are available. Actually the line-power flows are calculated by power flow calculation using nodal-power flows. These input data are essential for further calculations. However, in the real-world applications, power flows are not always available, thus they can be provided (estimated) by a state estimator installed in the control centre. The usage of these methods in the real-world applications also requires recalculation of the factors for any small change in the system. The topology is rarely changing, which is not so for producers’ and consumers’ operation. Since their frequent changes require new calculations, a very fast method is needed that is able to obtain the solution between two snapshots of the system state. Moreover, the essence of the flow-tracing methods is the ability to capture the changes in the system when identifying new flow paths for new operating states. In other words, these methods are developed to deal with changeable systems where it is necessary to assess new operating states. It should be noted that recalculations of the shares due to generators’ and loads’ changes can be avoided by applying the Generalized Generation Distribution Factor (GGDF) method [8] that calculates influence distribution factors, i.e. the generators’ impacts on the line flows. However, all methods, including ones for sensitivity analyses, require recalculation of the shares when the system topology is changed. The sensitivity analysis is not addressed in the proposed paper.
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Since the various types of transmission system have some common characteristics, knowledge of one technical field can be used in other fields. The idea is to solve problems in a proper manner for a variety of transmission networks, since the deregulation and liberalization of transmission systems poses similar problems to all types of transmission network. The new flow-tracing approach can serve as a useful system tool and can provide a better insight into system operation and control in the market environment, and open access to a transmission network. For instance, this method is useful for congestion management and transmission-service pricing.
2. Decoupled TGDF power-flow-tracing method Bialek proposed the TGDF power-flow-tracing method that enables investigation of the power-flow shares in supplying the loads. Due to introduction of additional nodes in order to take into the account also the transmission losses, the matrix dimensions become quite large, which requires high computational effort. To avoid the matrix expansion, the paper presents the Decoupled Topological Generation Distribution Factor method (DTGDF) that introduces four novelties mentioned in the introduction. 2.1. The TGDF method in brief To present the new modified approach of the flow tracing, a short introduction of the TGDF method is required. Fig. 1 demonstrates the conditions at the node i, where Gi is the production, Di is the consumption. Ji is the set of nodes that directly supply the node i and Xi is the set of nodes that are directly supplied by the node i. The symbols Sij and Sji are the active or reactive power flows on the line i–j, directed from the node i to the node j. Since the particular line flow changes because of a transaction loss LijZSij-Sji, the values Sij and Sji are different. Hence, Sij is the value at the node i, and Sji is the value at the node j. For each node, the total nodal flow Si can be defined. It is equal to the sum of all the inflows (1) or outflows (2) from the node i: Si Z
X
Sis C Gi
i Z 1; 2; .; n;
(1)
s2Ji
Fig. 1. Conditions at node i.
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Fig. 2. Additional node on line i–j.
Si Z
X
Sij C Di
i Z 1; 2; .; n;
(2)
j2Xi
In the subsequent text, (2) identifies the flow paths from the producers to the consumers. For each line the TGDF approach introduces additional node, Fig. 2, with power consumption or production equals to the transmission loss Lij. The active power loss is always positive, thus only consumptions at additional nodes are concerned. Direction of the reactive power depends on generators and loads connected to nodes as well as on the line charging, which can be inductive or capacitive depending on its loading. According to the possibilities of reactive power flows direction, Fig. 3, additional nodes introduced by the TGDF method can also incorporate producers, Fig. 2. Introduction of additional nodes is justified since a power balance at terminal nodes of lines is preserved. Although the paper is focused on the flow tracing in different networks that requires only the concept of active-power-flow tracing in electric systems, the paper also concerns the reactive power research in order to present all the novelties of the improved TGDF method. Reformulation of (2) leads to a matrix notation (3)
A$S Z D;
where S is the unknown vector of nodal active or reactive power flows, D is the vector of nodal consumptions and A is the distribution-matrix with its (i,g)th element aig, which is equal to: 8 1 i Zg > < aig Z KSgi =Sg g 2Xi ; i sg : (4) > : 0 g ;Xi ; i sg Due to introduction of additional nodes in order to allow for transmission losses, the dimension of the distribution-matrix A is enlarged to (nCp)!(nCp) where n is the number of system nodes and p is
Fig. 3. Reactive power flow on the line i–l.
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Fig. 4. Equivalent model of the line i–l.
the number of additional nodes, i.e. the number of lines in a system. Also the matrices S and D are enlarged to the dimension (nCp). According to the proportional-sharing principle [1–3] and assuming that the inverse distributionmatrix TZAK1 exists, then the share of the power flow on the line g–i that is supplying the load at the node r can be developed as dgi;r Z Sgi
tgr : Sg
(5)
The exact derivation of dgi,r can be found in [1–3]. 2.2. Equivalent model of a line As mentioned before, transmission losses of reactive power depend on the line charging. To avoid the possibility of capacitive charging of the lines, the proposed modification of the TGDF method introduces an equivalent model of a line, Fig. 4, where LliZI2ilXil presents the reactive power transmission loss at reactance Xil. Assuming that the voltage of the shunt admittance Bsh/2,il is equal to the nearby nodal voltage the total reactive power production of the line i–l can be calculated as Bsh=2;il ðUi2 C Ui2 Þ. Nodal voltages can be obtained by power flow calculation, by measurements or can be approximated as 1 p.u. (per unit) If Lgi =Bsh=2;il ðUi2 C Ui2 Þ! 1 then the line i–l produces reactive power. The relation Lgi =Bsh=2;il ðUi2 C Ui2 Þ! 1 denotes the line i–l as a consumer of the reactive power. Proposed solution unites the reactive nodal power flows and reactive-power flows produced by shunt admittances, Fig. 4. Calculation of dgi,r follows the TGDF procedure captured in (3)–(5). The active and reactive powers are treated independently using the same methodology and formulas. 2.3. Decoupled line power flow Second modification of the TGDF method is different consideration of the transmission losses. Instead of introducing additional node in the middle of the line i–j proposed by the TGDF method, this line can be decoupled into two lossless transmission lines transmitting the power flow Sji and the transmission losses LijZSijKSji, separately, Fig. 5. Although in reality, the transmission losses do not have consumer characteristics they can be transmitted by additional lines and treated as loads at terminal nodes from the mathematical point of view. Since this solution does not affect the power balances at the terminal nodes, it is acceptable; as it is acceptable the solution proposed by the TGDF method that also presents the losses as additional loads. From the physical point of view, both approaches are justified,
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Fig. 5. Decoupled power flow of line i–j.
Fig. 6. Flows in a 2-bus test system.
thus the power-flow allocation or loss allocation are not impeded and are performed in fair manner. A simple 2-bus test system in Fig. 6, that can be adjusted as presented in Fig. 7, shows that the nodal balances are not affected by proposed modification. This means that the expanded network can be considered as lossless, which simplifies the mathematical treatment. In contrast to the Bialek’s introduction of additional nodes [1–3], the described novelty enables matrix decoupling presented in the subsequent text. 2.4. Matrix decoupling It should be stressed that the TGDF method and proposed modified approach introduce additional nodes in order to allow for transmission losses. Since the increased matrix-dimension requires extra time for computation, the presented modified approach suggests decoupling of the matrices in (3).
Fig. 7. Decoupled flows in a simple 2-bus test system.
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Eq. (3) can be rewritten to show the dimensions of the matrices AðnCpÞ!ðnCpÞ !SðnCpÞ!1 Z DðnCpÞ!1 ;
(6)
where n is the number of nodes in the system and p is the number of additional nodes. Matrices A, S and D can be further decoupled as " 0 # " 0 # " 0 # 00 Sn!1 Dn!1 An!n An!p ! 00 Z ; (7) 00 0p!n Ip!p ðnCpÞ!ðnCpÞ Sp!1 ðnCpÞ!1 Dp!1 ðnCpÞ!1 which can be rewritten as A 0 $S 0 C A 00 $S 00 Z D 0 ;
(8)
0$S 0 C I$S 00 Z D 00 :
(9)
The distribution submatrix A 0 describes relations among lossless-system nodes and the submatrix S 0 is comprised of their nodal flows. The submatrix A 00 includes elements related to the power flows on additional lines and the nodal flows of additional nodes that are equal to transmission losses and are also captured in the submatrix S 00 . Since the additional lines transmit only losses from the system nodes to the additional nodes, the distribution submatrix A 00 carries information about the system losses. The submatrix 0 encompasses zero elements since all power flows on additional lines are directed from the system nodes to the additional nodes. Since the additional nodes are not mutual connected by the transmission lines, the submatrix I is the uniform matrix. The submatrix D 0 presents the loads at the system nodes and D 00 includes the loadings at the additional nodes that are equal to the transmission losses, Fig. 5. Reformulation of (8) gives A 0 $S 0 Z D 0 K A 00 $S 00 ;
(10)
which can be rewritten as (3). The power-flow shares in supplying the loads can be calculated using (5). The proposed modification of the TGDF method requires less mathematical effort due to smaller matrix dimensions obtained by presented matrix decoupling. 2.5. Load distribution factor and load share factor As already mentioned, dgi,r is the share of the power flow on the line g–i that is supplying the load at the node r. It is also possible to reformulate the solution as LDFgi;r Z dgi;r
Dr D Z tgr r ; Sgi Sg
(11)
where tgr is the element of the inverse distribution-matrix T and LDFgi,r is the Load Distribution Factor, i.e. the share of the load at the node r in the power flow of the line g–i. The advantage of the new form is its independency of transmission losses, which is shown for the simple 2-bus test system in Fig. 6 that can be decoupled as shown in Fig. 7. The comparison of different factor forms is obtained by the results in Table 1. It can be noticed that the share factor in (5) depends on transmission losses, thus the values
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Table 1 Load shares in a simple 2-bus test system d12,2
d21,2
GDF12,2
GDF21,2
0.9091
1.0000
1.0000
1.0000
d12,2 and d21,2 are different. GDF remains constant along the line, since it does not depend on transmission losses. Therefore, GDFs can be successfully applied to transmission loss allocation. Since the modified TGDF approach treats the transmission losses as the loads, they also have their LDFs on the lines. Instead of exercising (7) the sum of the loss shares on the lines can be calculated as X X LDFgi;load ; (12) LDFgi;loss Z 1 K where SLDFgi,load is the sum of all load shares on the line g–i and SLDFgi,loss is the sum of all loss shares on the line g–i. In this way the actual use of the network by consumers is obtained. The new method also obtains the consumers’ participation in the production units. The Load Share Factor LSFb,k that represents the share of the kth consumer in the bth production unit can be calculated as P c2Xb LDFcb;k $Sbc LSFb;k Z ; (13) Gb where LDFcb,k represents the share of the kth consumer on the transmission line c–b, Sbc is the flow on the line c–b, Gb is the production at node b, and Xb is the set of nodes that are directly supplied by the node b. Like the LDFs, the LSFs are also positive, they take values between 0 and 1, and the sum of all the consumers’ shares in a specific generator node is equal to 1. 0% LSFb;k % 1; h X
LSFb;k Z 1;
(14)
(15)
kZ1
where h is the total number of consumers in a system. To calculate the Generator Distribution Factors GDFs, i.e. the shares of the generators in the linepower flows and the Generator Share Factors GSFs, i.e. the shares of the generators in the consumptions, (1) should be used in a similar development such as from (3) to (15). Exact derivation is outlined in the Appendix A.
3. Results The new decoupled TGDF power-flow-tracing method has been tested on an electric-power system that consists of 18 nodes, 22 transmission lines, five producers and seven consumers. Fig. 8 presents the LDFs of all the consumers in the system. The results show that, in general, nearby production units tend
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Fig. 8. 18-bus test system, Load Distribution Factors.
to supply consumers, and electric power does not actually flow to remote locations. We can conclude that a particular flow takes the path through the network governed by Kirchoff’s laws. The results show that each consumer is supplied by a specific set of producers in the locality, Fig. 9. We can see that some areas overlap others, which means that more than one producer supplies a certain consumer. In this case the LDFs can be used to fairly allocate the operation and maintenance costs, which constitute the transmission charges. For example, the total line costs on the line 2–4 can be divided among all the consumers that are supplied by this line. According to the results, only the consumers at nodes 4 and 5 are obliged to pay: 97.3 and 2.7% of the costs, respectively. The LDFs also represent the consumers’ shares in the transmission losses, thus their allocation among all the consumers can be obtained in a transparent and simple manner. For example, the transmission losses of the line 1–14 can be distributed among the consumers at nodes 1, 4, 5 and 13 according to their LDFs on that line. In this case, the consumers are obliged to pay 41.6, 16.1, 0.4 and 41.1% of costs of those losses, respectively. Information about the LSFs, Fig. 10, can be used to divide the production costs among all the consumers in the system. For instance, the production costs of the generator at the node 16 can be proportionally distributed among consumers 4, 5 and 16, which are supplied by this generator. They are obliged to pay 93.2, 2.6 and 4.2% of those costs, respectively.
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Fig. 9. 18-bus test system with consumers’ areas.
Fig. 10. 18-bus test system, consumers’ participations in the production units.
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The new method also offers the opportunity to calculate the production shares in the line flows and their participation in the consumers’ supply, as it is presented in Appendix A. In this way all the market players in the system can be identified and the transmission costs can be fairly distributed among all the market players according to their usage of the network.
4. Conclusion In this article, we present the modified TGDF power-flow tracing method. It was developed to cope with the new problems that are arising from the deregulation and liberalization of transmission networks. The new method, which is based on a matrix calculation, analytically obtains the flow paths from sources to sinks, i.e. from producers to consumers. In the electric-energy market it can be used for congestion management, transmission-service pricing and security analyses. For other transmission and distribution networks it can offer a basis for the fair distribution of transmission costs, including transmission losses, among consumers. The common characteristics of networks provide the new approach with an opportunity to become, with slight modifications, a new tool in network operation, planning and control in other types of networks that are also facing the challenges of free access. Furthermore, it can present researchers with new challenges to solve the problems posed by the market environment.
Acknowledgements The authors gratefully acknowledge the financial support from The Ministry of Education, Science and Sport of the Republic of Slovenia.
Appendix A The Generator Distribution Factors GDFs and the Generator Share Factors GSFs can be obtained by developing (1). Introduction of the equivalent model of a line, Fig. 4, and the line decoupling, Fig. 5, lead to the extended matrix notation BðnCpÞ!ðnCpÞ $SðnCpÞ!1 Z GðnCpÞ!1 ;
(A1)
where G is the vector of nodal productions and B is the distribution-matrix with its (i,g)th element big, which is equal to: 8 1 i Zg > < big Z KSig =Sg g 2Ji ; i sg : (A2) > : 0 g ;Ji ; i sg
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Matrices B, S and G can be further decoupled as " 0 # " 0 # " 0 # Sn!1 Bn!n 0n!p Gn!1 $ Z ; 00 00 0p!1 ðnCpÞ!1 Bp!n Ip!p ðnCpÞ!ðnCpÞ Sp!1 ðnCpÞ!1
(A3)
which can be rewritten as B 0 $S 0 C 0$S 00 Z G 0 ;
(A4)
B 00 $S 0 C I$S 00 Z 0:
(A5) 0
00
0
00
The distribution submatrices B and B are constructed similar as A and A according to (A2). The submatrix G 0 presents the productions at the system nodes. (A4) can be rewritten as (A1) and according to the proportional-sharing principle [1–3] and assuming that the inverse distribution-matrix WZBK1 exists, the GDFs and GSFs can be obtained as GDFgi;r Z wgr P GSFb;k Z
Gr ; Sg
c2Jb
GDFcb;k $Sbc ; Db
(A6)
(A7)
where wgr is the element of the inverse distribution-matrix W, Jb is the set of nodes that directly supply the node b, GDFgi,r is the Generator Distribution Factor, i.e. the share of the generator at the node r in the power flow of the line g–i and GSFb,k is the Generator Share Factor that represents the share of the kth generator in the bth consumption.
References [1] Bialek J. Topological generation and load distribution factors for supplement charge allocation in transmission open access. IEEE Trans Power Syst 1997;12(3):1185–93. [2] Bialek J. Tracing the flow of electricity. IEE Proc Gen, Transm Distrib 1996;143(4):313–20. [3] Bialek J. Allocation of transmission supplementary charge to real and reactive loads. IEEE Trans Power Syst 1998;13(3): 749–54. [4] Kirschen D, Allan R, Sˇtrbac G. Contributions of individual generators to loads flows. IEEE Trans Power Syst 1997;12(1): 52–60. [5] Kirschen D, Sˇtrbac G. Tracing active and reactive power between generators and loads using real and imaginary currents. IEEE Trans Power Syst 1999;14(4):1312–9. [6] Kirschen D, Ahmed S, Sˇtrbac G. Allocating transmission system usage on the basis of traceable contributions of generators and loads to flows. IEEE Trans Power Syst 1998;13(2):527–34. [7] Grgicˇ D, Gubina F, Banicˇ I. A method for determining the generators’ share in a consumer load. IEEE Trans Power Syst 2000;15(4):1376–81. [8] Ng WY. Generalized generation distribution factors for power system security evaluations. IEEE Trans Power Apparatus Syst 1981;PAS-100(3):1001–5.
A flow-tracing method for transmission networks Milosˇ Pantosˇ*, Ferdinand Gubina Faculty of Electrical Engineering, University of Ljubljana, Trzˇasˇka 25, SI-1000 Ljubljana, Slovenia Received 3 February 2004
Abstract In this article, we present a new network flow-tracing method. This new method is designed to trace flow paths across a network from selected consumers to a specified producer. Flow tracing gives fundamental insight into the relations between consumers and producers. The new method is based on a matrix calculation that considers the transmission losses in a simple way. We tested the new flow-tracing method on an 18-bus electric-power test system. Since all the transmission systems have some common characteristics, knowledge from one technical field can be imposingly used in the other fields. The idea is to spread the knowledge and to find some basic principles of observed phenomena enabling to solve problems in a proper manner. In this way, a better insight into the system operation and control can be obtained, especially nowadays when deregulation and liberalization of transmission systems are introduced. q 2004 Elsevier Ltd. All rights reserved.
1. Introduction Throughout both the developed and developing worlds, the needs of the population are increasing in terms of the supply of utilities like gas, oil, hot water, and electricity. As a result of geography, weather conditions and the location of natural resources, high-consumption areas are usually supplied from distant production canters. When supplying these utilities, the main criteria are the preservation of quality, uninterrupted delivery and the minimizations of transmission losses combined with minimized costs. This applies to all forms of transmission networks, including pipelines, gas networks, heat networks and electric-power networks. At first sight, the question ‘which producer supplies a particular consumer?’ in a transmission network seems to be a very complicated one; however, with
* Corresponding author. Tel.: C386 1 4768 240; fax: C386 1 426 46 51. E-mail address: [email protected] (M. Pantosˇ). 0360-5442/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2004.11.009
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the requirement for open access to networks and the need to follow network flows and losses in order to remunerate producers in a transparent and fair way, this question needs to be answered. Transmission-loss management is becoming a significant part of the field of network operation and control, which are being influenced by deregulation and the liberalization of network use. In the energy market, for instance, the financial flow follows the contracts between consumers and producers, while the electrical energy flows to the consumer from the producers in compliance with Kirchoff’s laws. For instance, in the market environment, transmission-service pricing has an important role because of the need for a transparent and fair division of the network costs and transmission losses to the consumers according to their network usage. A few methods of electric-power flow tracing were already proposed. The Topological Generation Distribution Factor (TGDF) method [1–3] is based on matrix calculation of producers’ and consumers’ shares in the line flow. Since it considers transmission losses by introducing virtual nodes on every branch, the calculation becomes very complex and time consuming. The Domain Generation Distribution Factor (DGDF) method [4–6] is based on a generator’s domain, i.e. a set of buses supplied by the same set of generators. The obtained clusters are viewed as new nodes, connected together with tie branches. Such a simplification leads to a significant change in the generator’s contribution to some line flows due to a slight change in the system’s topology. The Nodal Generation Distribution Factor (NGDF) method [7] is based on search algorithm that needs additional algorithm in case of circular flows, thus it is time consuming method. To overcome these problems, the paper presents a modification of the TGDF calculation. In order to avoid the matrix expansion, it introduces four novelties: equivalent model of a line, different consideration of the transmission losses, decoupled extended distribution-matrix and finally Load Distribution Factor and Load Share Factor. Since electric networks consist of a number of producers and consumers that are connected by transmission lines, the features of the proposed method can be effectively demonstrated. Thus the proposed decoupled TGDF method was tested on an 18-bus electricpower test system. The power-flow tracing methods assume all the line- and nodal-power flows are available. Actually the line-power flows are calculated by power flow calculation using nodal-power flows. These input data are essential for further calculations. However, in the real-world applications, power flows are not always available, thus they can be provided (estimated) by a state estimator installed in the control centre. The usage of these methods in the real-world applications also requires recalculation of the factors for any small change in the system. The topology is rarely changing, which is not so for producers’ and consumers’ operation. Since their frequent changes require new calculations, a very fast method is needed that is able to obtain the solution between two snapshots of the system state. Moreover, the essence of the flow-tracing methods is the ability to capture the changes in the system when identifying new flow paths for new operating states. In other words, these methods are developed to deal with changeable systems where it is necessary to assess new operating states. It should be noted that recalculations of the shares due to generators’ and loads’ changes can be avoided by applying the Generalized Generation Distribution Factor (GGDF) method [8] that calculates influence distribution factors, i.e. the generators’ impacts on the line flows. However, all methods, including ones for sensitivity analyses, require recalculation of the shares when the system topology is changed. The sensitivity analysis is not addressed in the proposed paper.
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Since the various types of transmission system have some common characteristics, knowledge of one technical field can be used in other fields. The idea is to solve problems in a proper manner for a variety of transmission networks, since the deregulation and liberalization of transmission systems poses similar problems to all types of transmission network. The new flow-tracing approach can serve as a useful system tool and can provide a better insight into system operation and control in the market environment, and open access to a transmission network. For instance, this method is useful for congestion management and transmission-service pricing.
2. Decoupled TGDF power-flow-tracing method Bialek proposed the TGDF power-flow-tracing method that enables investigation of the power-flow shares in supplying the loads. Due to introduction of additional nodes in order to take into the account also the transmission losses, the matrix dimensions become quite large, which requires high computational effort. To avoid the matrix expansion, the paper presents the Decoupled Topological Generation Distribution Factor method (DTGDF) that introduces four novelties mentioned in the introduction. 2.1. The TGDF method in brief To present the new modified approach of the flow tracing, a short introduction of the TGDF method is required. Fig. 1 demonstrates the conditions at the node i, where Gi is the production, Di is the consumption. Ji is the set of nodes that directly supply the node i and Xi is the set of nodes that are directly supplied by the node i. The symbols Sij and Sji are the active or reactive power flows on the line i–j, directed from the node i to the node j. Since the particular line flow changes because of a transaction loss LijZSij-Sji, the values Sij and Sji are different. Hence, Sij is the value at the node i, and Sji is the value at the node j. For each node, the total nodal flow Si can be defined. It is equal to the sum of all the inflows (1) or outflows (2) from the node i: Si Z
X
Sis C Gi
i Z 1; 2; .; n;
(1)
s2Ji
Fig. 1. Conditions at node i.
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Fig. 2. Additional node on line i–j.
Si Z
X
Sij C Di
i Z 1; 2; .; n;
(2)
j2Xi
In the subsequent text, (2) identifies the flow paths from the producers to the consumers. For each line the TGDF approach introduces additional node, Fig. 2, with power consumption or production equals to the transmission loss Lij. The active power loss is always positive, thus only consumptions at additional nodes are concerned. Direction of the reactive power depends on generators and loads connected to nodes as well as on the line charging, which can be inductive or capacitive depending on its loading. According to the possibilities of reactive power flows direction, Fig. 3, additional nodes introduced by the TGDF method can also incorporate producers, Fig. 2. Introduction of additional nodes is justified since a power balance at terminal nodes of lines is preserved. Although the paper is focused on the flow tracing in different networks that requires only the concept of active-power-flow tracing in electric systems, the paper also concerns the reactive power research in order to present all the novelties of the improved TGDF method. Reformulation of (2) leads to a matrix notation (3)
A$S Z D;
where S is the unknown vector of nodal active or reactive power flows, D is the vector of nodal consumptions and A is the distribution-matrix with its (i,g)th element aig, which is equal to: 8 1 i Zg > < aig Z KSgi =Sg g 2Xi ; i sg : (4) > : 0 g ;Xi ; i sg Due to introduction of additional nodes in order to allow for transmission losses, the dimension of the distribution-matrix A is enlarged to (nCp)!(nCp) where n is the number of system nodes and p is
Fig. 3. Reactive power flow on the line i–l.
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Fig. 4. Equivalent model of the line i–l.
the number of additional nodes, i.e. the number of lines in a system. Also the matrices S and D are enlarged to the dimension (nCp). According to the proportional-sharing principle [1–3] and assuming that the inverse distributionmatrix TZAK1 exists, then the share of the power flow on the line g–i that is supplying the load at the node r can be developed as dgi;r Z Sgi
tgr : Sg
(5)
The exact derivation of dgi,r can be found in [1–3]. 2.2. Equivalent model of a line As mentioned before, transmission losses of reactive power depend on the line charging. To avoid the possibility of capacitive charging of the lines, the proposed modification of the TGDF method introduces an equivalent model of a line, Fig. 4, where LliZI2ilXil presents the reactive power transmission loss at reactance Xil. Assuming that the voltage of the shunt admittance Bsh/2,il is equal to the nearby nodal voltage the total reactive power production of the line i–l can be calculated as Bsh=2;il ðUi2 C Ui2 Þ. Nodal voltages can be obtained by power flow calculation, by measurements or can be approximated as 1 p.u. (per unit) If Lgi =Bsh=2;il ðUi2 C Ui2 Þ! 1 then the line i–l produces reactive power. The relation Lgi =Bsh=2;il ðUi2 C Ui2 Þ! 1 denotes the line i–l as a consumer of the reactive power. Proposed solution unites the reactive nodal power flows and reactive-power flows produced by shunt admittances, Fig. 4. Calculation of dgi,r follows the TGDF procedure captured in (3)–(5). The active and reactive powers are treated independently using the same methodology and formulas. 2.3. Decoupled line power flow Second modification of the TGDF method is different consideration of the transmission losses. Instead of introducing additional node in the middle of the line i–j proposed by the TGDF method, this line can be decoupled into two lossless transmission lines transmitting the power flow Sji and the transmission losses LijZSijKSji, separately, Fig. 5. Although in reality, the transmission losses do not have consumer characteristics they can be transmitted by additional lines and treated as loads at terminal nodes from the mathematical point of view. Since this solution does not affect the power balances at the terminal nodes, it is acceptable; as it is acceptable the solution proposed by the TGDF method that also presents the losses as additional loads. From the physical point of view, both approaches are justified,
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Fig. 5. Decoupled power flow of line i–j.
Fig. 6. Flows in a 2-bus test system.
thus the power-flow allocation or loss allocation are not impeded and are performed in fair manner. A simple 2-bus test system in Fig. 6, that can be adjusted as presented in Fig. 7, shows that the nodal balances are not affected by proposed modification. This means that the expanded network can be considered as lossless, which simplifies the mathematical treatment. In contrast to the Bialek’s introduction of additional nodes [1–3], the described novelty enables matrix decoupling presented in the subsequent text. 2.4. Matrix decoupling It should be stressed that the TGDF method and proposed modified approach introduce additional nodes in order to allow for transmission losses. Since the increased matrix-dimension requires extra time for computation, the presented modified approach suggests decoupling of the matrices in (3).
Fig. 7. Decoupled flows in a simple 2-bus test system.
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Eq. (3) can be rewritten to show the dimensions of the matrices AðnCpÞ!ðnCpÞ !SðnCpÞ!1 Z DðnCpÞ!1 ;
(6)
where n is the number of nodes in the system and p is the number of additional nodes. Matrices A, S and D can be further decoupled as " 0 # " 0 # " 0 # 00 Sn!1 Dn!1 An!n An!p ! 00 Z ; (7) 00 0p!n Ip!p ðnCpÞ!ðnCpÞ Sp!1 ðnCpÞ!1 Dp!1 ðnCpÞ!1 which can be rewritten as A 0 $S 0 C A 00 $S 00 Z D 0 ;
(8)
0$S 0 C I$S 00 Z D 00 :
(9)
The distribution submatrix A 0 describes relations among lossless-system nodes and the submatrix S 0 is comprised of their nodal flows. The submatrix A 00 includes elements related to the power flows on additional lines and the nodal flows of additional nodes that are equal to transmission losses and are also captured in the submatrix S 00 . Since the additional lines transmit only losses from the system nodes to the additional nodes, the distribution submatrix A 00 carries information about the system losses. The submatrix 0 encompasses zero elements since all power flows on additional lines are directed from the system nodes to the additional nodes. Since the additional nodes are not mutual connected by the transmission lines, the submatrix I is the uniform matrix. The submatrix D 0 presents the loads at the system nodes and D 00 includes the loadings at the additional nodes that are equal to the transmission losses, Fig. 5. Reformulation of (8) gives A 0 $S 0 Z D 0 K A 00 $S 00 ;
(10)
which can be rewritten as (3). The power-flow shares in supplying the loads can be calculated using (5). The proposed modification of the TGDF method requires less mathematical effort due to smaller matrix dimensions obtained by presented matrix decoupling. 2.5. Load distribution factor and load share factor As already mentioned, dgi,r is the share of the power flow on the line g–i that is supplying the load at the node r. It is also possible to reformulate the solution as LDFgi;r Z dgi;r
Dr D Z tgr r ; Sgi Sg
(11)
where tgr is the element of the inverse distribution-matrix T and LDFgi,r is the Load Distribution Factor, i.e. the share of the load at the node r in the power flow of the line g–i. The advantage of the new form is its independency of transmission losses, which is shown for the simple 2-bus test system in Fig. 6 that can be decoupled as shown in Fig. 7. The comparison of different factor forms is obtained by the results in Table 1. It can be noticed that the share factor in (5) depends on transmission losses, thus the values
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Table 1 Load shares in a simple 2-bus test system d12,2
d21,2
GDF12,2
GDF21,2
0.9091
1.0000
1.0000
1.0000
d12,2 and d21,2 are different. GDF remains constant along the line, since it does not depend on transmission losses. Therefore, GDFs can be successfully applied to transmission loss allocation. Since the modified TGDF approach treats the transmission losses as the loads, they also have their LDFs on the lines. Instead of exercising (7) the sum of the loss shares on the lines can be calculated as X X LDFgi;load ; (12) LDFgi;loss Z 1 K where SLDFgi,load is the sum of all load shares on the line g–i and SLDFgi,loss is the sum of all loss shares on the line g–i. In this way the actual use of the network by consumers is obtained. The new method also obtains the consumers’ participation in the production units. The Load Share Factor LSFb,k that represents the share of the kth consumer in the bth production unit can be calculated as P c2Xb LDFcb;k $Sbc LSFb;k Z ; (13) Gb where LDFcb,k represents the share of the kth consumer on the transmission line c–b, Sbc is the flow on the line c–b, Gb is the production at node b, and Xb is the set of nodes that are directly supplied by the node b. Like the LDFs, the LSFs are also positive, they take values between 0 and 1, and the sum of all the consumers’ shares in a specific generator node is equal to 1. 0% LSFb;k % 1; h X
LSFb;k Z 1;
(14)
(15)
kZ1
where h is the total number of consumers in a system. To calculate the Generator Distribution Factors GDFs, i.e. the shares of the generators in the linepower flows and the Generator Share Factors GSFs, i.e. the shares of the generators in the consumptions, (1) should be used in a similar development such as from (3) to (15). Exact derivation is outlined in the Appendix A.
3. Results The new decoupled TGDF power-flow-tracing method has been tested on an electric-power system that consists of 18 nodes, 22 transmission lines, five producers and seven consumers. Fig. 8 presents the LDFs of all the consumers in the system. The results show that, in general, nearby production units tend
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Fig. 8. 18-bus test system, Load Distribution Factors.
to supply consumers, and electric power does not actually flow to remote locations. We can conclude that a particular flow takes the path through the network governed by Kirchoff’s laws. The results show that each consumer is supplied by a specific set of producers in the locality, Fig. 9. We can see that some areas overlap others, which means that more than one producer supplies a certain consumer. In this case the LDFs can be used to fairly allocate the operation and maintenance costs, which constitute the transmission charges. For example, the total line costs on the line 2–4 can be divided among all the consumers that are supplied by this line. According to the results, only the consumers at nodes 4 and 5 are obliged to pay: 97.3 and 2.7% of the costs, respectively. The LDFs also represent the consumers’ shares in the transmission losses, thus their allocation among all the consumers can be obtained in a transparent and simple manner. For example, the transmission losses of the line 1–14 can be distributed among the consumers at nodes 1, 4, 5 and 13 according to their LDFs on that line. In this case, the consumers are obliged to pay 41.6, 16.1, 0.4 and 41.1% of costs of those losses, respectively. Information about the LSFs, Fig. 10, can be used to divide the production costs among all the consumers in the system. For instance, the production costs of the generator at the node 16 can be proportionally distributed among consumers 4, 5 and 16, which are supplied by this generator. They are obliged to pay 93.2, 2.6 and 4.2% of those costs, respectively.
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Fig. 9. 18-bus test system with consumers’ areas.
Fig. 10. 18-bus test system, consumers’ participations in the production units.
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The new method also offers the opportunity to calculate the production shares in the line flows and their participation in the consumers’ supply, as it is presented in Appendix A. In this way all the market players in the system can be identified and the transmission costs can be fairly distributed among all the market players according to their usage of the network.
4. Conclusion In this article, we present the modified TGDF power-flow tracing method. It was developed to cope with the new problems that are arising from the deregulation and liberalization of transmission networks. The new method, which is based on a matrix calculation, analytically obtains the flow paths from sources to sinks, i.e. from producers to consumers. In the electric-energy market it can be used for congestion management, transmission-service pricing and security analyses. For other transmission and distribution networks it can offer a basis for the fair distribution of transmission costs, including transmission losses, among consumers. The common characteristics of networks provide the new approach with an opportunity to become, with slight modifications, a new tool in network operation, planning and control in other types of networks that are also facing the challenges of free access. Furthermore, it can present researchers with new challenges to solve the problems posed by the market environment.
Acknowledgements The authors gratefully acknowledge the financial support from The Ministry of Education, Science and Sport of the Republic of Slovenia.
Appendix A The Generator Distribution Factors GDFs and the Generator Share Factors GSFs can be obtained by developing (1). Introduction of the equivalent model of a line, Fig. 4, and the line decoupling, Fig. 5, lead to the extended matrix notation BðnCpÞ!ðnCpÞ $SðnCpÞ!1 Z GðnCpÞ!1 ;
(A1)
where G is the vector of nodal productions and B is the distribution-matrix with its (i,g)th element big, which is equal to: 8 1 i Zg > < big Z KSig =Sg g 2Ji ; i sg : (A2) > : 0 g ;Ji ; i sg
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Matrices B, S and G can be further decoupled as " 0 # " 0 # " 0 # Sn!1 Bn!n 0n!p Gn!1 $ Z ; 00 00 0p!1 ðnCpÞ!1 Bp!n Ip!p ðnCpÞ!ðnCpÞ Sp!1 ðnCpÞ!1
(A3)
which can be rewritten as B 0 $S 0 C 0$S 00 Z G 0 ;
(A4)
B 00 $S 0 C I$S 00 Z 0:
(A5) 0
00
0
00
The distribution submatrices B and B are constructed similar as A and A according to (A2). The submatrix G 0 presents the productions at the system nodes. (A4) can be rewritten as (A1) and according to the proportional-sharing principle [1–3] and assuming that the inverse distribution-matrix WZBK1 exists, the GDFs and GSFs can be obtained as GDFgi;r Z wgr P GSFb;k Z
Gr ; Sg
c2Jb
GDFcb;k $Sbc ; Db
(A6)
(A7)
where wgr is the element of the inverse distribution-matrix W, Jb is the set of nodes that directly supply the node b, GDFgi,r is the Generator Distribution Factor, i.e. the share of the generator at the node r in the power flow of the line g–i and GSFb,k is the Generator Share Factor that represents the share of the kth generator in the bth consumption.
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