# Exact Cross-Term Decomposition Method for Loss Allocation in Contemporary Distribution Systems

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## Abstract

High penetration of distributed generation (DG) units in contemporary distribution systems can contribute significantly towards loss reduction in distribution feeders. The loss allocation method should reward the benefit of loss reduction to DG owners (DGOs), instead of diverting the same to the load points which actually never contribute towards loss reduction. This paper presents an exact cross-term decomposition method that decomposes each crossed term of power loss/remuneration among contributing network users while independently addressing their active and reactive power transactions. Separate allocation factors are proposed to decompose each individual crossed term involved. The allocation factors are not based upon heuristic, as in several existing methods, but are derived analytically. The method employs superposition principle to evaluate contributing currents of DGs. Thereafter, the losses/remunerations are allocated fairly among network users by suggesting a suitable remuneration strategy for DGOs. The proposed method is investigated on standard test distribution system. The application results are promising compared with other established methods.

## Keywords

Active distribution systems Circuit theory-based method Distributed generations Loss allocation Superposition## List of symbols

- \(\hbox {CDG}(ij)\)
Set of contributing DG currents in branch

*ij*- \(\hbox {CN}(ij)\)
Set of contributing node currents in branch

*ij*- \(\hbox {CT}^{\mathrm{a}}(ij)/\hbox {CT}^{\mathrm{r}}(ij)\)
Crossed term for active/reactive component of contributing currents in branch

*ij*- \(\hbox {CT}^{\mathrm{a}}(ij, k))/\hbox {CT}^{\mathrm{r}}(ij, k)\)
Crossed term related to active/reactive component of contributing node current

*I*(*n*(*ij*,*k*)) in branch*ij*- \(\hbox {CT}_{\mathrm{DG}}^\mathrm{a} \left( {ij,p} \right) /\hbox {CT}_{\mathrm{DG}}^\mathrm{r} \left( {ij,p} \right) \)
Crossed term for active/reactive component of contributing

*p*th DG currents in branch*ij*- \({{\varvec{I}}}(ij)\)
Phasor current of branch

*ij*- \({{\varvec{I}}}(ij,k)\)
Contributing phasor current of \(k\hbox {th}\) node in branch

*ij*- \({\varvec{I}}_{\mathrm{c}}(ij)\)
Phasor current of branch

*ij*with DGs- \({\varvec{I}}_{\mathrm{DG}}(ij)\)
Phasor sum of contributing DG currents in branch

*ij*- \(L^{\mathrm{a}}(ij, k)/L^{\mathrm{r}}(ij, k)\)
Loss allocation factor to bifurcate CT\(^{\mathrm{a}}\)(

*ij*,*k*)/CT\(^{\mathrm{r}}\)(*ij*,*k*)- \({\hbox {MT}}_{\mathrm{DG}}(ij, p)\)
Mixed term related to

*p*th contributing DG current in branch*ij*- \({N}/{\hbox {NB}}/{\hbox {NDG}}\)
Total nodes/branches/DGs in the distribution system

- PLoss
Total real power loss in the system

- \(\hbox {Ploss}(ij)\)
Power loss of branch

*ij*- \(\hbox {ploss}(ij, k)\)
Power loss of branch

*ij*allocated to \(k\hbox {th}\) contributing node- \(\hbox {ploss}(k)\)
Power loss allocated to \(k\hbox {th}\) contributing node

*R*(*ij*)Resistance of branch

*ij*- \(R^{\mathrm{a}}(ij, p)/R^{\mathrm{r}}(ij, p)\)
Remuneration allocation factor to bifurcate \(\hbox {CT}_{\mathrm{DG}}^\mathrm{a} \left( {ij,p} \right) /\hbox {CT}_{\mathrm{DG}}^r \left( {ij,p} \right) \)

- \(R_{\hbox {DG}}(ij)\)
Remuneration of DG through the branch

*ij*- \(R_{\hbox {DG}}(ij, p)\)
Remuneration of

*p*th DG through the branch*ij*- \(\mathfrak {R}\{x\}/\hbox {Im}\{{{\varvec{x}}}\}\)
Real/imaginary part of complex quantity

**x**- ST(
*ij*) Sum of squared term of contributing node currents in branch

*ij*- \(\hbox {ST}_{\mathrm{DG}}(ij, p)\)
Squared term of contributing

*p*th DG current in branch*ij*

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