THE introduction of competition in the electricity sector

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1 1170 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 3, AUGUST 2006 Branch Current Decomposition Method for Loss Allocation in Radial Distribution Systems With Distributed Generation Enrico Carpaneto, Member, IEEE, Gianfranco Chicco, Member, IEEE, and Jean Sumaili Akilimali Abstract The allocation of the system losses to suppliers and consumers is a challenging issue for the restructured electricity business. Meaningful loss allocation techniques have to be adopted to set up appropriate economic penalties or rewards. The allocation factors should depend on size, location, and time evolution of the resources connected to the system. In the presence of distributed generation, the variety of the power flows in distribution systems calls for adopting mechanisms able to discriminate among the contributions that increase or reduce the total losses. Some loss allocation techniques already developed in the literature have shown consistent behavior. However, their application requires computing a set of additional quantities with respect to those provided by the distribution system power flow solved with the backward/forward sweep approach. This paper presents a new circuit-based loss allocation technique, based on the decomposition of the branch currents, specifically developed for radial distribution systems with distributed generation. The proposed technique is simple and effective and is only based on the information provided by the network data and by the power flow solution. Examples of application are shown to confirm its effectiveness. Index Terms Branch current decomposition, distributed generation (DG), distribution systems, losses, loss allocation. I. INTRODUCTION THE introduction of competition in the electricity sector has changed the interactions between suppliers and consumers, leading to increased attention to the economic aspects in the electricity supply management. In addition, significant energy efficiency improvement and cost reductions for various technologies have made the adoption of small generation units economically competitive. Further incentives toward increasing adoption of renewable sources have been included in various regulations for driving the energy systems evolution toward extended adoption of local generation sources. All these aspects result in increasing the penetration of the distributed generation (DG) in distribution systems. For what concerns the power losses in the electricity networks, the combined effect of competition and DG expansion is making the handling of the corresponding costs much more challenging than in the past, raising questions about the responsibility and allocation of the losses to suppliers and consumers. In fact, in a vertically integrated power structure, the cost of losses was included in the overall electricity production costs. Manuscript received September 15, 2005; revised December 21, Paper no. TPWRS The authors are with the Dipartimento di Ingegneria Elettrica, Politecnico di Torino, I Torino, Italy ( enrico.carpaneto@polito.it; gianfranco. chicco@polito.it; jean.sumailiakilimali@polito.it). Digital Object Identifier /TPWRS With the introduction of competitive markets, generation, transmission, and distribution of electricity have been separated into autonomous businesses, and the costs of losses have to be specifically allocated among the entities participating with the electricity provision. Conceptually, loss allocation is a difficult task, because losses in the distribution system branches are nonlinear functions of generations and loads. It is impossible to calculate the exact amount of losses in advance, without running a power flow. At the same time, even after computing the power flow solution, there is a strong interdependence among all the users, expressed by the presence of cross-terms due to the fact that losses are a nearly quadratic function of the power flows. Hence, allocating the losses to the market participants cannot be carried out in a straightforward way. Some principles for effective loss allocation [1] are recalled here. A loss allocation technique should be: easy to understand and based on real data of the network; carefully designed to avoid discrimination between users; able to recover the total amount of the losses; consistent with the rules of competitive electricity markets; economically efficient, avoiding cross-subsidization between users; able to send out economic signals aimed at increasing the efficiency of the network; able to provide correct signals concerning the size and location of loads and DG sources in the network; applicable to different situations, e.g., following the time evolution of the generation and load patterns. The presence of local generators has changed the distribution systems from passive networks, with unidirectional power flows in the branches, into active networks with bidirectional power flows. The allocation of losses has to consider the nature of each entity, which may be either a consumer or a generator, at different time instants. Different methods for loss allocation have been reported in the literature. Early formulations of uniform or demand-squared-based loss allocation [2] are only related to the demand side and as such are not suitable for being used in the presence of DG. In general, a first distinction can be made between loss allocation methods dedicated to transmission and to distribution systems. The difference between these two classes of methods basically lies in the role given to the slack node. In transmission systems, the generator located in the slack node compensates for all the losses and is explicitly considered in the mechanism of loss allocation /$ IEEE

2 CARPANETO et al.: BRANCH CURRENT DECOMPOSITION METHOD FOR LOSS ALLOCATION 1171 In radial distribution systems, the location of the slack node at the root node of the distribution tree is naturally unique, and the slack usually represents the connection to a higher voltage network. With the evolution of DG, the simultaneous presence of several generators in the distribution systems could raise the question of the possibility of using the same loss allocation mechanisms defined for the transmission systems. More specifically, it should be clarified whether or not the slack node has to be considered as a generator participating in the loss allocation. This aspect is fundamental, since the slack node delivers a significant amount of the generated power. A possible framework for distribution systems with DG includes only local generators and loads as participants to the loss allocation. On one hand, suitably located DG units could reduce the amount of losses. On the other hand, the possible benefit in terms of loss reduction cannot be directly associated with any individual DG unit, since it depends on the system structure and on the location and power of every generator and load. The critical nature of the loss allocation problem is made evident by the fact that early formulated loss allocation mechanisms, even adopted at the regulatory level, have been found to be inconsistent. An example is the substitution method [3], in which the impact of a generator (or load) unit is evaluated by computing the difference between the total losses occurring with and without the unit. The inconsistency of such an approach has been clearly shown in [4, Sec. 8.4]. This paper provides various contributions to the analysis and improvement of the loss allocation mechanisms, from the viewpoint of their applicability to distribution systems with DG. The common characteristics and the limits of application of the existing allocation methods are summarized and discussed. The critical aspects of some loss allocation mechanisms are highlighted, showing that it is possible to formulate mechanisms able to cover the total system losses but whose behavior leads to evident paradoxes. This paper identifies the methods able to correctly represent the generation and load characteristics and to provide consistent technical and economic signals. A key aspect of this paper is the proposal of an original, simple, and efficient allocation method, specifically formulated for radial distribution systems with DG and requiring only data from the network structure and the power flow solution, without the need for computing additional quantities. Section II contains an overview of the loss allocation methods. Section III provides a comparison among the loss allocation techniques. Section IV presents the proposed method. Section V shows how the concepts used for developing the proposed method can be applied to correct one of the previously defined circuit-based methods by eliminating the paradox that can affect its allocation results. Numerical results are provided throughout this paper for assisting the discussion and highlighting the features of the various loss allocation techniques. The last section contains the concluding remarks. II. OVERVIEW OF THE LOSS ALLOCATION TECHNIQUES A comprehensive classification of the loss allocation mechanisms referred to transmission systems has been provided in [5], where different types of procedures are identified: pro-rata; marginal; unsubsidized marginal; proportional sharing; circuit-based; transaction-based. These classification concepts can be fully extended to distribution systems. However, some methods based on a priori assumptions (i.e., pro-rata, unsubsidized marginal, and proportional sharing) are considered to be too approximated for the purpose of this paper. As an example, the explicit assumption of allocating 50% of the losses to the loads and 50% to the generators is not suitable to be applied to the case of distribution systems, with or without DG, when the slack node is not considered among the loss allocation subjects. The method presented in [1], formulated for distribution systems with DG, contains a number of predefined assumptions for allocating some partial contributions to the total losses and is not included in the discussion as well. In addition, addressing transaction-based loss modeling as in [6] [8] is outside the scope of this paper. In this section, the analysis is focused on two types of existing methods: a) circuit-based methods, such as the Z-bus loss allocation (ZLA) [9] and the succinct method (SMLA) [10]; b) marginal or derivative-based loss allocation mechanisms, such as the marginal loss coefficients (MLC) [11] and the direct loss coefficients (DLC) [11]. The characteristics of these methods are summarized and compared, showing the existence of a possible paradox for the SMLA method. A way for eliminating this paradox has been identified and will be explained in Section V. A. Circuit-Based Methods The circuit-based methods are defined on the basis of the system structure, expressed by the bus impedance matrix and of the results of a power flow calculation. For a transmission or distribution system with nodes (slack at node 0) and branches, the components of the matrix are denoted as, for and. The power flow solution provides the complex node voltage, the net input complex power, and the net input current for any node. The following loss allocation methods are considered here: 1) ZLA (Z-bus loss allocation [9]), where the losses allocated to each node are expressed as where the asterisk denotes conjugation; 2) SMLA [10]), developed for allocating only the variable loss due to the series impedance branch, whereas the almost invariable loss due to the shunt admittance branch must be allocated in average terms among all users. Let us consider and as the sending and ending nodes of branch, and let us assume to be the impedance of branch (1)

3 1172 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 3, AUGUST 2006 Fig. 1. Two-node system., for. The losses allocated to each node are (2) This method presents a paradox according to which it is unable to provide a meaningful loss allocation under specific circumstances concerning the reactive power loads. Here the paradox is illustrated and discussed, with the help of the two-node system shown in Fig. 1. For this system, supplying a constant power load with, the expressions of the losses allocated to node 1 are Fig. 2. Loss allocation in the two-node system. (3) Practically, the loss allocation for the SMLA is based on the scalar product between the load current and the voltage variation. According to the implementation of the method, the coefficients that multiply and in (3) could be seen as active and reactive loss coefficients. Yet, the active power coefficient also depends on reactive power, and vice versa. However, a further step could be done by recognizing that the term and its opposite could be simplified. 1 This step is not performed in the original method. Fig. 2 shows an example for the case and p.u. ( ). Taking the characteristic angle of the line as a parameter, if, the reactive loss coefficient changes its sign. In these conditions, if the current load is assumed as the sum of two loads with the same active power, the SMLA penalizes the load with the lower value of reactive power, with an evident paradox. The two cases are presented in Fig. 3. In both cases, the method allocates exactly the total losses. The allocated losses are indicated by the projections of the load current phasors onto the voltage drop phasor. Let the current phasors and correspond to a purely resistive load (A) with active power and to a resistive-inductive load (B) with active power and reactive power, respectively. From Fig. 3, the losses allocated to the two loads are proportional to the segments OA and OB. The phasor diagram of Fig. 3(a) refers to the case and leads to the paradox of allocating more losses to the load A absorbing a lower current. Note that 1 As it will be shown in Section V, this missing step is exactly the cause of the paradox occurring in the SMLA. Fig. 3. Phasor interpretation of the reactive power paradox. in particular cases, like highly inductive load B connected in parallel to a resistive load A at the same node, the global loss allocation coefficient for load B could be negative. The paradox does not occur in the case, indicated in the phasor diagram of Fig. 3(b). A way for eliminating the paradox and building a consistent version of the SMLA, based on the same concepts indicated in Section IV for the development of the proposed method, is shown in Section V. B. Derivative-Based Methods The derivative-based methods use information based on the power flow Jacobian to build a set of loss allocation factors able to follow the trend of variation of the losses when the generated or absorbed power varies. The net input (active and reactive) node power vector contains all the net active powers at the PQ and PV nodes, as well as all the net reactive powers at the PQ nodes. On the reactive power side, any node participating to the voltage control (within the generator reactive power limits) is not included in the loss allocation. Two basic methods are as follows.

4 CARPANETO et al.: BRANCH CURRENT DECOMPOSITION METHOD FOR LOSS ALLOCATION ) MLC [11]), for which the total losses are decomposed in the form where is the column vector containing the loss allocation factors corresponding to each component of the net power vector, and the superscript indicates the vector transposition. The vector is computed starting from the power flow solution, by first solving for the auxiliary vector the linear system where is the vector containing the power flow state variables (voltage angles of all the nodes excluding the slack and voltage magnitudes at the PQ nodes), is the power flow Jacobian matrix containing the derivatives of the power flow equations with respect to the power flow state variables, and the column vector contains the derivatives of the total losses with respect to the power flow state variables. Since the product does not match the total losses, a reconciliation process is needed to compute the loss allocation vector as In a real system, typically, as indicated in [11]. 2) DLC, based on the expression of the total losses by using the Taylor series expansion around the initial operating point of the power flow computation [11]. The total losses are decomposed in the form where is the vector containing the loss allocation factors corresponding to each component of the net power vector. The vector is computed starting from the power flow solution, by first solving for the auxiliary vector the linear system where is the average Jacobian matrix, computed by averaging the terms of the Jacobian matrix at the current operating point with the ones of the Jacobian matrix evaluated at no-load, and is the power flow Hessian matrix at the current operating point. If the losses can be expressed in function of the load powers as a quadratic form with the second-order components only, the product matches the total losses, and no reconciliation is needed. Otherwise, the method is able to reproduce the total losses only approximately. (4) (5) (6) (7) (8) Fig. 4. Single-feeder test circuit. III. COMPARISONS AMONG THE LOSS ALLOCATION TECHNIQUES The loss allocation techniques introduced above are analyzed in this section in the light of their applicability to the distribution systems. A. Network Representation and Matrix Singularities For the application of the ZLA method, it is important to know the bus impedance matrix, which is defined only if the admittance matrix is non-singular. For distribution systems composed of overhead lines only, the shunt admittance can be negligible, so that the bus admittance matrix becomes singular, and the bus impedance matrix is not defined. The ZLA method cannot be applied in this case. In addition, it may be very sensitive to the variation of its parameters in systems where the shunt capacitances are relatively low. The SMLA method is able to work also in the absence of shunt parameters, since it does not contain directly the bus impedance matrix coefficients, but the differences among its coefficients, corresponding to indeterminate forms that can be always defined as limit cases. B. Slack Node Impact on the Loss Allocation Let us consider a simple real system with one generation (slack) node and a single feeder (cable line) supplying 12 identical and equally spaced loads. The base voltage is 22 kv, and the base power is 1 MVA. The feeder portion between two nodes is characterized by a resistance reactance and susceptance p.u. Each nodal load has a PQ model with p.u. and p.u. (see Fig. 4). Fig. 5 shows that the ZLA and the SMLA method allocate a significant part of the losses to the slack node. Fig. 6 highlights that the portion of losses allocated to the slack node increases with the number of loads supplied by the feeder. One of the key differences between the loss allocation methods used for transmission (SMLA, ZLA) and distribution systems (MLC, DLC) is based on the treatment of the slack node. The slack node current is the sum of the load currents (shunt parameters are treated as fictitious loads added to the bus loads) and can then be expressed as the sum of contributions of the other node currents, as After dividing the losses allocated to the slack node among the others nodes according to their respective currents, the ZLA and SMLA method can be used for loss allocation in distribution systems without allocating losses to the slack node. (9)

5 1174 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 3, AUGUST 2006 Fig. 5. Loss allocation on the single-feeder circuit test. Fig. 7. Total network losses and losses allocated to the reactive power generator. losses reach their minimum value. It does not happen for the SMLA because of the paradox illustrated in Section II-A. Fig. 6. Portion of total network losses allocated to the slack node. C. Sensitivity Concepts in the Loss Allocation Methods The loss allocation factors can be seen as sensitivity factors indicating the change of the total real losses for a marginal variation in the real or reactive generation or consumption at node of the system, that is, and. In particular, the MLC coefficients provide nearly twice the total losses (thus needing reconciliation), whereas the coefficients of the DLC method (that contains the term in its definition) provide directly the loss allocation factors. In order to show that the sensitivity concept is essential to obtain a meaningful interpretation of the loss allocation in the presence of variable loads, the test feeder shown in Fig. 4 has been modified, by substituting the load at node 7 with a variable reactive power generation. As shown in Fig. 7, for the DLC, MLC, and ZLA methods, the amount of losses allocated to the reactive power generator is consistent with the variation of the total losses of the network, being equal to zero when the total IV. PROPOSED BRANCH CURRENT DECOMPOSITION METHOD FOR LOSS ALLOCATION The previous analysis has shown that the derivative-based (MLC and DLC) methods are effective in dealing with distribution systems with DG and can provide correct sensitivity information. However, both methods require the computation of the power flow Jacobian matrix, and the DLC requires a further calculation of the elements of the Hessian matrix, with a more computationally intensive procedure. The ZLA method is also able to produce correct loss allocation coefficients, but it needs the calculation of the bus impedance matrix coefficient in the presence of shunt parameters. However, the power flow calculation for distribution systems (with or without DG) can be effectively carried out by using the well-known backward/forward sweep method [12], [13], which does not require computing the power flow Jacobian nor the bus impedance matrix coefficients. These considerations are the basis for developing a loss allocation method based on the power flow results (voltages, currents, and powers) and able to provide meaningful and intuitive results without the need of calculating further matrices (e.g., bus impedance, Jacobian, or Hessian). For this purpose, the authors have developed a new loss allocation method, called branch current decomposition loss allocation (BCDLA), which adopts a very simple and effective mechanism. Let us consider a radial distribution system, in which the root node is assumed as slack and its voltage angle is the angle reference. Let us define as the set of the branches that connect the node to the root node and as the set of nodes supplied from branch (located downward from branch ). Starting from the Cartesian representation of the current flowing through branch (10)

6 CARPANETO et al.: BRANCH CURRENT DECOMPOSITION METHOD FOR LOSS ALLOCATION 1175 and considering the output current from node it is possible to express the branch losses as (11) (12) where is the branch resistance. The components and are then, respectively, written as the sum of the real and imaginary parts of the currents injected in the portion of the distribution system located downward with respect to the branch, obtaining (13) The losses associated to branch are assigned to the nodes located in the path from branch to the root as follows: Fig. 8. Path from node k to the root with identification of the current contributions. Fig. 9. Modified circuit for the calculation of the virtual voltage W. capacitors) connected to node p. However, there is no need for computing the currents, since the forward sweep uses only the available actual branch currents. The node currents are expressed in terms of the net active and reactive output power from node as (14) (17) The total losses are then allocated to each node relationship by using the By representing the voltage at the coordinates th node with the Cartesian (18) (15) the components of the node current can be obtained from where the terms and can be seen as the real and imaginary components of a virtual voltage at node, such that (19) (20) (16) From (15), the losses allocated to node become All the virtual voltages, for, can be easily and simultaneously calculated as the total voltage drop on the path from a single forward sweep on a modified system with the same structure as the original network but with the branch series impedances replaced by their resistive terms. Figs. 8 and 9 show an example related to one of the virtual voltages, for which the relevant portion of the system used in the forward sweep includes the path from the root to node. The node and branch currents are the same as obtained from the power flows, but the branches are represented by using only the resistive terms, without considering the branch reactance. The ideal current generators introduced in Fig. 9 contain the currents flowing from the respective node to other paths and graphically represent the need for maintaining the node current balance. For instance, the subsystem indicated as Sub(p ) contains the portion of the system linked to node p through all the branches not belonging to the path, as well as all the shunt elements (e.g., (21) and can be expressed in function of the net active and reactive power as with (22) (23) (24) so that and are the coefficients expressing the loss allocation in terms of the active and reactive power components.

7 1176 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 3, AUGUST 2006 Fig. 10. Three-node system example. In order to explain the characteristics of the proposed method, let us consider a simple three-node system [see Fig. 10(a)] with branch impedances and power injections p.u. (load), p.u. (load), and p.u. (local generator). The slack node voltage is p.u. The voltages resulting from the power flow are and p.u. The corresponding currents are and p.u. The total losses are p.u. The virtual voltage, for, is calculated by performing the forward sweep on the modified circuit of Fig. 10(b), in which the branches are represented by the resistances and and the currents and, obtaining and p.u. The allocated losses, computed from (16), are and and their sum gives exactly the total losses. The proposed method has been successfully applied to several distribution systems. Some results are shown in Section VI. Besides its simplicity, other positive features of the proposed BCDLA method have to be mentioned, to elect it as a powerful alternative to the other methods. It is a circuit-based method that takes full advantage of the power flow results, with no additional assumptions nor approximations. The implementation is very simple, since is requires only an additional forward sweep on the modified network. It provides automatically the total losses with no need for reconciliation. The sensitivity information is implicitly included, since the BCDLA method provides the same behavior of the allocated losses as the MLC and DLC methods, without the need for computing additional derivatives (Jacobian and Hessian matrices). The loss allocation factors are defined for all the active and reactive power components at every node. A conceptual difference between the BCDLA and MLC (or DLC) methods emerges when there are voltage-controllable local generators connected to the system, in any case in which at least one of these generators is operating in the voltage-support mode [16], [17]. In fact, the MLC and DLC methods cannot provide the loss allocation factors related to the reactive power at the PV nodes, whereas the BCDLA method provides the loss allocation coefficients for all active and reactive powers at all system nodes (slack excluded). However, both the MLC method (after reconciliation) and the BCDLA method (by definition) are able to represent exactly the total losses. Hence, small differences occur in the loss allocation factors in the presence of PV nodes, when the local generator operates in the voltage-support mode for a part of the day and in voltage-following mode (with reactive power fixed at the maximum or minimum limit) for the remaining part of the day. In this case, using the MLC or DLC would lead to a discontinuous time evolution of the reactive power loss allocation factor, whereas the corresponding factor in the BCDLA method never experiences these discontinuities. The proposed loss allocation method can be easily extended to radial distribution systems with multiple voltage levels and transformer interconnections. In this case, the transformer shortcircuit impedance is treated as a branch impedance and the shunt admittance as an equivalent load; the corresponding losses can then be assigned to the transformer owner. Other possible extensions of the proposed loss allocation method include its application to unbalanced distribution systems and the development of variants to be used for weakly meshed distribution systems. V. SMLA METHOD REVISITED The same concepts used for developing the proposed BCDLA method can be applied to correct the formulation of the SMLA method in order to remove the paradox shown in Section II-B. The simple example shown on the two-node system can help in understanding the concept. The SMLA method is based on the expression of the total losses, whereas the BCDLA method is based on the expression. The two expressions provide the same amount of the total losses, due to the presence of cross-terms associated with the branch reactance, which contribute with opposite signs to the total losses but appear individually whenever the losses are allocated among different loads connected to the same node (leading to the paradox shown in Section II-B). For instance, let us decompose the current with respect to a unique reference frame, as. Then, according to the SMLA formulation, the total losses can be written as (25) and the terms depending on the branch reactance can be eliminated, obtaining (26) Applying the same concepts, it is possible to rewrite the formulation of the SMLA method by omitting the reactive terms of the bus impedance matrix and using the actual branch current available from the power flow, obtaining the following expression for the losses allocated to bus (with the same notation of Section II-A): (27) being again and the sending and ending node of branch, respectively. This formulation has been successfully tested to

8 CARPANETO et al.: BRANCH CURRENT DECOMPOSITION METHOD FOR LOSS ALLOCATION 1177 TABLE I DATA OF THE LOCAL GENERATION SYSTEMS Fig. 11. Modified 32-node test system. confirm the elimination of the paradox shown in Section II-B for the SMLA method. VI. APPLICATIONS OF THE BCDLA LOSS ALLOCATION METHOD A. Example of Application The proposed loss allocation method has been applied to various distribution systems with DG. A first test system has been the widely used four-node system with and DG [4], [11], [14], where the DG power has been increased from 0 to 500 kw. The results (not shown here for brevity) are identical to those presented in [11] for the same test system and operating conditions. However, this system has purely resistive branches, so that even the original SMLA method behaves correctly, since the paradox conditions do not occur. Further tests have been performed on more general systems, including real systems, focusing the attention on the following aspects: 1) possibility of defining positive or negative loss allocation factors, in order to give a clear message on the effectiveness of the DG to the loss reduction; 2) ability of the allocation method to clearly reflect the contribution of each supplier/consumer to the power losses; 3) consistent treatment of all the shunt components, including those represented in the branch model and capacitors for power factor correction, that are treated as reactive power loads and can be associated with positive or negative loss allocation factors, depending on their location in the system. Practically, the losses allocated to the shunt parameters of the branch model are attributed to the distributor, whereas the losses allocated to power factor correction capacitors (or other shunt components) are attributed to their owner. All these aspects are fully covered by the BCDLA method. An example is provided here for a test system, by considering the evolution of loads and generations over a day. The test system has been set up with the network structure of the 32-node distribution system defined in [15] (see Fig. 11), adding three small-hydro local generators at nodes with their corresponding transformers connected, respectively, at nodes 8, 17, and 32 (see Table I) and introducing specific load Fig. 12. Total losses in the modified 32-node test system. profiles at the system nodes. In such a way, the analysis carried out on the system provides the daily evolution of the allocated losses. The base power is 1 MVA. The local generators at nodes 33 and 35 have a flat generation profile, whereas the local generator at node 34 has a two-level generation profile, with active power of 0.4 p.u. from 8.00 A.M. to 6.00 P.M. and 0.24 p.u. in the other hours. The generator at node 35 is modeled as a PV node with the maximum and minimum reactive power limits indicated in Table I (these limits have been restricted with respect to the real values to make more evident the differences among the methods), whereas the others two generators (at nodes 33 and 34) are modeled as PQ nodes with. The active power load patterns have been defined by using typical load profiles for residential, industrial, and commercial customers, associating each node with a different share of these load profiles. Then, each node has a different load pattern, whose maximum value is close to the active power load defined in [15]. The reactive power of each load at each instant has been assumed to be one half of the corresponding active power. Multiple power flows have been run at each hour of the day. Fig. 12 shows the evolution of the total losses. Fig. 13 indicates the allocated losses at 5.00 A.M. and A.M., respectively. Here a positive allocated loss is associated with a penalty, whereas a negative allocated loss corresponds to a reward. Then, the local generators are penalized during the hours of low consumption (e.g., 5 A.M.) and rewarded when the consumption increases, and their presence allows for reducing the network losses (e.g., 10 A.M.). Fig. 14 shows the active and reactive loss allocation coefficients at the same hours. The different role of the contribution

9 1178 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 3, AUGUST 2006 Fig. 13. Allocated losses (p.u.) at different hours for the modified 32-node test system. Fig. 15. Allocated losses (p.u.) at different nodes. Fig. 14. Loss allocation coefficients at low and high consumption hours. of the three generators to the reactive power is represented by the values and sign of the reactive coefficients. Fig. 15 shows the losses allocated to a selected set of nodes. These losses qualitatively follow the evolution of the load, either directly (for the load nodes) or inversely (for the generation nodes). Fig. 16 shows the daily evolution of the voltage magnitude, loss allocation factors, and allocated loss at node 35, whose local generator is able to partially control the voltage magnitude during the day. The difference between the BCDLA and the MLC methods clearly emerge any time the generator operates as a PV node, and the reactive coefficient for the MLC method becomes zero. This corresponds to a slight difference in the losses allocated to node 35, and also to the other nodes, since the total losses allocated by the two methods are always identical. B. Method Comparisons A summary of the main relationships between the BCDLA method and the other methods tested leads to the following observations. Fig. 16. Voltage magnitude, loss allocation coefficients, and losses allocated to node 35, showing the differences between the MLC and the BCDLA methods in the presence of voltage-support mode operation. In the absence of PV nodes operating in the voltage-support mode, the MLC method performs as the BCDLA method. In systems with non-negligible shunt parameters, the ZLA method (in its version that does not consider the loss allocation to the slack node) performs as the BCDLA method. The revisited SMLA method (in its version that does not consider the loss allocation to the slack node) always performs as the BCDLA method. The application of the DLC method leads to small differences in the real cases, due to its approximated evaluation of the total losses (that could be eliminated by performing reconciliation, not included in the conceptual development of the method). The superiority of the BCDLA method depends on the fact that, even in the cases in which the other methods can provide the same results, this happens at the expense of additional calculations requiring extra information (i.e., coefficients of bus impedance or Jacobian or Hessian matrix) with respect to the those available from the power flow solution.

10 CARPANETO et al.: BRANCH CURRENT DECOMPOSITION METHOD FOR LOSS ALLOCATION 1179 VII. CONCLUSION The new, simple, and efficient branch current decomposition method for calculating the loss allocation factors in radial distribution systems with DG has been presented. This paper has provided the conceptual framework under which the proposed method has been developed, the details of its definition, and an example of application. The application of the proposed BCDLA method has shown very effective performance and good ability to easily and specifically address the details of the location and time-domain evolution of generations and loads. The relationships among the proposed method and other methods of different type have been discussed, also resulting in revisiting the SMLA method to avoid a paradox implicitly included in its formulation. On these basis, the BCDLA method has emerged as a powerful alternative to the other methods. Some extensions of the proposed loss allocation method are under development, in order to apply it to radial unbalanced distribution systems, to develop dedicated variants for weakly meshed systems, and to use the loss allocation scheme as a basis for developing suitable pricing mechanisms. REFERENCES [1] P. M. Costa and M. A. Matos, Loss allocation in distribution networks with embedded generation, IEEE Trans. Power Syst., vol. 19, no. 1, pp , Feb [2] C. N. Macqueen and M. R. Irving, An algorithm for the allocation of distribution system demand and energy losses, IEEE Trans. Power Syst., vol. 11, no. 1, pp , Feb [3] The Electricity Pool in England and Wales, Guidance Note for Calculation of Loss Factors for Embedded Generation in Settlement, Apr. 1992, Paper SSC1390OP. [4] N. Jenkins, R. Allan, P. Crossley, D. Kirschen, and G. Strbac, Embedded Generation. London, U.K.: Inst. Elect. Eng., 2000, IEE Power and Energy Series 31. [5] A. J. Conejo, J. M. Arroyo, N. Alguacil, and A. L. Guijarro, Transmission loss allocation: A comparison of different practical algorithms, IEEE Trans. Power Syst., vol. 17, no. 3, pp , Aug [6] A.-G. Expósito, J. M. R. Santos, T. G. García, and E. A. R. Velasco, Fair allocation of transmission power losses, IEEE Trans. Power Syst., vol. 15, no. 1, pp , Feb [7] E. De Tuglie and F. Torelli, Nondiscriminatory system losses dispatching policy in a bilateral transaction-based market, IEEE Trans. Power Syst., vol. 17, no. 4, pp , Nov [8] G. Gross and S. Tao, A physical-flow-based approach to allocating transmission losses in a transaction framework, IEEE Trans. Power Syst., vol. 15, no. 2, pp , May [9] A. J. Conejo, F. D. Galiana, and I. Kockar, Z-Bus loss allocation, IEEE Trans. Power Syst., vol. 16, no. 1, pp , Feb [10] W. L. Fang and H. W. Ngan, Succinct method for allocation of network losses, Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 149, no. 2, pp , Mar [11] J. Mutale, G. Strbac, S. Curcic, and N. Jenkins, Allocation of losses in distribution systems with embedded generation, Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 147, no. 1, pp. 7 14, Jan [12] M. E. Baran and F. F. Wu, Optimal sizing of capacitors placed on a radial distribution system, IEEE Trans. Power Del., vol. 4, no. 1, pp , Jan [13] E. Bompard, E. Carpaneto, G. Chicco, and R. Napoli, Convergence of the backward/forward sweep method for the load-flow analysis of radial distribution systems, Elect. Power Energy Syst., vol. 22, pp , [14] G. Strbac and N. Jenkins, Calculation of cost and benefits to the distribution network of embedded generation, in Proc. Inst. Elect. Eng. Colloq. Economics Embedded Generation, Oct. 29, 1998, pp. 6/1 6/13, Ref. No. 1998/512. [15] M. Baran and F. F. Wu, Network reconfiguration in distribution systems for loss reduction and load balancing, IEEE Trans. Power Del., vol. 4, no. 2, pp , Apr [16] P. A. N. Garcia, J. L. R. Pereira, and S. Carneiro Jr., Voltage control devices models for distribution power flow analysis, IEEE Trans. Power Syst., vol. 16, no. 4, pp , Nov [17] E. Carpaneto, G. Chicco, M. De Donno, and R. Napoli, Voltage controllability of distribution systems with local generation sources, in Proc. Bulk Power System Dynamics and Control VI, Managing Complexity in Power Systems: From Micro-grids to Mega Interconnections, Cortina d Ampezzo, Italy, Aug , 2004, pp Enrico Carpaneto (M 86) received the Ph.D. degree in electrotechnical engineering in from the Politecnico di Torino (PdT), Torino, Italy, in Currently, he is an Associate Professor of electric power systems at PdT. His research activities include power systems and distribution systems analysis, competitive electricity markets, and power quality. Dr. Carpaneto is a Member of AEI. Gianfranco Chicco (M 98) received the Ph.D. degree in electrotechnical engineering from the Politecnico di Torino (PdT), Torino, Italy, in Currently, he is an Associate Professor of electricity distribution systems at PdT. His research activities include power system and distribution system analysis, competitive electricity markets, load management, artificial intelligence applications, and power quality. Jean Sumaili Akilimali received the B.S. degree in sciences appliquées (option: Electricité) at the University of Kinshasa, Kinshasa, Democratic Republic of Congo, in 1998 and the M.S. degree in electrical engineering at the Politecnico di Torino (PdT), Torino, Italy, in Currently, he is pursuing the Ph.D. degree in electrical engineering at PdT. His research activities include distribution systems analysis and distributed generation applications.