Distribution System Losses Allocation Based on Circuit Theory Distribution System Losses Allocation Based on Circuit Theory with Distributed Generation with Distributed Generation

Electricity energy is the most convenient and ef ﬁ cient form of energy transmission and conversion. Nonetheless, losses take place while electrical power is traveling from power stations through transmission and distribution networks to the loads. Despite the high ef ﬁ ciency of electrical transmission and distribution networks, the huge amounts of power moving through these networks make the ﬁ nancial loss due to the lost energy very high and its management becomes a critical consideration for both utility providers and regulatory bodies. The ef ﬁ cient management of losses encompasses two major functions: the ﬁ rst is running the system in a way that minimizes the losses; the second is to fairly allocate the losses to the power system users. Equitable allocation of the losses is a pivotal aspect of ensuring the fair operation of the power system. This paper presents a new method for loss allocation in the distribution system. The method is based on circuit theory and is free of any assumptions or arbitrariness and allocates losses in a transparent and explainable way. The paper presents the details and the derivation of the new components of the proposed methodology along with application to two test cases. The two test cases used are the 28-bus radial system and the 70-bus meshed system. The method is compared with three of the common loss allocation methods and proved to be suitable for all distribution systems with any con ﬁ guration.


Introduction
L oss allocation refers to the methodology used to attribute the incurred losses among various entities within the power system, including consumers, generators, and distributors.Ensuring the right balance in loss allocation mechanisms is crucial for maintaining economic viability and for the satisfaction of all power system users.The interest in loss allocation started at the transmission level with the power system restructuring, where the power system from a vertically integrated system, monopolized by a single authority providing a service, into a market with different competing entities, trading the electrical energy as a commodity, at both generation and distribution levels.The transmission remained as an integrated and independent system run by an independent system operator.
In the trial to recover the cost of losses in the transmission system, the independent system operator was faced by two main issues.The first is that who will pay for this loss: The generation side or the distribution side, or should both share it, and in what percentage for each.There is no physical law to determine how much the generator or the load shares in the power loss caused by power flow from a generator to a load.Therefore, this issue is determined by the regularity bodies; different shares are used by different regulators; some make 50% each, some make it 30% for generation and 70% for loads while others make it all on the loads.The second concern is about how to apportion the losses allocated on each side to its individual participants.The second issue, allocating losses to individual participants, poses serious challenges.
The first attempt was to allocate losses proportional to power, the pro rata method (PR) (Ilic et al., 2013).
The PR method divides the total power loss between the generators and loads according to a predefined ratio, which may differ for different systems.Then the total loss share of the generation is apportioned between the individual generators in proportion to their power.Similarly, the total loss share of the loads is allocated to individual loads in proportion to their demand.A PR method has been used in the electricity market of mainland Spain with 100% of losses allocated to consumers (Exposito et al., 2000).The PR method is very simple, but the main criticism against it is that it does not account for either the nonlinearity of the power loss or the distance of the load from the source.To take the distance between the load and the source into account, the MegaWatt Mile (MWM) method was introduced (Lee et al., 2001), where the loss is allocated in proportion to the product of the active power and the distance it travels to the load.However, the fair allocation of transmission losses faced serious challenges.The challenges faced by the MWM method have arisen from the highly looped transmission network, which makes it difficult to determine where the power to each load came from and the resulting difficulty in determining the distance it traveled.Moreover, the nonlinearity of the power loss in terms of the power flows poses a further hurdle to loss allocation between partial flows within a line (Hota et al., 2022).Hence, for the sake of fairness and transparency of loss allocation, the loss allocation method should be able to clearly determine the flows caused by each load/generator in each line and to determine the shares of each of the partial flows in each line in an explainable and transparent manner.These challenges have led to several methods based on different techniques for loss allocation.It is to be noted that most of these methods have been introduced for loss allocation in transmission systems, while distribution system losses were not a great concern because of the simple pattern of unidirectional power flow.However, with the growing integration of renewable energy and other forms of distributed generation (DG) at the distribution system level, the load flow patterns have changed introducing a wide range of new concerns including protection coordination, voltage control, and power loss allocation (Elmitwally et al., 2016).
Loss allocation in the distribution network becomes a challenge because of the spread of DG, which results in a drastic change in the power flow magnitudes and directions of power flows through the system conductors.Unlike the case of the simple radial distribution where all the power flows from the utility grid down to the loads, the presence of the DGs results in power injection at a number of nodes scattered across the distribution system, which makes identification of which load power supplies from not a straightforward task.Moreover, a customer with DG facilities becomes not only a consumer but also a producer, and the term 'prosumer' is introduced to identify such customers.Such a customer will draw power from the grid when his own power generation is less than his demand and will supply the excess power to the grid when he has a surplus.In the first case, when the customer load exceeds generation, the customer will definitely cause an increase in the total power loss in the distribution system.However, when the customer generation is greater than the load, the net effect of the customer may be a reduction in the power loss in the system.This is because the excess generation will flow into the distribution system opposite to the main power flow from the utility causing a reduction in the power flow through the distribution system, hence reducing the losses.
The changing power flows in distribution systems to resemble that in the transmission system has been the motive for most utilities as well as researchers to use the loss allocation methods of the transmission system for the same purpose in distribution systems.Of course, in some cases, some modifications were made to account for the special features of the distribution system such as the unbalanced loading on the three phases and the weakly meshed structure.Therefore, in the following survey, the loss allocation methods for transmission systems are reviewed, as they represent the main approaches for loss allocation, followed by the methods for distribution system loss allocation.

Loss allocation methods
Proportional sharing (PS) methods consider the network node as an ideal 'mixer' for incoming flows, making it impossible to know which of the incoming electrons goes toward which outgoing line (Conejo et al., 2002).This concept cannot be proved or refuted; it rather agrees with common sense and the widely accepted fact that electricity is inherently indistinguishable.The essence of the PS principle is the assumption that, at any node, the inflows distribute themselves proportionally to the outflows.In other words, each inflow contributes the same fraction to each of the outflows.This fraction equals the ratio of the individual inflow to the total inflow at the node.This assumption enables tracing the flow from loads/ generators through the network.Hence, each load/ generator shares of losses through the branches can be determined based on their contributions to the flows in the branches.Loss allocation to loads by this method starts at a sink node, which receives power from all neighboring nodes; and then moves to nodes preceding it on all the feed paths until reaching the source nodes.As the method assumes lossless lines, it is advised to carry out the whole process using power flows at the receiving ends of the lines at first and repeat it a second time using the flows at the sending ends of the lines.The final loss allocation is obtained by having the average of the values obtained in the two rounds.Loss allocation to generators uses the same steps with the only difference being that the process starts at source nodes moving toward the sink nodes.The reasonable assumption, observance of physical laws, and the easy tracing of power flow through the network have given the PS principle wide acceptance and several methods for transmission loss allocation (Bialek, 1997;Elmitwally et al., 2015;Kirschen and Strbac, 1999;Strbac et al., 1998) have been introduced based on it.PS has the advantage of avoiding volatility, negative losses, and allocation imbalance.However, the criticisms of the method are: the arbitrary division of losses between generation and load, the assumption of lossless network while tracing, the power flow and the linear apportioning of line loss individual partial flows neglecting the nonlinearity of power loss, the need to perform power flow tracing twice, and allocating active power loss and reactive power loss separately neglecting the cross effects of active/reactive power flows on reactive/active power losses.Also, there is a lack of solid physical basis and circuit theory.Improvements to the PS techniques have been introduced in (Abdelkader, 2006a(Abdelkader, , 2007a(Abdelkader, , 2007b).An efficient algorithm for power flow tracing that considers power at both ends and the loss allocation is determined by the difference between the flow caused by each load at both ends of the line (Abdelkader, 2006a).Also, a method for complex power tracing and loss allocation to loads is presented in (Abdelkader, 2007b) and to generators and loads in (Abdelkader, 2007a).
The nonlinearity of power loss and apportionment of losses to a group of currents flowing through the same line represented a major concern to loss allocation.A physical solution to this problem has been introduced in (Abdelkader, 2006b), where the line is assumed to be divided into a number of individual subconductors equal to the number of currents with an area proportional to the individual currents.This allowed determining the losses caused by each of the individual currents in a clear explainable manner with zero arbitrariness.The resulting loss allocation equation was also proved by game theory and the Shapley value for fair allocation in (Molina et al., 2010).
A circuit theory approach has been used in (Abdelkader, 2010;Abdelkader and Flynn, 2010) to determine the generator's and load's responsibilities for transmission losses.In (Abdelkader and Flynn, 2010) proof of that losses in line fed from both ends has two components: the first is due to the load current and the second is due to the voltage difference between the two ends.It was also proved that these two components can be calculated separately despite the nonlinearity of power loss.Generalization of this concept to a general power with any number of generators and loads has been introduced in (Abdelkader, 2010), where loads share in the transmission losses were determined by assuming that all generation busbars are voltage sources, while the generation shares of losses were determined due to the circulating currents between generators with all loads disconnected.
Other methods based on circuit theory were also introduced using Z bus (Conejo et al., 2001), where all power injections and loads were converted into currents, and these currents were used with the real part of Z bus to determine the loss allocated to each node in the system.However, the existence of Z bus is not guaranteed for all systems, especially distribution systems with short lines having negligible capacitances.Moreover, relying on shunt capacitances to form the required connection to the ground required obtaining Z bus , while treating generators as current sources change the current flows to be from both generators and loads to ground through capacitances as explained in (Abdelkader et al., 2014).The result is inaccurate and even incorrect loss allocation.As a solution to these problems and inspired by the transformer model, which separates the losses into two major parts, one is voltagedependent, and the other depends only on the load current; a method for flow tracing has been introduced in (Abdelkader et al., 2014).In that method the network equations are put in the form of multiterminal transformer equations keeping the system features unchanged; the generation buses are treated as voltage sources connected to the multiterminal primary side of the transformer, whereas load buses are treated as loads drawing current from the multiterminal secondary side.

Distribution system loss allocation methods
The maturity and availability of transmission loss allocation methods make it an immediate candidate for use in distribution system losses by making the required adaptation to fit the distribution system case.This is why many methods for distribution system losses are either modified versions of the methods previously introduced for the transmission system or used as it is in some cases.However, the distinct features of distributed features of being mainly radial or weakly meshed, having mostly unbalanced loading, and lack of high-quality sophisticated measuring instruments, like those available in transmission systems, have given rise to a wide range of methods based on different techniques for different purposes and different system structures/conditions.A branch-oriented power summation algorithm is utilized in (Atanasovski and Taleski, 2011) to decompose the receiving end power of each as a sum of the injected powers of all nodes considering the DG output negative and the load positive injections.A relationship is then established between branch losses and the node injections and used for loss allocation in the radial distribution system with DGs.The method is a variant of PS with the only difference in treating the cross terms of power terms as a quadratic function of node injections.One issue with this method is neglecting the series reactance and shunt capacitance of the lines while doing the power summation.The same method was developed in (Atanasovski and Taleski, 2012) to consider the daily variations of loads and DG outputs by statistical representation of the daily load and generation curves.The uncertainty of load and generation in a radial system has been modeled using fuzzy logic to consider the losses in DG allocation using a genetic algorithm (Ganguly and Samajpati, 2015).
Flow decomposition was also used in (Ghofrani-Jahromi et al., 2013) for loss allocation in a radial system with DGs in a three-step process, where the total active and reactive power allocation to DGs by tracing the power flow starting from the source node.Next, the power flow starting from load nodes and the total losses are allocated to loads.Finally, the allocated loss are normalized to make the total loss allocated equal to the actual loss.This is a typical PS algorithm the same as that used in the transmission system, and the same technique is used also in (Kumar et al., 2017) despite the claims of being a circuit theory-based method.
The fair axioms of game theory have been used to prove the fair distribution of losses to partial flows through a line in (Molina et al., 2010), which was found to be the same as that obtained on a physical basis in (Abdelkader, 2006b).This has paved the road to different applications of game theory methods for loss allocation in distribution systems (Amaris et al., 2018;Farsani et al., 2015;Kumar andKhatod, 2021, 2023;Shaloudegi et al., 2012;Sharma and Abhyankar, 2016).Game theory is used with Locational Marginal Pricing to develop a method for remunerating DGs for their contribution to loss reduction in the distribution system (Shaloudegi et al., 2012) and for contribution to loss and emission reduction in (Farsani et al., 2015).A cooperative game approach for determining the fair allocation of loss penalties and loss reduction rewards to a weakly meshed distribution system participants is introduced in (Sharma and Abhyankar, 2016) by deriving an analytical solution using the Shapley value.Also, in (Amaris et al., 2018) an analytical solution to the loss allocation problem, in both radial and meshed distribution networks, is derived but this time using Aumann-Shapley approach.Authors of (Kumar andKhatod, 2021, 2023) used the t-value concept of a cooperative game theory to allocate losses in radial and weakly meshed distribution networks with DGs (Kumar and Khatod, 2021) and for loss allocation in unbalanced three-phase distribution networks (Kumar and Khatod, 2023).Due to the requirement of applying the t-value concept, loss allocation to loads is done separately from the allocation of remuneration to generators.
Recognizing the nonlinear nature of power loss and the resulting cross terms between partial flows sharing the same line and/or between phase currents in a three-phase unbalanced distribution system, several methods have been developed to deal with that.Quadratic and geometric methods were used in (Usman et al., 2019) to consider the effect of phase loading on other phases.However, this method avoided allocating neutral losses to phase currents.Authors of (Ayres et al., 2013) presented a method for estimating the time variation of power loss in a distribution system with DG using the information of only one solved power flow where second-order power flow sensitivities were used to model power loss to avoid the power flow solution at each variation.
The authors in (Kumar et al., 2019) developed a circuit theory-based method for loss allocation in active distribution systems, emphasizing fair power loss distribution, reactive power effects, and equitable compensation for DG owners.The study in (Bhand and Debbarma, 2021) introduced a graphbased transaction-tracing loss allocation scheme for efficient peer-to-peer energy trading in 3Phase-4Wire systems, validated on a 33-bus network.A blockchain-based peer-to-peer electricity trading market with voltage-constrained adjustment and power loss allocation mechanisms to improve efficiency, accuracy, and transparency in distribution networks is proposed in (Xu and Wang, 2023).The objective of (Koochaki et al., 2023) is to present a power tracking method for accurate loss allocation across any network topology, addressing the complexities of increased transactions in energy networks.A bidirectional loss allocation method based on Virtual Contribution Theory and Power Flow Tracing to accurately and efficiently calculate power losses and fairly allocate them between generators and loads in active distribution networks is suggested in (Zhao et al., 2023).The work in (Moret et al., 2021) suggested involving transmission and distribution system operators as active market participants to manage grid constraints and line losses, analyzing policies for fair outcomes in decentralized electricity markets with multi-bilateral trades.
The research gap in loss distribution for active distribution systems is the inadequacy of traditional approaches to deal with the complexities presented by modern power networks.These include bidirectional power flows, high penetration of distributed energy resources, and increasingly meshed network topologies.Current methodologies fall short in ensuring fairness, computational efficiency, and real-time dynamic allocation.Furthermore, the need for transparency and stakeholder understanding, alongside the impact of evolving regulatory and market structures, remains underexplored.The wide range of proposed approaches to allocating distribution losses underscores the importance of the problem and the urgent need for a solution that meets the needs of all participants in the distribution system.Many of these methods are scientifically robust and employ advanced techniques.They often fail to address the practical requirements of distribution system users.To achieve universal acceptance, the loss allocation method must be simple and comprehensible to the diverse range of distribution system users.In addition, it should be grounded in solid physical and theoretical principles, capable of deconstructing allocated costs to their most granular levels, and able to explain the rationale behind each component of the cost.In this context, this paper introduces a method for loss allocation in distribution systems, based on straightforward circuit theory.This approach is designed to be easily understandable, facilitating the explanation and justification of the loss allocated to each customer.

The proposed method
For the loss allocation to be fair and explainable requires solving two major issues.The first is to determine the flow caused by each individual transaction through each of the distribution lines, i.e., tracing the flow of the individual loads through the distribution network.The second is an acceptable method that transparently apportions the losses through each line to the partial flows through it.In the following sections, a proposed general flow tracing method for all distribution system configurations is presented.General proofs of the validity of the proposed method to all distribution system configurations have been derived mathematically for general cases of each of the possible configurations.The flow tracing process decomposes the branch currents into a number of components, each of which is attributed to an individual load.This decomposition is represented by the current contribution matrix, C, that determines the branch currents as combinations of fractions of the load currents.Also, a method for loss apportioning between partial flows is also introduced.

General method for flow tracing in distribution systems
The goal of the flow tracing process is to determine the current contribution matrix, C, which determines the components of each branch current caused by each load current as shown in ( 1 where I br is the branch current vector; I L is the node current vector; N br is the number of branches; and N is the number of nodes.
The method proposed for determining the contribution of load currents to branch currents is based on determining the contributions of load currents to the voltage drops at the distribution system nodes.Hence, the contributions to voltage drop across the branches can be obtained.Dividing the voltage drop caused by a load current in a given branch by the impedance of that branch determines the current flow caused by the load current into that branch.While doing this, all sources are assumed to be ideal having zero internal impedance.This assumption is necessary to make sure that the current flows determined through the branches are due to the load currents only with no contribution from the circulating currents between the sources.
The steps of the proposed method can be described as follows: Determine the branch-to-node incidence matrix A. This is the N Â N br matrix, with its rows corresponding to the system nodes and columns corresponding to the system branches.
Form the branch admittance matrix, y br , which is a diagonal N br Â N br matrix with its diagonal elements are the admittances of the distribution system branches.
Determine the bus admittance matrix: Determine the bus impedance matrix: Load current contributions to voltage drops at all the load nodes are determined as where diagðI l Þ is the N Â N diagonal matrix with its diagonal elements are the load currents.The ij th element of DV bus will be the voltage drop caused by the i th load current at the j th node.Contributions of load currents to the voltage drop across the system branches can then be calculated as Bearing in mind that I br ¼ y br DV br yields The load current contribution matrix to branch currents for a general distribution system can thus be determined as follows: Equation ( 7) provides a means for determining load current contributions to branch currents for any distribution system whether it is radial, ring, or meshed in a structured manner.In the following, the load contribution matrices for the three above cases using the proposed method and compared with those obtained above to test the validity of the proposed method.The symbolic matrices involved are manipulated using Mathematica software.In a ring distributor, as shown in Fig. 1, each load current will have two possible paths from the source to the node where it is tapped off.Hence, the current will divide itself between the two paths.There are two big differences between this case and radial/ tree distributor cases.The first is that the current contribution factors are no more 1's and 0's, it becomes fractions.The impedance of the distributor sections will have its say in the current contribution factors as the current divides itself in inverse proportion to the impedances of the available two paths.Moreover, if the different sections of the distributor have different R=X ratios, the load current contribution factors may be complex numbers.
The distributor has five sections with four loads tapped off all nodes other than the source node, which are four nodes in this case.Assuming the positive direction of current flow is from the source to node 1 as indicated by the arrow on the figure., branch currents are determined by the superposition of individual load current contributions to each branch.Consider Fig. 2 where the load current i 1 is shown and all other load currents are set to zero.In this case, the current i 1 flows from the substation to load over the distributor and back to the substation through the ground.Node 1 divides the ring into two parallel paths as shown.Hence, i 1 divides between the two parallel paths into i 1þ , flowing in the assumed positive direction through branch 1, and i 1À , flowing through branches 2, 3, 4, and 5 opposite to the assumed positive direction, as shown in the figure.
According to the current divider rule, i 1þ and i 1À can be calculated as follows: through branches 2; 3; 4; and 5; where Z t is the total impedance of the distributor, In a similar way, the divisions of other load currents over the corresponding parallel paths can be determined as follows: through branches 1 and 2; i 2À ¼ À z 1þ z 2 Z t i 2 ; through branches 3; 4; and 5: through branches 1; 2; and 3; i 3À ¼ À z 4 þ z 5 Z t i 3 ; through branches 4 and 5: through branches 1; 2; 3; and 4; i Thus, the load current contribution matrix will be as follows:.
4.2.2.Derivation of the C matrix using the proposed method For the system of Fig. 1, the incidence matrix A and the branch admittance matrix, y br can be as follows: Using A and y br , the bus impedance matrix can be obtained as below: Plugging the values of A, y br , and Z bus in (7), the contribution matrix C can be found to be the same as that derived above using current division and superposition.

Loss apportionmentto partial currents sharing a conductor
In (Abdelkader, 2006a), a loss allocation formula was derived based on physical laws by assuming that all partial currents in a conductor distribute themselves at the same density over the conductor area.Loss allocated to the current I k flowing through a conductor having resistance r c and carrying a total current, including I k , of I c is determined as follows: In this work, the same equation has been proved on a purely physical basis with no assumptions.The definition of the voltage difference between two points is the work done (the energy consumed) in moving 1 C of charge between the two nodes.Hence, if the charge is transferred by the flow of a current I k across a voltage difference, DV is q k , the energy dissipated due to the flow of the current I k can be calculated as follows: Bearing in mind that a conductor has resistance r c and carrying a current I c the voltage difference between its ends DV ¼ I c r c and the power loss equals the time derivative of the work done in transferring the charge, the power loss is found to be In terms of power, the loss allocated to the k th load can be derived from (11) as follows: where P t is the total power flow through the conductor and P loss;t is the total power loss through the conductor.The loss allocation formula presented in (Abdelkader, 2006a) has been obtained by the proposed arbitrary and assumption-free current division over the conductor surface, which is a clear double proof of its validity.It is also proved that power loss caused by a partial current through a conductor equals the product of the magnitude of that partial current and voltage drop across the conductor, which also resulted in the same formula for loss allocation.All these have proved to be perfect for the DC case.Although the AC case is different; the conductor impedance is not pure resistive, there is the reactance; the total current is not the algebraic sum of the partial currents with different phase angles.However, this issue has been resolved in (Abdelkader, 2006a), where the individual current projections on the direction of the total current have been used in the same way as in the DC case.It is worth noting that it has been proved in (Abdelkader, 2007b) that using current projections for loss allocation is the only way that satisfies the fair allocation as defined by Shapley values.Hence, it is adopted in our work and the loss in a conductor is apportioned to the individual flows in terms of currents or in terms of the parameters in equations ( 13) and ( 14), respectively.
where P k , Q k are the active power and reactive power of the k th load, and P c , Q c are the total active power and reactive power flows through the conductor.

Loss allocation to loads/DGs
It might have become clear that loss allocation in the distribution system involves two steps.The first is to trace the load/DG currents throughout the distribution system to determine the flow in each caused by each load/DG current through each of the distribution system branches.The second is to apportion the losses in each of the distribution system branches between the partial currents caused by the system loads through it.These two steps are explained in detail in the previous two sections.This section presents the method of determining the shares of each load/DG in the total system losses.The section presents the application of the two processes in a structured manner to determine the loss shares of the individual loads/ DGs.
The data required for the method is just the updated system configuration and branch parameters as well as the load current/power measurements.In the following, index i will be used for nodes whereas j for branches, I l is the load current vector, I c is the branch current vector, I lc is the matrix of partial flows with its element I lc ði; jÞ is the partial current in line j due to load current i, and rðjÞ, xðjÞ are the resistance and reactance of branch j.P loss;i is the total loss allocated to load i. P l ðjÞ is the power loss in the branch j, and dP l ði; jÞ is the fraction of P l ðjÞ caused by load i.
With the distribution system topology and branch parameters known, the incidence matrix A and branch admittance matrix y br can be determined and used to determine the load-to-branch contribution matrix as explained in sec.4.1 using (7).Hence, the partial flow matrix can be determined as follows: where diagðI l Þ is a diagonal matrix with its diagonal elements being the load current.It is worth mentioning that if matrix C is multiplied by the load current vector I l , the result will be the branch currents vector I c , but if multiplied by the load current diagonal matrix will yield the partial flow matrix I lc .The loss portion of the i th load current through the j th conductor is determined as follows: The total loss allocated to the i th load is obtained by the summation of the partial losses allocated to it in each conductor, that is It is possible to determine the loss allocated to all of the load currents in one go using matrices as follows: where dP l is a column vector with N l elements representing the allocated losses to each of the load nodes; RðI lc Þ is an N l Â N br matrix of the real parts of the partial currents caused by loads through the branches and is determined as explained above; IðI lc Þ is an N l Â N br matrix of the imaginary parts of the partial currents caused by loads through the branches and is determined as explained above; RðI c Þ is a column vector of N br values of the real parts of the net branch currents; IðI c Þ is a column vector of N br values of the imaginary parts of the net branch currents; and R c is the branch resistance matrix, N l Â N br diagonal matrix.

Case studies
This section presents the application of the proposed method to determine the loss allocation to loads/DGs in standard test systems.The 28-bus and the 70-bus distribution systems are used as test systems.The 28-bus system is a radial system, while the 70-bus is a meshed distribution system.

Case 1: The 28-bus test system
The single line of the 28-bus test system is shown in Fig. 3.It is clear from the figure that the system is radial.The branch data are listed on the left of Table 1.The bus data, including the specified bus loads and the voltages obtained from the power flow solution, are listed on the right of Table 1.The standard data of the 28-bus system are used, and the power flow solution is carried out using the Mat-Power toolbox on MATLAB.The power flow solution results are listed in Table 2 with the base values being 1 MVA and 11 kV.Bus 1 is the utility bus supplying power to all other buses.Power at all buses other than bus 1 are loads.Load node currents are listed in Table 2.
The power flow solution results are used to determine the loss allocation to the load nodes using the proposed new method as well as three of the common methods: the PS, the PR, and the MWM methods.While calculating loss allocation using the MWM method, the resistances of the lines were used as an indication of the line length.Both active power and reactive power losses are allocated to loads using the proposed method introduced in Section 4: the PS method, the PR method, and the MWM method.The PS is applied with the current projection as introduced in (Conejo et al., 2002).
The system under study has a total load of 761.104MW and 776.419MVAr.The total losses are 68.819kW and 46.042 kVAr.The total power loss represents 9.043% of the total active power load on  the system and 5.93% of the total reactive power load.Table 3 lists both the active power loss and reactive power loss allocated to each load node of the system as determined by the four methods.It can be noticed that all of the four methods allocate exactly the total losses both active and reactive.This means that the total allocated losses equal the actual total losses and no further adjustments are required.
It is clear from the table that the PR method allocates the same loss to loads having the same power regardless of their place, while this is not the case for all the other methods.
For a more comprehensive comparison of the methods, the active power of the loads and the active power loss allocated by each of the methods are plotted in Fig. 4. The active power load is plotted on the main vertical axis, to the left, whereas the active power loss is on the secondary axis.The allocated power loss is presented as a percentage of the load power.The load active power is presented by the solid line.The percentage power loss allocated by the proposed method is presented by the dashed line and that allocated by the PR method is presented by the diamond markers.The dotted line presents the MWM method, and the dashed dot red line presents the PR results.From the figure, it can be observed that the two methods give almost the same loss allocation.The %age difference between the two methods is limited to À0.4%e1.2% as can be noticed from Fig. 4.
The following observations can be made from the figure: The PR method allocates an identical percentage of power loss to all loads, irrespective of their location within the distribution network.Specifically, this percentage corresponds to the proportionate total loss relative to the overall system load.While this approach might initially appear equitabledsince each MW of load incurs the same share of distribution lossesdit may not be perceived as fair or acceptable by all consumers.In particular, consumers located near the supply source are likely to object to bearing the same loss burden as those situated at the farthest ends of the distribution system.This uniform allocation fails to account for the varying distances and associated transmission losses experienced by different consumers, potentially leading to dissatisfaction and perceptions of inequity among those closer to the supply point.
The MWM method allocated losses, as expected, show dependence on the distance from the source node.It can be noticed from the figure the loss allocated to the nodes near the source node is less than that allocated to the far nodes.However, it can also be noticed that the loss allocated to the nodes near the source is less than the losses allocated to the same nodes by both the proposed and the PS methods.The opposite is true for the far nodes, which have the losses allocated by the MWM method greater than those allocated to it by the new and the PS methods.Hence, the MWM method favors the near nodes on account of the far nodes which bear more than it caused.The reason for that is that an MW traveling one mile through a heavily loaded line causes more losses than an MW traveling a mile along a lightly loaded line, while the MWM mile gives the same weight for all the MW miles.
It is evident that the PR method disproportionately allocates power loss to nodes situated closer to the source, assigning them a greater share of the losses than they contribute.Conversely, nodes located farther from the source are allocated less power loss than they are responsible for generating.However, the MWM method reverses this pattern by assigning a higher proportion of power loss to the distant nodes and a lesser amount to the nodes near the source.This discrepancy in loss allocation by both methods leads to outcomes that are neither fair nor justifiable to customers.The inherent imbalance in the allocation process makes it challenging to provide a transparent and equitable explanation to consumers, thus undermining the credibility and acceptance of the loss allocation methodologies.
It can also be noticed that the losses allocated to individual loads by the new method and those allocated by the PS method are almost equal.This is a very important notice, which verifies that the two methods allocate the right loss to each load.This is because the correct result of any problem should be obtained whatever the method used is.In other words, if different methods yield the same result for a particular problem indicates that the result is correct, and the methods are right.Fig. 5 depicts the %age difference between the methods, that is the difference between the power loss allocated as a % age of the power loss allocated by the new method.
As can be noticed it is limited to À0.3%e1.2%,which is too small to be considered a difference between the two methods.
Fig. 6 illustrates the load power at each bus as a percentage of the total system load, alongside the loss allocated to each bus as a percentage of the total system loss, as determined by both the PS and the new methods.A clear agreement between the two methods is evident.It is also observed that buses with identical loads are allocated different losses  based on their location within the distribution system.For instance, buses 6, 7, and 8, despite having equal loads, exhibit varying loss allocations: bus 7 is allocated more loss than bus 6, and bus 8 is allocated more loss than bus 7.This variation reflects the impact of distance from the supply node, with bus 7 following bus 6, and bus 8 following bus 7 along the main trunk of the distribution system.Similar patterns are observed for buses 11, 12, 14, and 15 on the same lateral branch.For buses 26, 27, and 28, bus 26 has a larger loss allocation compared with buses 27 and 28.This is attributable to the fact that bus 26 is located on a different lateral and is not followed by buses 27 and 28, as depicted in Fig. 3.
Comparable results are observed for reactive power loss allocation, where the PS method yields values nearly identical to those obtained by the new method, mirroring the pattern seen in active power loss allocation.The difference in reactive power loss allocation between the two methods ranges from À1.2% to 0.35% relative to the reactive power loss allocated by the new method.This indicates that the PS method allocates reactive power loss with the same level of accuracy as it does for active power loss.However, it is important to note that these findings apply specifically to radial distribution systems.The scenario may differ for meshed distribution systems.The subsequent section will present the results of loss allocation in a meshed distribution system.

Case 2: The 70-bus test system
The singleeline diagram of the 11 kV 70-bus test system is shown in Fig. 7.The system has utility connections at both bus 1 and bus 70.The system is normally operated as a radial system with the dotted branches open.However, in this test case, all the normally opened branches are closed for comparison of the proposed method and the PS method.
The system also has distributed generation, Fig. 6. %Age load and %age active power loss allocated for the 28-bus system.Fig. 7.The 70-bus test system.photovoltaic (PV) systems at buses 13, 26, 41, and 61.Branch and bus data are taken from (Gad and Gabbar, 2020).The standard data of the system are used, where the power flow is carried out to determine the bus voltages and branch flow.Branch flows are used by the PS method to allocate losses to the different buses as described in (Conejo et al., 2002).The determined bus voltages are used along with the bus-specified powers to determine the load current at each node and then use these currents to allocate losses to the different loads using the proposed method.To avoid overlengthening the paper, these data along with the solved power flow and detailed loss allocation will be made available on request, and only some figures will be used for comparing the proposed method and the PS method.
For this test case, the loss allocation carried out by the new method is compared with the loss allocation determined by the PS method.The objective of this comparison is not to validate or justify the new method, as its foundation rests on solid physical and theoretical principles.Rather, the primary aim is to demonstrate the necessity for the new method as being the one based on sound theory rather than assumption, even if it seems irrefutable.It is worth noting that neither the PR nor the MWM methods will be tested in this instance, as they have previously proven inadequate in accurately allocating losses in the very simple case of a radial system; consequently, their accuracy is expected to be even poorer in more complicated looped systems.
Fig. 8 illustrates the percentage of active power loss allocated to each load by both the new method and the PS method.The discrepancy between the two methods ranges from À15 to 10%, which is substantially largerdapproximately tenfold or moredthan the differences observed in the radial system.Consequently, the accuracy of active power loss allocation using the PS method is significantly compromised in looped systems or distribution networks with distributed generation.The new method demonstrates a consistent pattern by allocating higher power losses to loads with greater power demands and to those located farther from the source nodes.This consistency underscores the method's reliability in reflecting the physical realities of power distribution networks.
Fig. 9 illustrates the results of reactive power loss allocation using both the new method and the PS method.The graph depicts the reactive power loss allocated to each load by both methods, expressed as a percentage of the reactive power demand of each respective load.The figure presents the reactive power demand for each load.It is evident from the figure that the discrepancy between the reactive power losses allocated by the two methods is more pronounced than that observed for active power losses.Specifically, the difference in loss allocation between the two methods ranges from a decrease of 15% to an increase exceeding 30%.
Fig. 10 presents the load power at each bus as a percentage of the total system load, alongside the corresponding loss allocated to each bus, also expressed as a percentage of the total system loss.A comparison between the PS method and the newly proposed method reveals a significant discrepancy between the two approaches.Similar to the findings in the radial system, buses with identical loads exhibit different loss allocations due to their varying locations within the network.Nevertheless, despite the differences observed between the methods, the percentage of loss allocated consistently follows the same trend as the percentage load distribution.
The results of the allocation of reactive power losses are also depicted in Fig. 11, revealing a significant difference in the percentage allocation of reactive power losses between the two methods.This dissimilarity extends not only to the magnitudes but also to the profile and distribution of losses across different buses.The discrepancy arises from the fact that the PS allocates losses locally, without considering what is happening at the first neighboring node.This approach fails to account for the sources of flows passing through a node.Considering that these flows may originate from different sources with unequal voltages, a portion of the flow may result from voltage differences between the sources.Such flows come out from one source, traveling through the network to get into another source, and are unrelated to loads in any way.This type of flow is a major contributor to disruptive loss allocation, as highlighted by (Abdelkader et al., 2014).The PS method does not address this issue.Even when all the sources have the same voltage, they all contribute to supplying all  loads.The current of each load will come from two or more sources over different routes, which the PS method cannot deal with as it treats every node independently as a perfect mixer to all the inflows regardless of which route it came over.The new method, being based on physics and circuit theory, can accurately determine where each flow comes from and where it goes and hence can accurately determine the loss incurred by each flow.

Conclusion
This paper presented a method for loss allocation in distribution networks including distributed generations.A circuit theory-based technique to determine the contribution of individual node injections into the flows through the distribution network branches.Active/reactive power losses through branches are then apportioned between the partial flows caused by node injections.Hence, the proposed method has a sound theoretical basis, and therefore the claim of having the highest accuracy, transparency, and explainability in loss allocation can be easily proved and accepted.The proposed method makes it possible to fairly determine the loss shares of loads and to determine the distributed generation's contributions to loss increase/reduction in a transparent manner.For this purpose, the distributed generations injections are treated the same way as the load powers with the only difference being that they are considered as negative loads.Unlike most of the available methods, the method introduced in this work has no specific assumptions.It is rather based on physical grounds and circuit theory.
The proposed method has been applied to different systems with different numbers of buses and configurations and compared with the most common methods.The comparison shows the shortages of the common methods as the system complexity and interconnectivity increase, while the proposed method keeps a consistent performance in all system configurations being free of any assumptions.

4. 2 .
Flow tracing in a ring distribution system 4.2.1.Derivation of the C matrix using basic circuit laws

Fig. 5 .
Fig. 5.The %age difference between the loss allocation of the new and the PS method.

Fig. 10 .
Fig. 10.%Age load and %age active power loss allocated for the 70-bus system.

Table 1 .
Branch and bus data for the 28-bus system.

Table 2 .
Load currents and line flows for the 28-Bus system.

Table 3 .
Loss allocation for loads of the 28-bus system.