POLITECNICO DI TORINO. Dipartimento di Ingegneria Elettrica. Dottorato di Ricerca in Ingegneria Elettrica XX Ciclo (DOTTORATO EUROPEO) DOCTORAL THESIS

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1 POLTECNCO D TORNO Dipartimento di ngegneria Elettrica Dottorato di Ricerca in ngegneria Elettrica XX Ciclo DOTTORATO EUROPEO DOCTORAL THESS CURRENT DECOMPOSTON-BASED LOSS PARTTONNG AND LOSS ALLOCATON N DSTRBUTON SYSTEMS Supervisors: Prof. Gianfranco CHCCO Prof. Enrico CARPANETO Candidate: Jean SUMAL AKLMAL May 8

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3 CURRENT DECOMPOSTON-BASED LOSS PARTTONNG AND LOSS ALLOCATON N DSTRBUTON SYSTEMS Jean SUMAL AKLMAL Doctoral Thesis, May 8 Dipartimento di ngegneria Elettrica Politecnico di Torino, Corso Duca degli Aruzzi 4, 9 Torino, taly Supervisors: Prof. Gianfranco CHCCO, Prof. Enrico CARPANETO Doctoral Course Coordinator: Prof. Mario CHAMP Astract The allocation of the system losses to suppliers and consumers is a challenging issue for the restructured electricity usiness. Meaningful loss allocation techniques have to e adopted in order to send correct signals to the maret taing into account the location and characteristics of loads and generations, including the local sources forming the distriuted generation DG. The allocation factors should depend on size, location and time evolution of the resources connected to the system. n the presence of DG, the variety of the power flows in distriution systems calls for adopting mechanisms ale to discriminate among the contriutions that increase or reduce the total losses. Some loss allocation techniques already developed in the literature have shown consistent ehaviour. However, their application requires computing a set of additional quantities with respect to those provided y the distriution system power flow solved with the acward/forward sweep approach. The initial part of this thesis addresses the issues related to loss allocation in radial distriution systems with DG, with a threefold focus. First, the ey differences in the formulation of the loss allocation prolem for radial distriution systems with respect to transmission systems are discussed, specifying the modelling and computational issues concerning the treatment of the Politecnico di Torino May 8

4 Current Decomposition-ased Loss Partitioning and Loss Allocation in Distriution Systems slac node in radial distriution systems. Then, the characteristics of derivative-ased and circuit-ased loss allocation techniques are presented and compared, illustrating the arrangements used for adapting the various techniques to e applied to radial distriution systems with DG. Finally, the effects of introducing voltage-controllale local generation on the calculation of the loss allocation coefficients are discussed, proposing the adoption of a reduced representation of the system capale of taing into proper account the characteristics of the nodes containing voltage-controllale DG units. A ey contriution of this thesis is the formulation of a new circuit-ased loss allocation technique ased on the decomposition of the ranch currents, specifically developed for radial distriution systems with DG. The proposed technique called Branch Current Decomposition Loss Allocation BCDLA is only ased on the information provided y the networ data and y the power flow solution. Examples of application to confirm its effectiveness are provided, showing the time evolution of the loss allocation coefficients for distriution systems with variale load and local generation patterns. n the last part of the thesis, the concepts related to loss partitioning among the phase currents in three-phase distriution systems are revisited in the light of new findings. n particular, the presence of a paradox in the classical loss partitioning approach, ased on the use of the phase-y-phase difference etween the input and output complex power, is highlighted. The conditions for performing effective loss partitioning without the occurrence of the paradox are thus estalished. The corresponding results are then used to extend the BCDLA method for enaling its application to three-phase unalanced distriution systems with distriuted generation. Numerical examples on a three-phase line with grounded neutral and on the EEE 3-node test system are provided to assist the illustration and discussion of the novel conceptual framewor. Keywords: ranch current decomposition, us admittance matrix, us impedance matrix; circuitased techniques; derivative-ased methods; distriuted generation, distriution systems, loss allocation, loss partitioning, losses, paradox, unalanced systems, voltage-controllale units. Jean SUMAL AKLMAL Doctoral Thesis

5 Astract Pulications ased on this thesis Journal papers E. Carpaneto, G. Chicco, and J. Sumaili Ailimali, Loss Partitioning and Loss Allocation in Three-Phase Radial Distriution Systems with Distriuted Generation, EEE Trans. on Power Systems in press. E. Carpaneto, G. Chicco, and J. Sumaili Ailimali, Characterization of the loss allocation techniques for radial systems with distriuted generation, Electric Power System Research 78, 8, E. Carpaneto, G. Chicco, and J. Sumaili Ailimali, Branch Current Decomposition Method for Loss Allocation in Radial Distriution Systems with Distriuted Generation, EEE Trans. on Power Systems, 3, 7 79 August 6. Conference proceedings E. Carpaneto, G. Chicco and J. Sumaili Ailimali, Three-phase loss partioning in unalanced distriution system ranches, Proc. nternational Symposium of Electrical Engineering SEE 7, Targoviste, Romania 5-7 Octoer 7, in Scientific Bullentin of the Electrical Engineering Faculty, Valahia, University of Targoviste, 5. E. Carpaneto, G. Chicco and J. Sumaili Ailimali, Application of the circuit-ased loss allocation techniques to radial distriution systems, Proc. V World Energy System Conference, Torino, taly - July E. Carpaneto, G. Chicco and J. Sumaili Ailimali, Computational aspects of the marginal loss allocation methods for distriution systems with distriuted generation, Proc. EEE Melecon 6, Benalmádena Málaga, Spain 6-9 May Politecnico di Torino May 8

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7 ndex PREFACE... GLOSSARY...3 Acronyms...3 Symols...3 NTRODUCTON...5 CHAPTER LOSS ALLOCATON N DSTRBUTON SYSTEMS: STATE OF THE ART Classification of loss allocation methods Loss allocation for radial distriution systems Derivative-ased methods Circuit-ased methods Comments...9 CHAPTER ADVANCED SSUES CONCERNNG LOSS ALLOCATON N DSTRBUTON SYSTEMS Loss allocation paradox Adaptation of Transmission Loss allocation methods to Distriution Systems Slac node impact on the loss allocation Sensitivity concepts in the loss allocation methods Networ representation and matrix singularities DG units treatment Voltage-controlled DG unit Reduced representation of local generators...45 CHAPTER BRANCH CURRENT DECOMPOSTON METHOD FOR LOSS ALLOCATON N RADAL DSTRBUTON SYSTEMS Need of a new loss allocation method BCDLA method Numerical example BCDLA: compact formulation Fairness of the BCDLA method Comments on BCDLA...55 Politecnico di Torino May 8 V

8 Current Decomposition-ased Loss Partitioning and Loss Allocation in Distriution Systems CHAPTER V LOSS ALLOCATON N DSTRBUTON SYSTEMS: NUMERCAL RESULTS...57 V.. Loss allocation on a five-load feeder...57 V.. Loss allocation on the 33-node test system...59 V... Base case...59 V... Time evolution of loads, voltages and losses...6 V... Distriuted Generation effect...64 V..3 Usage of the reduced model...68 CHAPTER V THREE-PHASE LOSS PARTTONNG PARADOX...7 V.. ntroduction...7 V.. Circuit model for three-phase ranches...73 V... Carson s equations and definition of the impedances...73 V... Networ equations...74 V.3 Loss calculation and loss partitioning on Three-Phase Branches...77 V.4. Numerical Results...78 V.4. Application to a distriution line...78 V.4.. Application to the EEE 4-node Test System...86 V.5. Comments on loss partitioning...9 CHAPTER V LOSS ALLOCATON N THREE-PHASE RADAL DSTRBUTON SYSTEMS...95 V.. Extension of the BCDLA method to radial unalanced three-phase systems...95 V.. Loss Allocation in the EEE 3-Node Test System...97 V.3. Loss Allocation in the modified EEE 3-Node Test System with Distriuted Generation...99 V.4. Discussions... V.4.. Discussion on the capacitive effects... V.4.. Discussion on the losses allocated to the DG unit... CONCLUSONS...3 REFERENCES...7 APPENDX... A.. Data of the 33-node Test System... A. Characteristics of the unalanced EEE 3-node Test System... Overhead Line Configuration Data... Underground Line Configuration Data... Line Segment Data... Transformer Data... Capacitor Data...3 Regulator Data...3 Spot Load Data...3 Distriuted Load Data...3 Line configuration impedances...4 V Jean SUMAL AKLMAL Doctoral Thesis

9 Preface This wor has een developed from January 5 to Decemer 7 in the Dipartimento di ngegneria Elettrica of the Politecnico di Torino in the context of the European Ph.D. During this period had the opportunity to attend many interesting courses that helped me in increasing my nowledge and in preparing conference and journal papers related the suject of this thesis. n the context of the European Ph.D., had the possiility of spending two research periods at the nstitute for Systems and Computer Engineering of Porto NESC-PORTO, Porto, Portugal and at the School of Electronics and Electrical Engineering of the University of Manchester, Manchester, UK. All these opportunities helped me to enlarge my nowledge in the electrical engineering sciences and to write this thesis. would lie to than Professor Gianfranco Chicco, who supervised the development of the entire wor during all this period, and Professor Enrico Carpaneto for his deep insights and rilliant suggestions. would lie to express to oth of them my deepest gratitude for sharing with me their nowledge. Many thans also to the entire staff of the Dipartimento di ngegneria Elettrica for their assistance and indness. Lastly, ut y no means least-ly, would lie to than my family and all my friends for their support. Politecnico di Torino May 8

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11 Glossary Acronyms BCDLA DLC DG GMR MLC RCLP SMLA SMLAD SMLAD-R SMLA-R VPLA YLA YLAD ZLA ZLAD Branch Current Decomposition Loss Allocation Direct Loss Coefficients Distriuted Generation Geometric Mean Ratio Marginal Loss Coefficients Resistive Component-ased Loss Partitioning Succinct Method for Loss Allocation Succinct Method for Loss Allocation in Distriution systems Succinct Method for Loss Allocation in Distriution systems Revised. Succinct Method for Loss Allocation Revised. Voltage profile Loss Allocation modified Y-us Loss Allocation modified Y-us Loss Allocation for Distriution systems Z-us Loss Allocation Z-us Loss Allocation for Distriution systems Symols Component-y-component vector product * The superscript * indicates the conjugate operator ac The suscripts a, and c, are related to the 3 phases of a three-phase system B G The superscript is related to a generic ranch Set of ranches that connect node to the root node Conductance Politecnico di Torino May 8 3

12 Current Decomposition-ased Loss Partitioning and Loss Allocation in Distriution Systems H Hessian matrix i Complex current Current vector J Jacoian matrix The suscript is referred to a single node K Numer of nodes K Set of nodes supplied from ranch located downward from ranch L Losses μ, ~ μ Marginal Loss Coefficient N P p Q R R ρ S Numer of nodes Active power Power vector Reactive power Resistance Resistance matrix Resistivity of earth Complex power T The superscript T indicates the matrix or vector transposition operator Θ V V v X X Voltage angle Complex voltage Voltage magnitude Voltage vector Reactance Reactance matrix Y Complex admittance Y us Bus Admittance matrix Z Complex impedance Z us Bus mpedance matrix Z prim Primitive mpedance matrix 4 Jean SUMAL AKLMAL Doctoral Thesis

13 ntroduction The introduction of competition in the electricity sector has changed the interactions etween suppliers and consumers, leading to increased attention to the economic aspects in the electricity supply management. n addition, significant energy efficiency improvement and cost reductions for various technologies have made the adoption of small generation units economically competitive. Further incentives towards increasing adoption of renewale sources have een included in various regulations for driving the energy systems evolution towards extended adoption of local generation sources. All these aspects result in increasing the penetration of the Distriuted Generation DG in distriution systems. For what concerns the power losses in the electricity networs, the comined effect of competition and DG expansion is maing the handling of the corresponding costs much more challenging than in the past, raising questions aout the responsiility and allocation of the losses to suppliers and consumers. n fact, in a vertically integrated power structure, the cost of losses was included into the overall electricity production costs. With the introduction of competitive marets, generation, transmission and distriution of electricity have een separated into autonomous usinesses, and the costs of losses have to e specifically allocated among the entities participating to the electricity maret. Conceptually, loss allocation is a difficult tas, ecause losses in the distriution system ranches are non-linear functions of generations and loads. t is impossile to calculate the exact amount of losses in advance, without running a power flow calculation program. At the same time, even after computing the power flow solution, there is a strong interdependence among all the users, expressed y the presence of cross-terms due to the fact that losses are a nearly Politecnico di Torino May 8 5

14 Current Decomposition-ased Loss Partitioning and Loss Allocation in Distriution Systems quadratic function of the power flows. Hence, allocating the losses to the maret participants cannot e carried out in a straightforward way. Some principles for effective loss allocation [CoMa4, DaS5] are recalled here. A loss allocation technique should e: easy to understand and ased on real data of the networ; carefully designed to avoid discrimination etween users; consistent with the power flow solution; ale to recover the total amount of the losses; consistent to the rules of competitive electricity marets; economically efficient, avoiding cross-susidization etween users; ale to send out economic signals aimed at increasing the efficiency of the networ; ale to provide correct signals concerning the size and location of loads and DG sources in the networ; applicale to different situations, e.g., following the time evolution of the generation and load patterns. The presence of local generators has changed the distriution systems from passive networs, with unidirectional power flows in the ranches, into active networs with idirectional power flows. The allocation of losses has to consider the nature of each entity, that may e either a consumer or a generator, at different time instants. Different methods for loss allocation have een reported in the literature. Early formulations of uniform or demand-squared-ased loss allocation [Mar96] are only related to the demand side and as such are not suitale for eing used in the presence of DG. n general, a first distinction can e made etween loss allocation methods dedicated to transmission and to distriution systems. The difference etween these two classes of methods asically lies in the role given to the slac node. n transmission systems, the generator located in the slac node compensates for all the losses and is explicitly considered in the mechanism of loss allocation. n radial distriution systems, the location of the slac node at the root node of the distriution tree is naturally unique, and the slac usually represents the connection to a higher voltage networ. With the evolution of DG, the simultaneous presence of several generators in the distriution systems could raise the question on the possiility of using the same loss 6 Jean SUMAL AKLMAL Doctoral Thesis

15 ntroduction allocation mechanisms defined for the transmission systems. More specifically, it should e clarified whether or not the slac node has to e considered as a generator participating to the loss allocation. This aspect is fundamental, since the slac node delivers a significant amount of the generated power. A possile framewor for distriution systems with DG includes only local generators and loads as participants to the loss allocation. On one hand, suitaly located DG units could reduce the amount of losses. On the other hand, the possile enefit in terms of loss reduction cannot e directly associated to 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 prolem is made evident y the fact that early formulated loss allocation mechanisms, even adopted at the regulatory level, have een found to e inconsistent. An example is the sustitution method [EPEW9], in which the impact of a generator or load unit is evaluated y computing the difference etween the total losses occurring with and without the unit. The inconsistency of such an approach has een clearly shown in Section 8.4 of [JACK]. This thesis provides various contriutions to the analysis and improvement of the loss allocation mechanisms, from the viewpoint of their applicaility to distriution 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 possile to formulate mechanisms ale to cover the total system losses, ut whose ehaviour leads to evident paradoxes. The analysis of the techniques proposed in the scientific literature has permitted the identification of the methods ale to correctly represent the generation and load characteristics and to provide consistent technical and economic signals. One of the ey aspects developed here is the proposal of an original, simple and efficient allocation method, specifically formulated for radial distriution systems with DG and requiring only data from the networ structure and the power flow solution, without the need of computing additional quantities lie Jacoian and Hessian matrixes which are not necessary for the load flow calculation in radial distriution systems. The first part of this thesis is dedicated to the State of the-art. First, the loss allocation techniques presented in the literature are classified according to the methodologies and principles used for their formulation. Then an overview of the important methods is presented providing a comparison among the loss allocation techniques. The second part presents the advancements and original contriution to loss allocation in distriution systems highlighting the possile Politecnico di Torino May 8 7

16 Current Decomposition-ased Loss Partitioning and Loss Allocation in Distriution Systems occurrence of a paradox depending on the formulation of the loss allocation method. The third part deals with the three-phase systems providing a fair loss partitioning method and the extension of loss allocation to unalanced systems. The last part contains concluding remars and suggestions for future investigation. 8 Jean SUMAL AKLMAL Doctoral Thesis

17 Chapter Loss Allocation in Distriution Systems: State of the Art This chapter presents an overview of loss allocation in Distriution Systems. First the methods are classified according to different grouping criteria. Then the promising methods ale to provide consistent loss allocation in distriution systems with Distriuted Generation DG are highlighted... Classification of loss allocation methods Different proposals for the allocation of losses in electric networs have appeared in the last years for transmission and distriution systems. Clearly, the focus has een mainly on transmission systems, due to the large amount of power flows involved and the increasing numer of agents with open access to transmission networs. n this way, several proposals have appeared in the technical literature and some comparative studies were presented [CAAG, UnSa3]. For distriution systems some proposals have een made, mainly considering the presence of distriuted generation [CoMa4, Mar96, MSCJ, SEBM]. The loss allocation mechanisms can e classified in different ways according to the grouping criteria used. The first methods proposed did not tae into account the position of the users on the networ. They are ased on: uniform sharing proportional sharing Politecnico di Torino May 8 9

18 Current Decomposition-ased Loss Partitioning and Loss Allocation in Distriution Systems pro-rata sharing The uniform sharing method proposes the division of the losses in equal parts attriuting the same amount to all the participants. The proportional sharing methods allocate a portion of the total losses proportionally to the power of each user of the networ. According to the quadratic ehaviour of the losses, a variant of the proportional method divides the losses proportionally to the square of the power. The pro-rata sharing suggests the allocation of 5% of the losses to the generators and 5% to the loads. On the asis of the nature of the networ, it is possile to distinguish two ig families: methods developed for transmission systems methods developed for distriution systems The former methods have een formulated for meshed transmission systems with several generators of the same size order, while the latter methods are dedicated to distriution systems with radial or wealy meshed structure. n the distriution systems, the main supply is assured y the high level voltage networ. Referred to the methodologies used, the loss allocation techniques can e grouped in: Circuit-ased Marginal or derivative-ased The circuit-ased method use circuit laws principles, in opposition to the derivative-ased methods which analyse the effect of a marginal variation of load and generation. A relevant version of the circuit-ased method is the so-called tracing-ased method which exploit the power flow tracing concept [Bial96] n a competitive framewor, the loss allocation can e also ased on the type of contracts etween the operators of the maret defining the transaction-ased methods. Contracts can e grouped into two classes: ilateral contracts multilateral contracts Jean SUMAL AKLMAL Doctoral Thesis

19 Chapter Loss Allocation in Distriution Systems: State of the Art One of the most used models for loss allocation with transaction-ased methods is the game theory framewor constructed in the asis of a cooperative game in which the players are represented y the different transactions [PeJi4, LiCP7]. All these classification concepts can e applied to distriution systems. However, some methods ased on a priori assumptions i.e., pro-rata and proportional sharing are considered to e too approximated for the purpose of this thesis. n addition, addressing transaction-ased loss modelling as in [GRGR,DeTo,GrTa] is outside the scope of this thesis as the distriution systems are not concerned yet y total competitiveness. So, in this chapter the analysis is focused on two types of existing methods: the circuit-ased methods and the marginal or derivativeased loss allocation mechanisms... Loss allocation for radial distriution systems n a distriution system, the slac node represents the connection to a higher voltage networ usually to the transmission system, through a power sustation. The slac node exhiits some peculiar characteristics, since it is not associated to a physical generator, and the slac node voltage is defined according to the voltage control usually ensured through the underload tap-changer of the sustation transformer. Furthermore, in the whole electrical system usually the sustation sets the oundary etween the meshed transmission system and the radialoperated distriution system, characterized y different owners, networ structure and operators, and suject to different regulatory framewors. One of the characteristics of the loss allocation methods applied to radial distriution systems is that the slac node is uniquely defined i.e., it is the root node of the distriution tree. Since the power supplied from the slac node is generally prevailing over the power of each other possile local supply point, it is essential to estalish a correct loss allocation strategy among the suppliers and users connected to the distriution system. n this respect, all the distriution system users suppliers and consumers may e conceptually considered as sujects having a ilateral contract with a single provider, i.e., the distriution company. This enales for assuming the provider of the distriution service as a unique suject with the specific feature of sharing the ul amount of losses with all the other sujects, thus leading to a asic conceptual difference with respect to what occurs for transmission systems [CAAG, Mar96]. n fact, the loss allocation techniques formulated for transmission systems typically allocate part of the losses to the slac node, so they cannot generally e applied to radial distriution systems and Politecnico di Torino May 8

20 Current Decomposition-ased Loss Partitioning and Loss Allocation in Distriution Systems vice versa. The explicit pro-rata assumption of allocating 5% of the losses to the loads and 5% to the generators can e revisited in order to e applied to the case of distriution systems, with or without DG, considering that the losses have to e divided equally etween the higher level networ represented y the slac node and the distriution system customers loads and local generators.... Derivative-ased methods The derivative-ased methods compute the loss sensitivity factors exploiting the information extracted from the second-order derivatives of the equation representing the total losses. A general framewor related to the definition of the derivative-ased methods can e formulated y expressing the total losses L in function of the net node powers as a complete quadratic form neglecting higher-order terms: T T L L o + p + p Ap - where the column vector p contains all the active and reactive nodal net powers, A is a symmetrical real matrix T A = A, is a column vector, and the superscript T indicates matrix transposition. The term L o = L p= represents the losses with zero injected node power, due to the shunt currents and to the possile circulating currents at no-load introduced y the presence of voltage-controllale local generators with different voltage set-points. n normal operating conditions, the term L o is typically low compared to the other terms. The definition of derivative-ased loss allocation methods for distriution systems has een fully explained in [MSCJ] presenting two different inds of coefficients: Marginal Loss Coefficients and Direct Loss Coefficients.... Marginal loss coefficients The Marginal loss coefficients MLC measure the change in total active power losses L due to a marginal change in consumption/generation of active power P i and reactive power Q i at each node i in the networ ~ L L μ P = ~ μ i Q = P i Q - i i Jean SUMAL AKLMAL Doctoral Thesis

21 Chapter Loss Allocation in Distriution Systems: State of the Art Politecnico di Torino May 8 3 where P i μ ~ and Q i μ ~ represent the active and reactive power related MLC. f a generator taes part in voltage control y injecting the required reactive power PV node, there are no lossrelated charges for the reactive power to e allocated. This is reflected y ~ def i Q Q L ρ i = = if i is a PV node -3 Since in load flow calculations losses are deemed to e supplied from the slac node, the lossrelated charges for this node are zero. n other words, total power losses are insensitive to changes in active and reactive injections at the slac node, i.e., ~ ~ = = = = s Q s P Q L ρ P L ρ s s s is the slac node -4 Because of this assumption the choice of the slac node clearly has an impact on oth magnitude and polarity of MLC. Fortunately, in distriution systems this complication need not arise as the choice of the slac node is unique. The MLC are a function of a particular system operating point. As there is no explicit relationship etween losses and power injections the standard chain rule is applied in the calculation of MLC using intermediate state variales, voltage magnitudes and angles. Therefore, only a load flow solution for a particular system operating point system state at a certain quarter-hour, half-hour or hour is required to compute the MLC. Applying the standard chain rule, the following general system of linear equations can e estalished for calculating the MLC Θ Θ = Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ V L V L L L Q L Q L P L P L V Q V Q V Q V P V P V P V Q V Q V Q V P V P V P Q Q Q P P P Q Q Q P P P N N N N N N N N N N N N N N N N N N N N N N N N -5

22 Current Decomposition-ased Loss Partitioning and Loss Allocation in Distriution Systems Equation -5 can e written in a matrix form as A μ ~ = -6 Matrix A is the transpose of the Jacoian matrix in the Newton Raphson load flow and can e calculated on the asis of load flow results for a particular system operating point. The vector ρ ~ represents the MLC, whereas the right-hand vector represents sensitivities of total losses with respect to voltage angle and magnitude. Total system active loss L is given y L N N Gij i j i= j= = [ V + V V V cos Θ Θ ] i j i j -7 Therefore, the entries of vector in equation -6 are L Θ i L V i N Θ Θ = G V V sin i =,,N -8 N j ij i j [ V V cos Θ Θ ] i j = G i =,,N -9 j ij i j i j Note that there are no equations for any voltage-controlled node as y definition the MLC with respect to reactive power for any such node is zero. The result of applying the MLC calculated in accordance with the procedure outlined yields approximately twice the amount of losses. That is, N = [ ~ P + ~ μ Q ] L μ - P Q Therefore, reconciliation is needed in order to allocate the actual amount of losses exactly. To otain the vector of the reconciled MLC μ, a constant-multiplier reconciliation factor K o is applied. The factor K o is calculated as follows: 4 Jean SUMAL AKLMAL Doctoral Thesis

23 Chapter Loss Allocation in Distriution Systems: State of the Art K o = N = [ ~ μ P + ~ μ Q ] P L Q - The vector of reconciled MLC μ is then calculated as follows: μ K μ~ - = o... Direct loss coefficients The direct loss coefficient DLC method relates losses directly to nodal injections and therefore does not require reconciliation. n contrast to the MLC method which allocates marginal losses, the DLC method allocates the total losses. The ojective of this method is to derive a relationship such that losses can e expressed directly in terms of injections. Due to the complexity of AC load flow equations and their solution y iterative procedures, a closed form solution for losses is not feasile. Moreover, the formula used to compute losses contains system state variales whose values are only nown after the load flow solution has converged. As already indicated, the total power losses in an AC transmission networ are given y Equation -7 For a given change in the operating point the new total system losses can e evaluated using the Taylor series expansion around the initial operating point. The operating point is defined in terms of state variales V and Θ with P and Q representing the corresponding nodal power injections. The new loss formulation is therefore given y L = f Θ, V = f Θ + ΔΘ, V + ΔV L L T T Θ, V + [ ΔΘ ΔV ] + [ ΔΘ ΔV ] H [ ΔΘ ΔV ] +... f Θ V -3 where H is the Hessian matrix, ΔΘ and Δ V represent the change in operating point. Applying the following initial conditions to equation -3 V i =. Θi = i =,,N- -4 one otains Politecnico di Torino May 8 5

24 Current Decomposition-ased Loss Partitioning and Loss Allocation in Distriution Systems Θ, V = L = f -5 o where L o represents the system losses under the flat-start conditions. From equation -5 it follows that there are no flows through the circuits, which corresponds to zero initial nodal active power P and reactive power Q injections. The first derivative elements L, and Θ i L V i are also zero at flat start, while the Hessian matrix is symmetrical and contains only the real parts of the us admittance matrix. Therefore, the total networ losses can e represented as follows: L [ ΔΘ ΔV ] H [ ΔΘ ΔV ] T -6 t is important to point out that ΔΘ and ΔV in equation -6 represent the final deviations from the flat start values of voltage angle and magnitude, respectively. To express losses directly in terms of nodal injections, Equation -6 must e written in terms of nodal injections. This is accomplished y using an analogy with the well-estalished Newton-Raphson load flow algorithm T [ ΔΘ ΔV ] = [ ΔP ΔQ] T J -7 m where respectively: J m is an average Jacoian computed from the flat start and final Jacoians J o and J f m J J J = + o f -8 Note that Δ P and Δ Q in Equation -7 represent the actual nodal active and reactive power injections, respectively, as the initial P and Q values are assumed to e zero. Finally, the vector of changes in voltage angles and magnitudes on the extreme right of equation -6 can e replaced y the right-hand side of Equation -7 to otain equation -9. L m [ ΔΘ ΔV ] J [ P Q] T H -9 6 Jean SUMAL AKLMAL Doctoral Thesis

25 Chapter Loss Allocation in Distriution Systems: State of the Art The first three right-hand terms constitute the DLC. Thus, the vector of the DLC is given y equation [ ΔΘ ΔV ] H J γ = m - The vector γ can e sudivided into two su-vectors, γ = [ γ P γ P ] - where γ P e γ Q contain respectively the active and reactive power-related DLC. The loss allocated to a load or local generator connected to a node is expressed y the equation: L γ P P + γ Q Q = - The assumptions and approximations made in the computation of direct loss coefficients give rise to small differences etween the losses calculated from the application of the DLC and those calculated from load flow. However, in contrast to the MLC, there is no fundamental requirement for reconciliation in the case of the DLC. This is ecause the DLC method is ased on allocation of total losses. Furthermore, losses are, approximately, a quadratic function of power and equation -7 used as the asis for derivation of the DLC stops at the quadratic term.... Circuit-ased methods The circuit-ased methods are defined on the asis of the system structure, expressed y the us admittance matrix Y us or the us impedance matrix Z us, starting from the results of the power flow calculation. Some of them exploit the concept of power tracing in order to determine the contriution of the networ actors to the flow in each ranch. The asic concept of the electricity flow tracing [Bial96] is the attriution of the ranch flows to all nodal injections, a challenging tas in a meshed networ. n a radial networ, taing into proper account the specificity of the slac node, this attriution is straightforward. n fact, each ranch flow current or power is the sum of all the corresponding negative injections of all the downward nodes. n a radial system, cutting any ranch generates a partition of the nodes Politecnico di Torino May 8 7

26 Current Decomposition-ased Loss Partitioning and Loss Allocation in Distriution Systems 8 Jean SUMAL AKLMAL Doctoral Thesis into two distinct sets, namely, the upward node set, which includes the slac node, and the downward node set containing the nodes not supplied y the slac node. This topological feature identifies the ey role of the slac node, which does not participate to the flow attriution since it never elongs to the downward set of any ranch. n radial distriution systems, as the tracing is unique, the challenge consists in a fair partitioning of the cross-terms which appear when the flow in a ranch is the sum of several components. Let us consider the current in a generic ranch sum of K loads or local generators currents. = = K -3 The Joule losses in this ranch are expressed in Equation -4 = = * Re Re Z Z L -4 where j X R Z + = is the impedance of the ranch and the asteris denotes conjugation. Sustituting y its expression of -3, Equation -4 ecomes + = = = = K j j j K K Z R L * Re -5 The components of the first right-hand term can e directly attriuted to relative currents, while for each couple of currents and j the related components of the second right-hand term reported in Equation -6 have to e sudivided etween the two currents = + * * * Re Re j j j R Z Z -6 n order to compute the portion to e attriuted to each current, let us define the coefficients of partition λ and j λ

27 Chapter Loss Allocation in Distriution Systems: State of the Art * * = * + Re Re j λ j j λ Re j -7 with λ + λ j = -8 The splitting of the cross terms [GRGR] can e done on the asis of: incremental coefficients λ = λ j = -9 proportional allocation λ λ = j -3 j quadratic allocation λ = λ -3 j j geometric allocation = λ log j λ log -3 j Another methodology proposed in [ScMC5] exploits the voltage drop on the ranch in order to overcome the cross terms prolem, writing the losses in ranch as * K L = Re Δ V -33 = and the portion allocated to each current ecomes = * L Re Δ V -34 All the methods ased on the us admittance matrix Y us or the us impedance matrix Z us have een formulated for transmission networs. n this Section, we report the original versions, while the versions developed for eing applied to distriution systems are presented in Chapter V. For a system with N nodes and B ranches, the power flow solution provides the complex node voltage V, the net input complex power S = P + jq and the net input current Politecnico di Torino May 8 9

28 Current Decomposition-ased Loss Partitioning and Loss Allocation in Distriution Systems for any node =,, N. The aim of each loss allocation method is to sudivide the total system losses among its nodes writing: L -35 = N L where L represents the loss allocated to node. Given the power flow solution, the following efficient circuit-ased methods have een proposed.... Z-us loss allocation [CoGK] To provide loss allocation according to Z us loss allocation method, first the networ admittance matrix has to e non-singular. This requirement is met in AC networs, since transmission lines always have a shunt capacitance to ground. Starting with * N L = Re V -36 = the system real losses can e expressed either in through the admittance matrix Y as us L N N * * Re V Y j V j -37 = j= = us us. or, through the impedance matrix Z = Y = R + jx L N * N Re Z j j -38 = j= = As it turns out, of the two, the impedance matrix formulation is the only one that yields loss allocation numers that are reasonale, that is, meeting the characteristics suggested in the Jean SUMAL AKLMAL Doctoral Thesis

29 Chapter Loss Allocation in Distriution Systems: State of the Art ntroduction. This is due to the fact that the losses are directly related to the currents, which are the independent variales in Equation -38 L N * N N * N Re + Rj j Re jx j j -39 = j= = j= = t has een shown that the second term in -39 is equal to zero, so that the system losses can e expressed uniquely in terms of the complex currents and the resistance matrix R. Thus, L N * N Re R j j -4 = j= = and it follows that the loss component associated with us can e expressed as L * N = Re Rj j -4 j= As it can e seen from Equation -4, the loss component L encompasses N terms representing the coupling actions etween current injections at all N uses with the current injection at us. One characteristic of this natural separation of the system losses is that the loss terms depend primarily on the complex us current injections and the loss allocation terms are invariant with respect to the current reference angle, since Equation -4 is defined y terms with products of the current vector i and its conjugated i *. This dependence of the loss terms on currents rather than power injections is intuitively reasonale. t must e emphasized that no special assumptions or approximations have een necessary in deriving the loss allocation terms, L. This natural separation of the system losses is ased solely on a solved load flow and the exact networ equations as characterized y the impedance matrix.... Succinct Method for Loss Allocation [FaNg] Traditionally the networ losses can e divided into two parts, a variale loss due to the series impedance ranch and an invariale loss due to the shunt admittance ranch. The Succinct Politecnico di Torino May 8

30 Current Decomposition-ased Loss Partitioning and Loss Allocation in Distriution Systems Jean SUMAL AKLMAL Doctoral Thesis Method Loss Allocation SMLA has to do with the variale part only, while it proposes that the invariale part can e allocated averagely among all users. The current going along the series impedance ij ij ij jx R Z + = in the π -model of ranch connecting the nodes i and j can e written as = = + = N j i ij ij ij j i ij Z Z y X R V V j -4 where j ij ij ij ij ij X R X R y + = -4 Thus the power loss on the series part of ranch can e otained as * * * * j i N ij ij ij ij j i ij S V V V Z Z y V V S = = Δ = -43 Considering = ij ij ij Z Z y a -44 * = j i V V V c -45 The coefficient c depends on the operational conditions, whilst the coefficient a is independent of the operational conditions and can e determined y the networ parameters and topology, since the impedance matrix us Z is always a full matrix and thus the coefficient a is a nonzero constant. The series active power loss of ranch can e expressed y nodal injected power as: N N N Q f P e Q P c a S c a L * j Re Re + = = = = = = -46

31 Chapter Loss Allocation in Distriution Systems: State of the Art where e = Re a c -47 f = m a c -48 Equation -46 is the formula used to allocate the series power loss, L, in ranch among all the users including power plants and consumers. Hence, the total variale networ losses of a power system can e otained summing the series loss of all the B ranches as L = B N e = = P + f Q B = = α P + β Q -49 where B = e = α -5 B = f = β -5 The coefficients α and β are the variale loss factors for the user at node Loss allocation method adopting a modified us admittance matrix [DaS5] The power system can e defined as a set of nodes and ranches with current injection at some or all nodes. This method considers a us as a generation us only if it delivers real power to the networ. The corresponding node could then e classified as a source or generator, with suffix g. f current injection occurs at a us with zero or negative real power it is classified as a sin or load, including lumped shunt components, with suffix l. Buses with zero injection are eliminated. Fixed power system losses occur whether a load or a generator is connected or not, and charging for these should e done on the asis of annual maximum demand or some other fixed rate rather than actual usage. Loss allocation factors will e derived here for variale losses only though these would include the effect of increased current in ranches that supply Politecnico di Torino May 8 3

32 Current Decomposition-ased Loss Partitioning and Loss Allocation in Distriution Systems fixed losses. t has also een assumed that, since current flows from a set of sources to a set of sins, either the set of sources or the set of sins can e regarded as responsile for losses, ut not oth simultaneously. The variale loss in a ranch can therefore e attriuted in two different ways. The steady-state power-flow equations relating the node current injections and complex us voltages V can e expressed y i = Y us v -5 or in partitioned form as i g Ygg = il Ylg Ygl v g Y ll vl -53 where suscripts g and l refer to generators and loads respectively. The dependency of loss on oth source and sin can e clearly expressed through the us resistance matrix L T * = i Rus i where the superscript T denotes the vector transposition operation. Equation -54 is the asis of -54 Z us loss allocation method [CoGK] descried aove. The use of the conventional admittance matrix imposes that loss fractions e attriuted to oth load and generation uses simultaneously Branch currents attriuted to sins Any power injection into a sin can e explicitly represented y a current injection. Power injections due to generation uses, however, need to e eliminated. The admittance component representation of the power injection of a generation us is found y dividing the complex current injection y the corresponding complex usar voltage, oth of which are found y otaining a load-flow solution for the networ. The so calculated admittance would not e physically realisale, since it would contain a negative conductance in this case. f the nodal admittance matrix is set up to include component representations of sources at nodes, ut the effects of loads are retained as current injections, Equation -53 then ecomes Y = il Y gg lg Y Y gl ll + Y G v v g l Jean SUMAL AKLMAL Doctoral Thesis

33 Chapter Loss Allocation in Distriution Systems: State of the Art where Y G is a diagonal matrix, with dimension equal to the numer of generation uses, whose th diagonal term represents the admittance equivalent to the complex generation at us for the power flow solution, that is, G, = V Y -56 The generation us voltages v g can then e expressed in terms of v l as v g [ Ygg + YG ] Ygl vl = -57 and eliminating i l v g from Equation -55 yields the expression [ Y ll Y lg Y gg + Y G Y gl ] v l = -58 On the other hand, the ranch current vector i B can e expressed in terms of the nodal voltages as B B v g B B v g Y C = [ A A ] B i = g l -59 vl vl B where Y is a diagonal matrix whose terms are the series admittance of the transmission lines, B B B B B B B C is the ranch-node incidence matrix, [ A g Al ] = Y C and matrices A g and A l result from the partition of columns corresponding to generation and load uses, respectively. The sustitution of Equation -57 into -59 allows the ranch current vector i B to e expressed in terms of the injected source currents, that is, i B = B [ ] Ygg + YG A A B g l l Y gl v where l is the identity matrix, and using v l from Equation -58 with B B l l l -6 i = K i -6 B B [ ] Y gg + YG Y A A gl [ Y Y Y + Y Y ] B g K l = l ll lg gg G gl -6 l B K l is a matrix which relates the ranch currents to the load us current injections. Note that, since the component representations of lumped shunts, loads and current injections exactly match the load-flow source currents, the ranch currents are numerically equal to those otained from the solution of the steady-state networ equations. Politecnico di Torino May 8 5

34 Current Decomposition-ased Loss Partitioning and Loss Allocation in Distriution Systems Branch currents attriuted to sources The ranch current vector i B can e expressed in terms of the injected source currents y adopting a procedure similar to that previously outlined for sins. n this case, the admittance component representation of a load would e found y dividing the sin current injection y the sin usar voltage, oth of which are again otained from a load-flow solution for the networ. The calculated admittance would e physically realizale as a passive RLC networ, in this case. f the nodal admittance matrix is set up to include lumped shunt components and component representations of loads at nodes, ut to retain sources as current injections, the procedure previously adopted gives i g Y = Y gg lg Y Y gl ll + Y L v v g l -63 where Y L is a diagonal matrix, with dimension equal to the numer of load uses and whose th diagonal term is L, = V Y -64 Now the relationship etween the ranch currents and the currents injection of generation uses is given y B B g i = K i -65 g with B B g [ A g Al ] Y + Y [ Y Y Y Y Y ] B K g = gg gl ll + L ll L Ylg lg -66 B K g is a matrix which relates the ranch currents to the generation us current injections. Note that in all cases source and sin representations and source and sin currents are identical in the load-flow and admittance matrix solutions. Equations -6 and -65 could also e used to decompose power flows from sources to sins. This would give useful information such as which generators were supplying the fixed 6 Jean SUMAL AKLMAL Doctoral Thesis

35 Chapter Loss Allocation in Distriution Systems: State of the Art losses modelled as loads in the networ, and how the networ capacity is eing used y the generators and loads Attriuted losses The total active power losses can e expressed in matrix form as L = B T * B B B T * B B T * B T * B B i R i = Kl il R Kl i l = i l Kl R Kl il -67 where B R is a diagonal matrix, whose terms are the series resistance of the transmission lines B T * B B and K R K l l is a symmetrical matrix. Matrices B K l and B K g assign coefficients relating ranch currents to current injections of loads and generators, respectively, ut we need to attriute ranch losses to loads and/or generators. The active power transmission loss corresponding to the th ranch is given y the current and the series resistance, respectively of the * R L = where and th transmission ranch. R are From Equation -6, the current of the us current injection as th ranch is expressed in terms of the load B K l il = -68 where B l K is the th row of the matrix B K l. Thus the power loss of ranch is L T B * B B = Kl il R Kl i l T * B B B T * i = l Kl R K l i l -69 T * B B B where the product K l R Kl provides the participation factors of load uses in the active power loss corresponding to the th ranch. The loss fraction assigned to the th load us is given y the product of its nodal factor y the corresponding us current injection. Summing over all the ranches the parcels corresponding to attriuted loss. th load us would give its total Politecnico di Torino May 8 7

36 Current Decomposition-ased Loss Partitioning and Loss Allocation in Distriution Systems Oserve that, since the loss is expressed in terms of the complex current injections, the participation factors are also complex numers. The transmission loss fraction attriuted to us may e expressed more rigorously as * T * B Re K R K T B B i l l, l, i l To attriute losses to generators, a similar process is carried out for the ranch currents attriuted to sources. n this case, the power loss is expressed as T * L = i g B T * B B K g R K g i g -7 with the vector of participation of generators given y the complex vector T * i g B T * B B K g R K g and the portion of the generation uses in the active power loss of the th transmission ranch calculated from T * B T * B B Re i g K g, R K g, i g Whether losses are eing attriuted to the set of sources or the set of sins, it would e possile for the total loss attriuted to a memer of that set to e negative. At first this seems counter-intuitive, as losses are always positive, ut an example of this would occur if the components of a ranch current attriuted to a large inductive load and a small capacitive load were almost antiphase. Losses attriuted to the last-mentioned would e negative ut oth the total loss and the loss attriuted to the inductive load would e less than the corresponding values when the capacitive load was switched off. No special treatment is needed for reactive power compensation sources, i.e. if power is transferred to the networ the component is treated as a source; alternatively, if power is drawn from the networ it is treated as a sin. f reactive power compensation sources ehave as sins and variale losses are to e attriuted to the sins, a choice can e made as to whether to include reactive power sources in the set of sins to which the losses are attriuted. Consider the example of the capacitance and conductance to earth of a line which can e modelled y a lumped lossy shunt capacitor at each end of the line. These components could either e treated explicitly as sins at each node or their contriution implicitly included in the us admittance matrix. f the former choice were made, then the contriutions to networ variale losses of these notional components could e evaluated. The networ line losses can therefore e expressed 8 Jean SUMAL AKLMAL Doctoral Thesis