The most appropriate loss allocation techniques start from the power flow results and exploit the concept of marginal losses to formulate suitable indicators to express the positive or negative contribution of generators and loads to reduce the system losses. These techniques can generally be partitioned into derivative-based and circuit-based. The correctness in the formulation of the computational techniques depends on avoiding the occurrence of the loss allocation paradox identified in Carpaneto et al., 2006a.

3.1.1. Derivative-based methods

A general expression for the derivative-based methods can be built by approximating the total losses L in function of the net node power vector p in quadratic form (Carpaneto et al., 2008b)

L = L0 + b Tp + -2- p T Ap

symmetric matrix. Higher-order terms are neglected. The terms L0 and b are null in the absence of shunt circuit components (e. g., shunt line or transformer parameters) and of circulating currents depending on different voltage settings at different PV nodes (as the ones introduced by local generators operating in the voltage control range).

By indicating the derivative of the total losses with respect to the vector p as Lp analytical elaborations it is possible to obtain

2 L = (LTp + bT )p (17)

from which the vector V containing the loss allocation coefficients is defined in such a way to represent the total losses as L = VT p :

The expression (18) indicates how the loss allocation vector depends on the derivative of the total losses with respect to the load vector. In particular, the derivative Lp alone is unable to

provide a loss allocation vector, and reconciliation to the total losses is needed by dividing by 2 even in the case in which b = 0.

(L і L і A Lp + Lp F|P1 * P2 |

From another point of view, the exact variation of the losses defined in the quadratic form (16) with respect to load power variations can be expressed by considering two generic net power vectors p1 and p2, leading to the total losses L(1) and L(2), respectively, by using the average value of the derivatives calculated in the two configurations (Carpaneto et al., 2008b):

The expression (19) is independent of L0. If p1 = 0 and p2 = p, the equation providing the total losses L = L(2) becomes

If b = 0, the loss coefficient vector v = 12 Ap can be directly used to represent the total losses as L = VT p, with no need of reconciliation. If b # 0, the product VT p gives an approximation of the total losses.

The above illustration of the properties of the total losses is useful to discuss the formulation of some derivative-based methods proposed in the literature. In general, the matrix A and the vector b are not known. The methods are then elaborated by using the power flow state variables x (voltage magnitudes at the PQ nodes and voltage phase angles at all nodes, slack node excluded), and the Jacobian matrix Jx containing the derivatives of the power flow equations with respect to the vector x. Two methods based on expressing the total losses in quadratic form have been presented in Mutale et al., 2000:

1) the Marginal Loss Coefficients (MLC) method, in which an auxiliary vector о is calculated by solving the linear system

and reconciliation is needed to get L = ^T p, since the product оT p in real systems approximately represents half of the total losses, on the basis of the same concepts discussed in (18), obtaining the loss allocation vector

2) the Direct Loss Coefficients (DLC) method, using the Taylor series expansion of the total loss equation around the no-load conditions, in which an auxiliary vector у is calculated by solving the linear system

-T 1

JT Y = – H Ax (23)

in which the rationale of using the average Jacobian matrix Jx calculated from the Jacobian matrices in two configurations (the current operating point and no-load) is based on the same concepts discussed in (19), and the Hessian matrix H of the loss equation is calculated at the current operating point. If the conditions corresponding to b = 0 are satisfied, there is no need for reconciliation (^ = y ), otherwise the product yT P gives an approximation of the total losses.

In distribution systems with voltage-controllable distributed generation, the definition of the MLC and DLC methods is affected by the fact that the voltage magnitude of a PV node is not a state variable in the power flow equations, thus the loss allocation coefficients are undefined for PV nodes. However, as remarked in Section 2.3.3 for synchronous generators, the local generators could operate in voltage control mode only for a portion of the total time interval of operation, being constrained to the reactive power limit in other time periods. These aspects may cause a discontinuity in the time evolution of the MLC and DLC coefficients. However, a voltage-controllable local generation unit is typically connected to the grid through a local transformer, that can be considered as integral part of the local system. As such, it is possible to adopt a two-step technique of analysis (Carpaneto et al., 2008b). In the first step, the power flow is solved by taking into account the detailed characteristics of the local generator (including its voltage control system) and of the local transformer. In the second step, each generation unit (generator and transformer) is replaced by the net power injected into the distribution network calculated from the power flow, thus constructing a reduced network in which no PV node appears. The losses in the local transformer are part of the local system and are correctly excluded from the loss allocation. Any possible reactive power limit enforced or other specific modelling details are implicitly

embedded in the net power representation. The loss allocation is then calculated for the reduced network by using derivative-based or other methods. For the derivative-based methods, the Jacobian matrix to be used for loss allocation purposes has to be recalculated also when the power flow solution has already used a method requiring the construction of a Jacobian matrix, since the number of nodes in the reduced network is different with respect to the one of the original network. One critical aspect is the calculation of the no-load configuration to be used in equation (23) when one or more local generator operate as a PV nodes. In this case, being the loss allocation calculated on the reduced network, the effect of the voltage setting at PV nodes cannot be taken into account.