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Reviewing strategies for active power transmission loss allocation in power pools
Electrical Power and Energy Systems 26 (2004) 81–90 www.elsevier.com/locate/ijepes
Reviewing strategies for active power transmission loss allocation in power pools R.S. Salgadoa,*,1, C.F. Moyanoa, A.D.R. Medeirosb a
Universidade Federal de Santa Catarina, CTC/EEL, Floriano´polis, Santa Catarina, Brazil b Operador Nacional do Sistema ONS/NRS, Floriano´polis, Santa Catarina, Brazil Accepted 29 July 2003
Abstract Transmission losses have a considerable effect on the active power generation cost and thus, a strategy to allocate them fairly among the power system agents is essential to economic efficiency of the electric energy market. The present work focuses on the allocation of the active power transmission losses among the buses of a power system operating under pool condition. Three types of approaches, based on extensions of the conventional and optimal solutions of the power network equations, are analyzed here: (1) direct use of sensitivity relationships between the transmission losses and the bus power injections; (2) use of participation factors obtained from power flow solutions; and (3) integration of sensitivity relationships mentioned in the previous item. Numerical results obtained with a 19-bus power network are used to illustrate the main aspects of the loss allocations based on the application of the selected techniques. q 2004 Elsevier Ltd. All rights reserved. Keywords: Loss allocation; Power flow; Optimal power flow; Co-operative game theory; Marginal cost; Nodal participation factors
1. Introduction One of the main features of deregulated electricity markets is that a common transmission system is shared by multiple agents providing/consuming power. In these markets, competition among generation utilities is promoted through the open access to the grid, the costs involved in the use of the transmission system being recovered from the power system agents [1]. Several costs are implicitly associated with the power supply, one of which corresponding to the losses in the transmission structure. Active power losses typically represent a fraction of the total active power generation ranging from 4 to 8%. They depend on the power flows, which are not physically traceable; that is, the active and reactive powers can flow from a generator to a load bus through a number of alternative routes. * Corresponding author. Address: Universidade Federal de Santa Catarina, Centro Tecnologico - Depto. de Engenharia, Eletrica Caixa Postal 476, Cidade Universita´ria, Floriano´polis 88040-900, Santa Catarina, Brazil. E-mail address: [email protected] (R.S. Salgado). 1 R. Salgado is currently on sabbatical leave at the Brunel Institute of Power Systems, University of Brunel/London, UK. 0142-0615/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2003.07.001
There are at least two main reasons for adopting schemes of power loss allocation in deregulated electric power markets: the transparency of the cost recovery process and the availability of good signals to the power system agents. Two situations illustrate this statement. If, for reason of computational simplicity, the base generation dispatch and its clearing price are calculated through a suboptimal merit-order approach that initially neglects the transmission losses, the recovery of common costs with fairness requires an equitable loss division [2]. In case of optimally dispatched power networks, loss allocation methodologies provide information about the amount of power effectively supplied to each particular consumer. In both cases, the loss allocation depends only on the criterion adopted for purposes of revenue and payment reconciliation, not affecting any power system variable. However, this information can be useful to induce efficient use of the grid by participants, providing incentive to energy producers and consumers to improve their operation conditions. Several difficulties are associated with the division of the power losses among the power system agents. The transmission losses depend on the bus power injections, being usually represented by non-linear functions of
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the complex bus voltages. These functions can have a single value for different power flow solutions. The loss allocation does not affect the generation levels and power flows, but modifies the distribution of revenues and payments among suppliers and consumers [2]. There is also a question concerning which power system agents must supply the active power transmission losses; that is, the losses must be allocated only to the load buses or also to generation buses? In addition, the choice of a set of generating units to supply the loss can also be seen as an arbitrary decision. Therefore, the establishment of a transparent, practical and politically acceptable charging system, that recovers common costs with fairness while provides incentive for the efficient use of the transmission system by all agents, is a regulatory challenge [1]. A number of strategies for the transmission loss and/or loss cost allocation have been proposed in the literature, most of which are based on conventional or optimal power flow solutions. The simplest loss allocation scheme is the so-called pro-rata, which divides the power transmission losses proportionally to bus power injections. It does not take into account the electrical distance between buses and thus, buses close to or distant from the generation centers are similarly penalized. Other approaches propose the direct use of decomposed marginal costs [3] or incremental transmission loss coefficients [4] as nodal participation factors. The tracking of the transmission line power flows to determine the fractions of power flows corresponding to the generators and loads is proposed in Refs. [5 –7]. Ref. [8] evaluates nodal factors based on the current injections and bus impedance matrix. In a recent past, the integration of incremental transmission loss coefficients [2] or alternatively marginal costs [1,9 – 11] has been proposed. In spite of the arbitrariness (implicitly or sometimes explicitly) inherent to any loss allocation scheme, there is a number of desirable requirements, which once satisfied, increase the degree of equity of the allocation strategy. These are: † consistency with the solution of the steady state power network equations; that is, the magnitude of the bus power or current injections should be reflected in the corresponding allocation factors [8]; † smallest possible degree of dependence of the nodal loss factors on changes in the set of generation buses responsible for loss supply; † non-negativeness of the loss fraction attributed to each bus and non-existence of cross subsidy; that is, changes in the demand of a set of buses must be reflected basically in the corresponding nodal factors, not leading any consumer to subsidize another. These features are also useful for comparison purposes of the various approaches proposed in the literature. For the objective of the present paper, the following types of approach were studied:
† direct use of marginal costs or sensitivity relationships between the active power transmission losses and the bus power injections; † extension of the power flow results through bus impedance matrix; † integration of the sensitivity relationships mentioned in the previous item. Four techniques, based on Refs. [2,3,8,9] illustrate the analysis presented here. These strategies are relatively simple extensions of the conventional and the optimal power flow solutions, which are basic numerical tools in the power system steady state analysis. They are suitable for solving loss allocation problems in power systems operating under pool conditions. The main objective of our study is to investigate the similarities and differences between the loss allocation approaches in terms of the quality of the signal provided to power system agents. Numerical results obtained with a 19-bus network, equivalent from the Brazilian South –Southeast power system were used to illustrate the proposed study.
2. Theoretical review The following sections summarize three types of approaches for active power transmission loss allocation, which are the basis for the study presented in this paper. 2.1. Loss allocation through direct use of sensitivity coefficients Traditionally, many electric utilities have adopted loss allocation schemes based on incremental factors and/or marginal costs, which are relatively easy to compute. Nodal participation coefficients are obtained from a linear approximation of the power loss equation with respect to the bus active and reactive power injections. Each term of the linear approximation defines the fraction of the incremental losses allocated to the corresponding bus. The computation of sensitivity relationships is proposed in Ref. [4] for loss allocation purposes and in Ref. [12] to assess the bus participation in the use of the transmission system. Usually, the largest incremental transmission factors are attributed to buses electrically far from the generation centers. However, although these factors are easy to compute, this type of loss allocation strategy can be arbitrary and discriminatory [2]. Ref. [3] proposes the determination of the loss nodal factors through the decomposition of the Lagrange multipliers. For this purpose, an OPF problem is solved to minimize the active power generation cost, from which the marginal costs are obtained. In order to compute the nodal loss factors, the marginal costs are decomposed into parcels corresponding to the cost of the power supply, active power transmission loss and congestion. It is supposed that
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the marginal cost of the reference bus lr includes only the power supply and the losses (the fraction relative to the congestion is excluded). In this case, the Lagrange multipliers are expressed as l ¼ lGL þ lC and evaluated by solving the equation ð7 FÞt 21 ð7 f Þt ð7 FÞt 21 ð7 GÞt Vr V V V lr þ m l ¼ ð1Þ t t t t ð7u FÞ ð7u fr Þ ð7u FÞ ð7u GÞ
the power balance equations included in the equality constraints set. This allows the flexibility to select the buses among which the loss is shared. The computational effort required to obtain the OPF solution is moderate, which incentivizes the application of this allocation technique.
where F is the set of power balance equations of all buses excluding the reference bus, fr refers to the power balance equations of the reference bus, G is the set of inequality constraints, the matrix (whose inverse is required) is the Jacobian matrix of the power balance equations and m is the set of dual multipliers of the inequality constraints. Although the marginal cost principle leads to easily obtained nodal factors, this charging scheme might result in over-recovery [1]. To overcome this problem, adjustment strategies are proposed to scale down the losses to one half of the marginal cost based charge [5] or alternatively to scale the transmission loss coefficients such that the total value of the power loss is recovered [2]. It is also important to note that, the assumption concerning the marginal cost of the reference bus implies that the selection of this bus has considerable influence on the loss allocation. However, it is possible to select previously the set of generation buses that will supply the transmission loss (distributed slack buses) and thus, although this strategy could be considered discriminatory, to reduce the degree of arbitrariness inherent to the choice of the reference bus. The loss allocation procedure based on Ref. [3] and used here can be summarized in the following steps:
The careful analysis and extension of the solution of steady state power network equations provides basis to another type of loss allocation methodology. The strategies of this type found in the literature range from the decomposition of the transmission line flows to the use of the bus impedance matrix and current injections. Kirschen et al. [6] define two main concepts based on the analysis of the power flow, concerning the domain and corresponding link of generating units. The first establishes the set of load buses supplied by each generator (domain of the generator), which is classified according to the number of generating units related to the domain. The second concept refers to the transmission lines located on the bounds of different generators domains. The losses are divided proportionally among the generators according to their participation in the transmission lines within their domain. Similarly, Reta and Vargas [4] propose a strategy to assess the influence of each generator in the load supply, based on the superposition principle. Current sources and equivalent impedances are used to model the effect of generation on the demand supply. This procedure provides the fraction of the line current, corresponding to each generated power injection supplying the load. This fraction is used to evaluate the parcels of power loss corresponding to the loads. The loss participation factors are composed of linear and quadratic terms related to the current supplied to the load, established according to the influence of each component in the total line current. Wu and Chen [14] propose the attribution of loss components to the power demand and generation and the computation of coefficients relating the generated power to each power load, as in Refs. [6,13]. The main feature of these approaches is to model the active power transmission loss as power loads at fictitious buses located in the middle of the transmission lines. Under this assumption, the power system is considered lossless, but with an increased number of buses. The superposition principle is used to evaluate the voltage magnitude at the fictitious buses. Ref. [8] proposes a loss allocation methodology, applicable to nodal power system models, based on both the bus impedance matrix and current injections, for a given power flow solution. This approach was chosen to illustrate the study presented in this paper, but with the loss nodal factors derived in terms of power injections as follows. The complex power loss in a transmission system is given by [15,16]
† determine the OPF solution to minimize the active power transmission loss, from which the marginal costs are a by-product; † evaluate the fraction of the marginal cost relative to the transmission loss, according to Eq. (1); † compute the nodal loss participation factors by scaling the parcel of the marginal costs corresponding to the losses, such that their product by the power injections matches the total value of the active power transmission losses. The active power transmission loss is a monotonically increasing function of the active power demand. This ensures that non-negative loss parcels are obtained for all buses. In the absence of congestion, marginal costs correspond to the sensitivity of the power loss with respect to the power injections, such that there is no need of the decomposition stated by Eq. (1). Additionally, the use of scaling ensures that the precise value of the loss is matched. This methodology is dependent on the OPF formulation, and therefore there will be Lagrange multipliers only for
2.2. Loss allocation based on power flow analysis
Sl ¼ Ipt b Zb Ib
ð2Þ
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where Ib is the n-vector of the bus current injections, Zb is the n £ n bus impedance matrix (with Zb ¼ 21 Y21 ¼ Rb þ ŒXb ) and the superscript pt b ¼ ðGb þ ŒBb Þ denotes complex transpose conjugate. Eq. (2) can be decomposed in pt Sl ¼ Ipt b Rb Ib þ ŒIb Xb Ib ¼ Pl þ ŒQl
ð3Þ
where Pl and Ql are the active and reactive power transmission losses, respectively. Denote by Sb ¼ Pb þ ŒQb the n vector of the bus complex power injections and by Dv a diagonal matrix, whose elements are the inverse of the complex bus voltages Vi : Thus, the vector of bus current injections Ib is written in matrix form as Ib ¼ Dpv Spb and the active power transmission loss Pl can be expressed in terms of the bus power injections as Pl ¼ ðDpv Spb Þpt Rb ðDpv Spb Þ ¼ Stb ðDv Rb Dpv ÞSpb where ðDv Rb Dpv Þ ¼ Dr þ ŒDi and the matrices Dr and Di are symmetric ðDtr ¼ Dr Þ and anti-symmetric ðDti ¼ 2Di Þ; respectively. Therefore Pl ¼ Stb ðDr þ ŒDi ÞSpb which, in terms of active and reactive power injections, can be expressed as Pl ¼ Ptb Dr Pb þ ŒQtb Dr Pb þ ŒPtb Di Pb 2 Qtb Di Pb 2 ŒPtb Dr Qb þ Qtb Dr Qb þ Ptb Di Qb þ ŒQtb Di Qb
ð4Þ
According to the property of the matrices Dr and Di ŒQtb Dr Pb 2 ŒPtb Dr Qb ¼ 0 Ptb Di Pb ¼ 0 Qtb Di Qb ¼ 0 such that the rearrangement of the terms of Eq. (4) provides Pl ¼ Ptb Dr Pb þ Qtb Dr Qb 2 Qtb Di Pb þ Ptb Di Qb
ð5Þ
The last equation shows the dependence of the active power transmission losses on the power injection at every bus, indicating that all buses contribute to transmission losses. It is derived from the steady state power flow equations, without any approximation and thus provides the accurate value of the active power transmission loss. For a given solution (conventional or optimal) of the steady state power system equations, the following two main steps summarize the transmission loss allocation through this methodology: † determine the real part of the bus impedance matrix Rb ; † evaluate the nodal loss distribution factors from Eq. (5) as pp ¼ Dr Pb 2 Dti Qb and pq ¼ Dr Qb þ Dti Pb : Note that the terms pp and pq represent the transmission losses factors, the summation of their product by active and
reactive power injections, respectively, providing the total active power transmission loss given by Eq. (5). The application of this methodology is simple, the main difficulty being the evaluation of the real part of the bus impedance matrix. For this purpose, sparsity techniques are recommended [8]. The lack of flexibility to attribute loss parcels only to generation (or alternatively load) buses, or even to establish the fractions of the loss to be shared by the consumers and by producers seems to be its main pitfall. Besides, there is no guarantee that all nodal participation factors assigned to the buses are non-negative, which means that negative loss parcels can be attributed. 2.3. Loss allocation through integration of sensitivity coefficients The Lagrange multipliers corresponding to the OPF solution to minimize the power losses as well as the incremental transmission loss factors at any operation point are interpreted as first order sensitivity relationships between the active power transmission loss and the bus power injections. These amounts refer to infinitesimal increments and thus need to be integrated. Two types of approach based on the integration of sensitivity relationships are found in the literature. In the first, nodal loss factors are obtained by integrating the incremental transmission loss coefficients [17], or alternatively, infinitesimal loss distribution coefficients [2], evaluated in a sequence of conventional power flow solutions. Transmission loss recovery can also be seen as a cost allocation problem, which is usually stated in the form of dividing the cost of a jointly used facility among participants in a co-operative venture [1]. The Aumann – Shapley (AS) methodology, which is part of the cooperative game theory, is applicable to the problem of sharing the loss among the users of the transmission system. It is based on the integration of the marginal costs to provide the so-called AS values. In order to share the cost corresponding to the active power transmission loss, Refs. [10,11] propose to integrate the difference between the Lagrange multipliers obtained from two OPF solutions for minimum active power generation cost, one considering and the other neglecting the resistance of the transmission lines. In Ref. [9], the active power transmission loss is minimized such that the integration of the corresponding Lagrange multipliers provide nodal factors relating the loss to the power injections. The approaches described as follows, based on conventional and OPF solutions, were selected for the present study. According to Ref. [2], the loss change corresponding to an infinitesimal power demand change is reflected on the set of generation buses responsible for loss supply and/or the set of load buses. It is shown that an assignment a priori of the proportion of the total losses Pl to generation/load buses is
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necessary. This proportion is represented P by the components ri of vector r; which are such that ni¼1 ri ¼ 1; with the product rPl providing the fraction of the loss assigned to each bus. Similarly, the power demand supply is supposed to be shared among the generators according to prespecified distribution factors mi ; satisfying the condition Pn tot tot i¼1 mi ¼ 1: The product mPd (where Pd is the total active power demand and m is a vector with components mi ) gives the parcel of the demand supplied by each ith generating unit. Under these assumptions, the incremental active power balance equation is expressed as dPg 2 d Pd ¼ dPðdÞ (where for the sake of simplicity the bus voltage magnitudes are omitted) or, alternatively m dPtot d þ r dPl 2 dPd ¼ dPðdÞ and the infinitesimal increment of the loss attributed to the jth bus, denoted dLj ; is given by dLj ¼
n X a j 2 ai i¼1
aT r
mi dPdj
ð6Þ
where dPdj is the power demand increment at the jth bus, the n-vector a has components ai ¼ 1 2 ITLi (ITLi is the incremental transmission loss factor of the ith bus) and the index i refers to generation buses. Note that the amount dLj =dPdj can be seen as the sensitivity relationship between the loss and the active power injection at the jth bus. The precise contribution of the jth bus to the power transmission loss is obtained by integrating Eq. (6) over the interval corresponding to demand at bus j; that is Lj ¼
( n ðPdj X aj 2 ai 0
i¼1
aT r
) mi
dPdj
ð7Þ
where all terms have been previously defined. Although not shown in Ref. [2], this allocation strategy should take into account the variation of the active power loss with respect to reactive power changes. Since the active power transmission loss is dependent on the bus voltage magnitude (or reactive power distribution), it is expected that neglecting the reactive power can affect the value of the loss allocation coefficients and consequently result in inaccuracy in the loss recovery. The loss recovery through the computation of AS values, as proposed in Ref. [9], is based on the incremental costs resulting from the inclusion of a new participator in a coalition. In the case of the loss allocation, the Lagrange multipliers are used to this aim. The dependence of these factors on the operating points is overcome by calculating a sequence of marginal indices, resulting from small increments in the power injection, from zero up to the load level for which the loss allocation is desired. This means to integrate the marginal costs, such that the parcel of the active power transmission loss corresponding to the jth bus
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is given by Lj ¼
ð P dj 0
lpj dPdj þ
ðQdj 0
lqj dQdj
ð8Þ
Instead of performing the explicit integration indicated in Eqs. (7) and (8), successive conventional/OPF solutions are used to provide these indices. For this purpose, the power generation/demand patterns are divided in a number of intervals, which are added cumulatively. For each addition the conventional/optimal power flow equations are solved to provide the voltage angles and magnitudes, the losses and the marginal costs/incremental transmission loss coefficients. This process is repeated until the generation matches the load level for which the loss allocation is required. The summation of these loss sensitivity factors provides the desired integration. The arithmetic mean associated with the small increments represents the exact contribution of each bus to the losses. Note that, for purpose of this work the OPF solution minimizes the active power transmission loss, the Lagrange multipliers lp and lq being expressed in MW/MW and MW/Mvar, respectively. Thus, the loss fraction Lj of Eq. (8) is expressed in MW. The procedure adopted to perform this combination can be summarized in the following steps: † Divide the power demand of each bus into N intervals. For the ith bus, each interval corresponds to Pdi =N and Qdi =N: † Starting with null demand, increase each load interval cumulatively. The load of each bus will be increased by Pdi =N and Qdi =N for each addition. † For each load increment, determine the conventional/OPF solution (active power transmission loss minimization). This provides the incremental transmission loss factors/marginal costs. † For each bus, compute the arithmetic mean of the incremental transmission loss factors/marginal costs (AS values), which are the nodal loss factors. The allocation schemes based on integration of sensitivity relationships are highly accurate but demand a large computational effort, since a conventional/optimal power flow solution is necessary at each integration interval. In order to overcome this drawback, scaling intervals are recommended in Ref. [2] which provide nodal loss factors with good level of accuracy in a small number of intervals. As in the case of the strategies based on the direct use of Lagrange multipliers, these allocation methodologies depend on the formulation used to evaluate the sensitivity relationships (marginal costs), and then allow to select the buses among which the loss is allocated. In the present case, the nature marginally crescent of the active power transmission loss function guarantees the non-negativeness of the nodal loss parcels of each bus.
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The main difference between these approaches is the way the sensitivity relationships are integrated. The premise of co-operative game theory requires the integration of marginal costs obtained from a sequence of optimal power flow solutions. If the conventional Newton –Raphson power flow solution is used to this aim, the power dispatch is not optimal as the case of the OPF, and the result of the loss recovery is prone to inaccuracies.
3. Numerical results In order to illustrate the analysis proposed in this work, the loss allocation methodologies described in Section 2 were applied to a 19-bus network whose transmission line data and diagram are presented in Table 1 and Fig. 1, respectively. This network represents an equivalent of the power system of the South –Southeastern region of Brazil. Reactors with nominal values of 200, 150, 300, 100, 150 and 300 Mvar are located, respectively, at buses 13, 15, 16, 17, 18 and 19. With the aim of better illustrating the analysis presented here, the loss allocation proportional to the load buses active power demand is shown, in addition to the loss sharing obtained through the selected allocation techniques. For the sake of simplicity, the strategies used here to loss allocation purposes are referred to as: † PRP: methodology based on proportionality to the bus power demand; † ZBS: methodology based on bus impedance matrix and power injections; † LMB: methodology based on the use of marginal costs; † ISR: methodology based on the use of integrated
sensitivity relationships; † CGT: methodology based on co-operative game theory. An OPF solution to minimize the active power transmission losses was obtained at the load level for which the loss allocation was desired. Thus, the active and reactive power generations are optimally dispatched to supply the active power load and compensating transmission losses. This operation point and the bus impedance matrix were used to calculate the nodal loss factors of the ZBS method. The Lagrange multipliers corresponding to this solution were adjusted to allocate the losses according to methodology LMB and integrated to provide the loss factors according to the premise of co-operative game theory. For allocation technique ISR, the sensitivity relationships between the transmission loss and the power demand of the load buses obtained from a sequence of power flow solutions, from which the last is the same as that obtained from OPF, were used to compute the loss factors of load buses. The nodal loss factors resulting from the application of ISR and CGT were obtained in 30 integration steps. Except for the application of Zbus methodology, the loss was assumed to be shared only by the load buses. Thus, the OPF problem for minimum loss was formulated such that the active and reactive power balances at load buses are the only equality constraints. For application of the criterion of proportionality with respect to the bus active power demand (PRP methodology), the loss fraction of the ith bus was computed as ! Pdi pli ¼ Pl Ptd where Pdi is the active power demand of the ith bus, Ptd is the total active power demand and Pl is the total active power transmission loss.
Table 1 Transmission system data From
To
R (%)
X (%)
B (%)
From
To
R (%)
X (%)
1 1 1 1 2 2 3 3 4 4 4 5 5 6 6 7 7 7 8
11 11 2 2 3 3 4 4 5 12 12 6 6 7 7 8 8 9 9
3.06 3.06 3.16 3.16 1.72 1.72 4.63 4.63 3.86 1.58 1.58 0.96 0.96 3.25 3.25 1.54 1.54 0.65 1.62
11.65 11.65 16.21 16.21 8.54 8.54 23.78 23.78 19.87 8.24 8.24 10.02 10.02 16.50 16.50 8.52 8.52 8.23 9.32
27.02 27.02 27.84 27.84 14.34 14.34 40.80 40.80 34.00 13.75 13.75 8.40 8.40 28.69 28.69 12.50 12.50
8 9 9 10 11 15 13 14 15 15 16 16 15 17 17 18 18 18
9 10 11 11 15 13 14 15 16 16 1 1 17 18 19 12 12 19
1.62 2.04 4.68 3.53
9.32 10.46 23.95 18.09 1.25 2.01 1.94 1.33 2.73 2.98 1.25 1.25 2.05 3.09 2.05 1.25 1.25 3.09
16.12
0.16 0.15 0.11 0.22 0.24
0.16 0.25 0.16
0.25
B (%) 16.12 18.08 41.38 31.25 245.8 237.0 163.7 334.0 363.0
250.2 377.7 250.2
377.7
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The objective of these tests was to verify how the loss parcels were modified as a result of the load change. The cases reported in the following sections illustrate these situations. 3.1. Base case Table 2 presents the results of the loss allocation for the base case. The transmission loss corresponding to the OPF solution is 26.75 MW. Buses 11, 16, 17 and 18 are transfer buses and thus no fraction of transmission losses is allocated to these buses. Strategies PRP, LMB, ISR and CGT assign no transmission losses to the generating buses (1, 3, 6, 14, 15 and 19). The demand of buses 1, 3 and 6 is supplied by the generators located at these buses, whereas there is no load located at buses 14, 15 and 19. Thus, the transmission system is not used which implies null loss factors associated to these loads. From the row indicating Total, is noted that the value of the active power transmission losses corresponding to the power flow solution is recovered with satisfactory accuracy in all cases. From columns 6 and 7 of Table 2, it is observed that results provided by PRP and LMB are very close. In this particular case, the magnitude of Lagrange multipliers related to the active power balance are close to 1.00, whereas those corresponding to the reactive power balance are close to zero. For this reason, the scaling factors used to recover the value of the loss work as a kind of proportionality coefficients. Thus, loss parcels determined according to these techniques tend to reflect predominantly the magnitude of the active power load. Column 8 of Table 2 shows that according to the approach ZBS, 10.48 MW (39%) of the total active power transmission loss (26.75 MW) is assigned to the generators
Fig. 1. The 19-bus system.
Similarly, for the application of the LMB allocation strategy the Lagrange P multipliers P were scaled by the factor Pl =P0l (where P0l ¼ lpi Pdi þ lqi Qdi is the value of the active power transmission loss estimated via Lagrange multipliers). In this case, the loss parcel assigned to bus i is ! ! Pl Pl pl i ¼ l P þ l Q P0l pi di P0l qi di Three load conditions were used to analyze the loss allocation obtained from the different methodologies. In the first, the loss parcel of each bus is evaluated for a base case. In the second, loss fractions for all buses are calculated for a given increase in the active load of selected buses. This procedure is repeated in the third case, with an increase in the reactive power demand of a set of chosen buses. Table 2 Loss allocation for the base case Bus
Pg (MW)
Qg (Mvar)
Pd (MW)
1 2 3 4 5 6 7 8 9 10 12 13 14 15 19
490.56
2105.83
430.50 157.90 138.60 158.50 108.50 98.30 188.00 224.20 243.10 124.00 407.90 902.80
241.70 54.90 2.60 44.40 44.80 75.70 106.00 70.10 23.50 26.00 34.20 92.80
3182.30
501.30
Total
301.06
349.13
271.81
94.32
642.47 997.80 427.88
2134.50 2733.09 2515.22
3209.05
21466.48
Qd (Mvar)
CPU time (s)
PRP (MW)
LMB (MW)
1.68
1.66
1.69 1.15
1.68 1.14
2.00 2.38 2.59 1.32 4.34 9.60
2.03 2.44 2.56 1.34 4.27 9.49
26.75
26.61
0.05
0.06
ZBS (MW) 0.29 0.75 0.61 1.19 0.19 0.88 3.06 4.30 1.14 1.95 1.11 2.59 2.54 2.98 3.18
ISR (MW)
CGT (MW)
1.21
1.30
2.19 0.66
1.83 0.68
3.29 4.67 1.98 2.36 2.53 5.99
4.04 5.35 2.05 2.44 2.47 6.46
26.75
24.88
26.61
0.05
0.55
5.33
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and 16.27 MW (61%) is attributed to the load buses. It must be recalled that this allocation strategy does not allow to allocate the loss among the load buses only (or alternatively the generation buses only). Note that the loss fraction assigned to the generators is related to the net bus current (power) injection. The power injections at generation buses 1, 3 and 6 are 60.06 (490.56 – 430.50) MW, 162.46 (301.06 – 138.60) MW and 250.83 (349.13 – 98.30) MW whereas those at generation buses 14, 15 and 19, which have no load, are 642.47, 997.80 and 427.88 MW, respectively. Although the active power generated at buses 1, 3 and 6 is high, the load of these buses reduces the current injected in the transmission grid by these generators. This explains why the loss fractions assigned to buses 1, 3 and 6 are not so high. On the other hand, there is no load at buses 14, 15 and 19 and then higher values of current are injected in the transmission system. With respect to load buses, the demand of bus 13 is considerably greater than those of the other buses but its loss parcel is the third greatest. This can be attributed to the electric distance from this bus to the generators, since the analysis of the network topology reveals that this bus is basically supplied by generation buses 14 and 15. Some similarity between the results provided by CGT and ISR could be expected, since both are based on the principle of the integration of sensitivity coefficients. However, the analysis of the last two columns of Table 2 reveals some differences between the loss parcels resulting from the application of these methods. These differences can be explained by examining the solutions of the network equations used to estimate the sensitivity relationships between the losses and the power injections. In CGT, a sequence of optimal solutions is used, and thus the active and reactive powers are optimally scheduled along the whole integration path, the Lagrange multipliers representing then optimal values of the sensitivities. Conventional power flow solutions are used in the application of ISR approach, and thus the generated reactive power is not optimally dispatched, being scheduled simply to satisfy the power balance. In other words, different sets of equations (and therefore different operation points) are used to compute the sensitivities, which explains the different loss parcels. Interesting differences can be observed between allocations obtained through LMB and CGT methods. For example, from the loss parcel of bus 13 (column 7 of Table 2) it could be inferred a considerable influence of this bus on the transmission loss. However, the integration of the corresponding Lagrange multiplier (instantaneous sensibilities) applied for CGT technique provides a different loss fraction for this bus, which reflects its location in the network. This value represents the actual influence of bus 13 on the loss, since it is measured in terms of all possible coalitions between all agents of the power system, as predicated by co-operative game theory [18]. The opposite case is that of buses 7 and 8, for instance, which are
electrically far from the generators. According to method CGT, a larger loss parcel is attributed to these buses, indicating the real effect of their power load on the total loss. The last row of Table 2 shows the CPU time required by each allocation methodology. These results were obtained in an AMD Athlon XP 1700 Processor, with 256 MB (RAM) and 1466 MHz. The large CPU time required by CGT strategy is due to the number of OPF solutions (30 in the present case) necessary to integrate the Lagrange multipliers. 3.2. Changing the active power load Aiming at assessing the loss allocations with respect to cross subsidy in terms of transmission loss, the active power demand of buses mentioned in Section 3.1 was augmented by 100 MW; that is, buses 2 (from 157.90 to 257.90 MW), 4 (from 158.50 to 258.50 MW), 5 (from 108.5 to 208.5 MW) and 7 (from 188.00 to 288.00 MW). The total active power demand increased from 3182.30 to 3582.30 MW and the transmission losses from 26.75 to 37.30 MW (increment of 10.55 MW). Naturally, changes in the transmission loss allocation are expected, with emphasis in the loss fractions corresponding to buses with modified active power demand. Table 3 shows the optimal power dispatch and the loss allocation for the case of increased active power demand. These results present a considerable variation of the fractions corresponding to the buses whose demand was increased. The total parcel of the transmission losses attributed to these buses can be obtained by summing the values in rows corresponding to buses 2, 4, 5, and 7 (columns 6 –10). The fractions of the loss increase of 10.55 MW allocated for each methodology to these buses are: 6.49 MW (61.52%) for PRP, 6.57 MW (62.27%) for LMB, 7.04 MW (66.73%) for ZBS, 8.98 MW (85.12%) for ISR and 9.57 MW (90.71%) for CGT. These values are obtained by summing (for each methodology) the increase in the loss fraction (with respect to the base case) corresponding to these buses. Therefore, methodologies ISR and CGT assign the major part of the loss increase to the buses with active power demand increase, so as to reduce as much as possible the cross subsidy. Note that, because of the change in the power flows due to the load increase, the loss fraction assigned to every bus is modified, being unavoidable to attribute the whole amount of loss increase to the buses whose demand was modified. The increase in the loss fraction of buses 2, 4, 5 and 7 evaluated through the other techniques (PRP, LMB and ZBS) do not significantly reflect the increase in the load of these buses, which indicates a greater level of cross subsidy. 3.3. Changing the reactive power load Table 4 shows the optimal power dispatch and the loss fractions allocated to the buses for the case in which an increase of 100 MVAr is assigned to the load of buses 2
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Table 3 Loss allocation after increasing the active power at selected buses Bus
Pg (MW)
1 2 3 4 5 6 7 8 9 10 12 13 14 15 19
530.97
708.33 996.92 498.45
2137.09 2700.95 2509.51
Total
3619.60
21379.63
396.30
488.64
Qg (Mvar) 297.58 256.08
121.58
Pd (MW)
Qd (Mvar)
430.50 257.90 138.60 258.50 208.50 98.30 288.00 224.20 243.10 124.00 407.90 902.80
241.70 54.90 2.60 44.40 44.80 75.70 106.00 70.10 23.50 26.00 34.20 92.80
3582.30
501.30
PRP (MW)
LMB (MW)
3.31
3.30
3.32 2.68
3.33 2.66
3.70 2.88 3.12 1.59 5.24 11.59
3.79 2.96 3.09 1.61 5.15 11.42
37.44
37.31
(reactive power demand increases from 54.9 to 154.9 MVAr), 4 (from 44.40 to 144.40 MVAr), 5 (from 44.80 to 144.80 MVAr) and 7 (from 106.00 to 206.00 MVAr). The total reactive power demand is 901.3 against 501.3 MVAr corresponding to the base case. The increase in reactive power demand resulted in an increase in the active power transmission loss (from 26.75 to 29.55 MW, increase of 2.8 MW) with a major impact on the buses whose demand has been augmented. Note that, although the reactive power demand was increased in 400 Mvar (79.8% with respect to the base case), the active power loss varies only in 11.3%, emphasizing the second order effect of the reactive power distribution over the active power transmission loss. In this case, the parcels of the increase in the transmission losses attributed to buses 2, 4, 5, and 7 and the percentages of the loss increase of 2.80 MW that they
ZBS (MW) 0.47 2.10 1.06 3.25 1.16 1.74 5.72 4.86 1.26 1.96 1.14 1.89 3.48 3.42 3.94 37.44
ISR (MW)
CGT (MW)
3.13
3.40
4.59 2.07
4.36 2.29
6.54 5.47 2.30 2.48 2.85 6.17
7.37 6.04 2.29 2.48 2.72 6.34
35.58
37.30
represent are: 0.68 MW (24.29%) for PRP, 0.72 MW (25.71%) for LMB, 3.07 MW (109.64%) for ZBS, 2.46 MW (87.86%) for ISR and 3.45 MW (123.21%) for CGT. The results obtained through methods CGT and ZBS show that an amount greater than the loss increase is allocated to the buses with increased demand. There is no considerable change in the Lagrange multipliers corresponding to the buses 2, 4, 5 and 7 for OPF solution at this load level, with respect to those of the base case. However, in case of CGT, in which the reactive power is dispatched to minimum active power losses, the integrated Lagrange multipliers are significantly modified, such that the increment in transmission losses is mainly allocated at the buses with increased reactive power load. The reactive power load change considered here degrades the power factor of buses 2, 4, 5 and 7, and results in a lower voltage magnitude with respect to the base case.
Table 4 Loss allocation after increasing the reactive power at selected buses Bus
Pg (MW)
1 2 3 4 5 6 7 8 9 10 12 13 14 15 19
490.62
Total
302.33
349.16
Qg (Mvar) 269.09 233.42
248.81
644.33 996.93 428.19
2144.13 2626.19 2469.21
3211.55
21036.38
Pd (MW)
Qd (Mvar)
430.50 157.90 138.60 158.50 108.50 98.30 188.00 224.20 243.10 124.00 407.90 902.80
241.70 154.90 2.60 144.40 144.80 75.70 206.00 70.10 23.50 26.00 34.20 92.80
3182.30
901.30
PRP (MW)
LMB (MW)
1.86
1.84
1.86 1.27
1.85 1.26
2.21 2.63 2.86 1.46 4.79 10.61
2.28 2.68 2.81 1.46 4.68 10.38
29.55
29.24
ZBS (MW) 0.22 1.50 0.58 1.73 0.59 1.44 4.44 4.65 1.16 1.96 1.17 2.46 2.43 2.46 2.74 29.55
ISR (MW)
CGT (MW)
1.62
2.09
2.60 1.01
2.14 1.39
4.58 5.06 1.98 2.37 2.54 5.94
5.68 5.54 1.94 2.35 2.28 5.80
27.70
29.25
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However, the comparison between columns 6 and 7 of Tables 2 and 4 shows that loss allocations provided by PRP and LMB for these buses do not reflect effectively this reactive power load increase. In case of LMB method, it must be recalled that in case of the performance index considered here in the OPF, the Lagrange multipliers corresponding to the reactive power balance are considerably smaller than those related to the active power balance. Besides, the changes in the former multipliers are not so significant, which is the reason for the absence of the major differences previously mentioned. In case of ZBS and CGT methods, the transmission loss is re-allocated, being penalized the buses more responsible for the decrease in the voltage magnitude level.
4. Conclusions The methods presented in this study provide different results with respect to the recovery of the active power transmission loss. Concerning the theoretical concepts and the numerical results shown in this study, the following must be remarked: † Similar results with respect to the active power transmission loss allocation are obtained by the approaches based on the integration of sensitivity relationships or Lagrange multipliers, in terms of both the loss fractions assigned to the load buses and the recovery of the total active power transmission loss value. † The loss allocation based on the integration of sensitivity relationships depends on the integration path. This approach is based on the conventional Newton – Raphson power flow solution, which does not take into account the optimal distribution of reactive power. For this reason a small accuracy error can be expected in the recovery of the transmission loss value. † The application of methodology based on the bus impedance matrix allows to recover accurately the value of the transmission losses. However, loss fractions are attributed to the generation buses. Thus, although the computational effort demanded by this technique is moderate, there is no flexibility to allocate losses for load buses only (or generations buses only). † The transmission loss distribution factors obtained through the direct use of the Lagrange multipliers require additional adjustments to recover the transmission loss accurately.
Acknowledgements R. Salgado thanks Coordenac¸a˜o de Aperfeic¸oamento de Pessoal de Nı´vel Superior, CAPES/Brazil for financial support. C.F. Moyano thanks for the financial support granted by Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico, CNPq/Brazil.
References [1] Kattuman PA, Bialek JW, Abi-Samra N. Electricity tracing and cooperative game theory. In: Proceedings of the Power System Computation Conference, Trondheim– Norway. 1999. p. 238–43. [2] Galiana FD, Conejo AJ, Kockar I. Incremental transmission loss allocation under pool dispatch. IEEE Trans Power Syst 2002;17(1): 582 –91. [3] Finney J, Othman H, Rutz W. Evaluating transmission congestion constraints in system planning. IEEE Trans Power Syst 1997;12(3): 1143–50. [4] David AK, Yu CW. Pricing transmission services in the context of industry deregulation. IEEE Trans Power Syst 1997;12(1):539–48. [5] Bialek JW. Elimination of merchandise surplus due to spot pricing of electricity. In: Proceedings of the IEE—Part C, vol. 144.; 1997. p. 399 –405. [6] Kirschen D, Allan R, Strbac G. Contributions of individual generators to loads and flows. IEEE Trans Power Syst 1997;12(1):52 –60. [7] Macqueen CN, Irving MR. An algorithm for the allocation of distribution system demand and energy losses. IEEE Trans Power Syst 1996;11(1):147 –56. [8] Conejo AJ, Galiana F, Kockar I. Z-bus loss allocation. IEEE Trans Power Syst 2001;16(1):105– 10. [9] Medeiros ADR, Salgado RS, Zu¨rn HH. Uma Metodologia Para Obtenc¸a˜o de fatores de perdas nodais. In: Proceedings of Semina´rio Nacional de Produc¸a˜o e Transmissa˜o de Energia Ele´trica, in Portuguese, Campinas-SP/Brasil. 2001. (9): 67–72. [10] Tan XH, Lie TT. Allocation of transmission loss cost using cooperative game theory in the context of open transmission access. In: Power Engineering Society Winter Meeting, vol. 3.; 2001. p. 1215–19. [11] Tan X, Lie TT. Application of the shapley value on transmission cost allocation in the competitive power market environment. In: Proceedings of the IEE—Part C, vol. 149.; 2002. p. 15– 20. [12] Filho XV, Granvile S, Pereira MVF, Gorenstin BG, Mello JC, Melo CG. Ancillary services: how to allocate costs in an optimisation-based framework. CIGRE´ Symposium, Neptun 1997. [13] Reta R, Vargas A. Electricity tracing and loss allocation methods based on electric concepts. In: Proceedings of the IEE—Part C, vol. 148.; 2001. [14] Wu ZQ, Chen GZ. Mva power flow and loss analysis for electricity market. In: Proceedings of the IEE—Part C, vol. 148.; 2001. [15] Wood AJ, Wollenberg BF. Power generation, operation, and control. New York: Wiley; 1984. [16] Kusic GL. Computer-aided power system analysis. Englewood Cliffs, NJ: Prentice-Hall; 1986. [17] Franco PC, Galiana FD. Transmission loss allocation under combined pool and bilateral operation. In: Proceedings of XIII Congresso Brasileiro de Automa´tica, Floriano´polis/SC-Brasil. 2000. p. 181–6. [18] Young HP. Cost allocation: principles, methods, applications. The Netherlands: North Holland; 1983.
Reviewing strategies for active power transmission loss allocation in power pools R.S. Salgadoa,*,1, C.F. Moyanoa, A.D.R. Medeirosb a
Universidade Federal de Santa Catarina, CTC/EEL, Floriano´polis, Santa Catarina, Brazil b Operador Nacional do Sistema ONS/NRS, Floriano´polis, Santa Catarina, Brazil Accepted 29 July 2003
Abstract Transmission losses have a considerable effect on the active power generation cost and thus, a strategy to allocate them fairly among the power system agents is essential to economic efficiency of the electric energy market. The present work focuses on the allocation of the active power transmission losses among the buses of a power system operating under pool condition. Three types of approaches, based on extensions of the conventional and optimal solutions of the power network equations, are analyzed here: (1) direct use of sensitivity relationships between the transmission losses and the bus power injections; (2) use of participation factors obtained from power flow solutions; and (3) integration of sensitivity relationships mentioned in the previous item. Numerical results obtained with a 19-bus power network are used to illustrate the main aspects of the loss allocations based on the application of the selected techniques. q 2004 Elsevier Ltd. All rights reserved. Keywords: Loss allocation; Power flow; Optimal power flow; Co-operative game theory; Marginal cost; Nodal participation factors
1. Introduction One of the main features of deregulated electricity markets is that a common transmission system is shared by multiple agents providing/consuming power. In these markets, competition among generation utilities is promoted through the open access to the grid, the costs involved in the use of the transmission system being recovered from the power system agents [1]. Several costs are implicitly associated with the power supply, one of which corresponding to the losses in the transmission structure. Active power losses typically represent a fraction of the total active power generation ranging from 4 to 8%. They depend on the power flows, which are not physically traceable; that is, the active and reactive powers can flow from a generator to a load bus through a number of alternative routes. * Corresponding author. Address: Universidade Federal de Santa Catarina, Centro Tecnologico - Depto. de Engenharia, Eletrica Caixa Postal 476, Cidade Universita´ria, Floriano´polis 88040-900, Santa Catarina, Brazil. E-mail address: [email protected] (R.S. Salgado). 1 R. Salgado is currently on sabbatical leave at the Brunel Institute of Power Systems, University of Brunel/London, UK. 0142-0615/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2003.07.001
There are at least two main reasons for adopting schemes of power loss allocation in deregulated electric power markets: the transparency of the cost recovery process and the availability of good signals to the power system agents. Two situations illustrate this statement. If, for reason of computational simplicity, the base generation dispatch and its clearing price are calculated through a suboptimal merit-order approach that initially neglects the transmission losses, the recovery of common costs with fairness requires an equitable loss division [2]. In case of optimally dispatched power networks, loss allocation methodologies provide information about the amount of power effectively supplied to each particular consumer. In both cases, the loss allocation depends only on the criterion adopted for purposes of revenue and payment reconciliation, not affecting any power system variable. However, this information can be useful to induce efficient use of the grid by participants, providing incentive to energy producers and consumers to improve their operation conditions. Several difficulties are associated with the division of the power losses among the power system agents. The transmission losses depend on the bus power injections, being usually represented by non-linear functions of
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the complex bus voltages. These functions can have a single value for different power flow solutions. The loss allocation does not affect the generation levels and power flows, but modifies the distribution of revenues and payments among suppliers and consumers [2]. There is also a question concerning which power system agents must supply the active power transmission losses; that is, the losses must be allocated only to the load buses or also to generation buses? In addition, the choice of a set of generating units to supply the loss can also be seen as an arbitrary decision. Therefore, the establishment of a transparent, practical and politically acceptable charging system, that recovers common costs with fairness while provides incentive for the efficient use of the transmission system by all agents, is a regulatory challenge [1]. A number of strategies for the transmission loss and/or loss cost allocation have been proposed in the literature, most of which are based on conventional or optimal power flow solutions. The simplest loss allocation scheme is the so-called pro-rata, which divides the power transmission losses proportionally to bus power injections. It does not take into account the electrical distance between buses and thus, buses close to or distant from the generation centers are similarly penalized. Other approaches propose the direct use of decomposed marginal costs [3] or incremental transmission loss coefficients [4] as nodal participation factors. The tracking of the transmission line power flows to determine the fractions of power flows corresponding to the generators and loads is proposed in Refs. [5 –7]. Ref. [8] evaluates nodal factors based on the current injections and bus impedance matrix. In a recent past, the integration of incremental transmission loss coefficients [2] or alternatively marginal costs [1,9 – 11] has been proposed. In spite of the arbitrariness (implicitly or sometimes explicitly) inherent to any loss allocation scheme, there is a number of desirable requirements, which once satisfied, increase the degree of equity of the allocation strategy. These are: † consistency with the solution of the steady state power network equations; that is, the magnitude of the bus power or current injections should be reflected in the corresponding allocation factors [8]; † smallest possible degree of dependence of the nodal loss factors on changes in the set of generation buses responsible for loss supply; † non-negativeness of the loss fraction attributed to each bus and non-existence of cross subsidy; that is, changes in the demand of a set of buses must be reflected basically in the corresponding nodal factors, not leading any consumer to subsidize another. These features are also useful for comparison purposes of the various approaches proposed in the literature. For the objective of the present paper, the following types of approach were studied:
† direct use of marginal costs or sensitivity relationships between the active power transmission losses and the bus power injections; † extension of the power flow results through bus impedance matrix; † integration of the sensitivity relationships mentioned in the previous item. Four techniques, based on Refs. [2,3,8,9] illustrate the analysis presented here. These strategies are relatively simple extensions of the conventional and the optimal power flow solutions, which are basic numerical tools in the power system steady state analysis. They are suitable for solving loss allocation problems in power systems operating under pool conditions. The main objective of our study is to investigate the similarities and differences between the loss allocation approaches in terms of the quality of the signal provided to power system agents. Numerical results obtained with a 19-bus network, equivalent from the Brazilian South –Southeast power system were used to illustrate the proposed study.
2. Theoretical review The following sections summarize three types of approaches for active power transmission loss allocation, which are the basis for the study presented in this paper. 2.1. Loss allocation through direct use of sensitivity coefficients Traditionally, many electric utilities have adopted loss allocation schemes based on incremental factors and/or marginal costs, which are relatively easy to compute. Nodal participation coefficients are obtained from a linear approximation of the power loss equation with respect to the bus active and reactive power injections. Each term of the linear approximation defines the fraction of the incremental losses allocated to the corresponding bus. The computation of sensitivity relationships is proposed in Ref. [4] for loss allocation purposes and in Ref. [12] to assess the bus participation in the use of the transmission system. Usually, the largest incremental transmission factors are attributed to buses electrically far from the generation centers. However, although these factors are easy to compute, this type of loss allocation strategy can be arbitrary and discriminatory [2]. Ref. [3] proposes the determination of the loss nodal factors through the decomposition of the Lagrange multipliers. For this purpose, an OPF problem is solved to minimize the active power generation cost, from which the marginal costs are obtained. In order to compute the nodal loss factors, the marginal costs are decomposed into parcels corresponding to the cost of the power supply, active power transmission loss and congestion. It is supposed that
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the marginal cost of the reference bus lr includes only the power supply and the losses (the fraction relative to the congestion is excluded). In this case, the Lagrange multipliers are expressed as l ¼ lGL þ lC and evaluated by solving the equation ð7 FÞt 21 ð7 f Þt ð7 FÞt 21 ð7 GÞt Vr V V V lr þ m l ¼ ð1Þ t t t t ð7u FÞ ð7u fr Þ ð7u FÞ ð7u GÞ
the power balance equations included in the equality constraints set. This allows the flexibility to select the buses among which the loss is shared. The computational effort required to obtain the OPF solution is moderate, which incentivizes the application of this allocation technique.
where F is the set of power balance equations of all buses excluding the reference bus, fr refers to the power balance equations of the reference bus, G is the set of inequality constraints, the matrix (whose inverse is required) is the Jacobian matrix of the power balance equations and m is the set of dual multipliers of the inequality constraints. Although the marginal cost principle leads to easily obtained nodal factors, this charging scheme might result in over-recovery [1]. To overcome this problem, adjustment strategies are proposed to scale down the losses to one half of the marginal cost based charge [5] or alternatively to scale the transmission loss coefficients such that the total value of the power loss is recovered [2]. It is also important to note that, the assumption concerning the marginal cost of the reference bus implies that the selection of this bus has considerable influence on the loss allocation. However, it is possible to select previously the set of generation buses that will supply the transmission loss (distributed slack buses) and thus, although this strategy could be considered discriminatory, to reduce the degree of arbitrariness inherent to the choice of the reference bus. The loss allocation procedure based on Ref. [3] and used here can be summarized in the following steps:
The careful analysis and extension of the solution of steady state power network equations provides basis to another type of loss allocation methodology. The strategies of this type found in the literature range from the decomposition of the transmission line flows to the use of the bus impedance matrix and current injections. Kirschen et al. [6] define two main concepts based on the analysis of the power flow, concerning the domain and corresponding link of generating units. The first establishes the set of load buses supplied by each generator (domain of the generator), which is classified according to the number of generating units related to the domain. The second concept refers to the transmission lines located on the bounds of different generators domains. The losses are divided proportionally among the generators according to their participation in the transmission lines within their domain. Similarly, Reta and Vargas [4] propose a strategy to assess the influence of each generator in the load supply, based on the superposition principle. Current sources and equivalent impedances are used to model the effect of generation on the demand supply. This procedure provides the fraction of the line current, corresponding to each generated power injection supplying the load. This fraction is used to evaluate the parcels of power loss corresponding to the loads. The loss participation factors are composed of linear and quadratic terms related to the current supplied to the load, established according to the influence of each component in the total line current. Wu and Chen [14] propose the attribution of loss components to the power demand and generation and the computation of coefficients relating the generated power to each power load, as in Refs. [6,13]. The main feature of these approaches is to model the active power transmission loss as power loads at fictitious buses located in the middle of the transmission lines. Under this assumption, the power system is considered lossless, but with an increased number of buses. The superposition principle is used to evaluate the voltage magnitude at the fictitious buses. Ref. [8] proposes a loss allocation methodology, applicable to nodal power system models, based on both the bus impedance matrix and current injections, for a given power flow solution. This approach was chosen to illustrate the study presented in this paper, but with the loss nodal factors derived in terms of power injections as follows. The complex power loss in a transmission system is given by [15,16]
† determine the OPF solution to minimize the active power transmission loss, from which the marginal costs are a by-product; † evaluate the fraction of the marginal cost relative to the transmission loss, according to Eq. (1); † compute the nodal loss participation factors by scaling the parcel of the marginal costs corresponding to the losses, such that their product by the power injections matches the total value of the active power transmission losses. The active power transmission loss is a monotonically increasing function of the active power demand. This ensures that non-negative loss parcels are obtained for all buses. In the absence of congestion, marginal costs correspond to the sensitivity of the power loss with respect to the power injections, such that there is no need of the decomposition stated by Eq. (1). Additionally, the use of scaling ensures that the precise value of the loss is matched. This methodology is dependent on the OPF formulation, and therefore there will be Lagrange multipliers only for
2.2. Loss allocation based on power flow analysis
Sl ¼ Ipt b Zb Ib
ð2Þ
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where Ib is the n-vector of the bus current injections, Zb is the n £ n bus impedance matrix (with Zb ¼ 21 Y21 ¼ Rb þ ŒXb ) and the superscript pt b ¼ ðGb þ ŒBb Þ denotes complex transpose conjugate. Eq. (2) can be decomposed in pt Sl ¼ Ipt b Rb Ib þ ŒIb Xb Ib ¼ Pl þ ŒQl
ð3Þ
where Pl and Ql are the active and reactive power transmission losses, respectively. Denote by Sb ¼ Pb þ ŒQb the n vector of the bus complex power injections and by Dv a diagonal matrix, whose elements are the inverse of the complex bus voltages Vi : Thus, the vector of bus current injections Ib is written in matrix form as Ib ¼ Dpv Spb and the active power transmission loss Pl can be expressed in terms of the bus power injections as Pl ¼ ðDpv Spb Þpt Rb ðDpv Spb Þ ¼ Stb ðDv Rb Dpv ÞSpb where ðDv Rb Dpv Þ ¼ Dr þ ŒDi and the matrices Dr and Di are symmetric ðDtr ¼ Dr Þ and anti-symmetric ðDti ¼ 2Di Þ; respectively. Therefore Pl ¼ Stb ðDr þ ŒDi ÞSpb which, in terms of active and reactive power injections, can be expressed as Pl ¼ Ptb Dr Pb þ ŒQtb Dr Pb þ ŒPtb Di Pb 2 Qtb Di Pb 2 ŒPtb Dr Qb þ Qtb Dr Qb þ Ptb Di Qb þ ŒQtb Di Qb
ð4Þ
According to the property of the matrices Dr and Di ŒQtb Dr Pb 2 ŒPtb Dr Qb ¼ 0 Ptb Di Pb ¼ 0 Qtb Di Qb ¼ 0 such that the rearrangement of the terms of Eq. (4) provides Pl ¼ Ptb Dr Pb þ Qtb Dr Qb 2 Qtb Di Pb þ Ptb Di Qb
ð5Þ
The last equation shows the dependence of the active power transmission losses on the power injection at every bus, indicating that all buses contribute to transmission losses. It is derived from the steady state power flow equations, without any approximation and thus provides the accurate value of the active power transmission loss. For a given solution (conventional or optimal) of the steady state power system equations, the following two main steps summarize the transmission loss allocation through this methodology: † determine the real part of the bus impedance matrix Rb ; † evaluate the nodal loss distribution factors from Eq. (5) as pp ¼ Dr Pb 2 Dti Qb and pq ¼ Dr Qb þ Dti Pb : Note that the terms pp and pq represent the transmission losses factors, the summation of their product by active and
reactive power injections, respectively, providing the total active power transmission loss given by Eq. (5). The application of this methodology is simple, the main difficulty being the evaluation of the real part of the bus impedance matrix. For this purpose, sparsity techniques are recommended [8]. The lack of flexibility to attribute loss parcels only to generation (or alternatively load) buses, or even to establish the fractions of the loss to be shared by the consumers and by producers seems to be its main pitfall. Besides, there is no guarantee that all nodal participation factors assigned to the buses are non-negative, which means that negative loss parcels can be attributed. 2.3. Loss allocation through integration of sensitivity coefficients The Lagrange multipliers corresponding to the OPF solution to minimize the power losses as well as the incremental transmission loss factors at any operation point are interpreted as first order sensitivity relationships between the active power transmission loss and the bus power injections. These amounts refer to infinitesimal increments and thus need to be integrated. Two types of approach based on the integration of sensitivity relationships are found in the literature. In the first, nodal loss factors are obtained by integrating the incremental transmission loss coefficients [17], or alternatively, infinitesimal loss distribution coefficients [2], evaluated in a sequence of conventional power flow solutions. Transmission loss recovery can also be seen as a cost allocation problem, which is usually stated in the form of dividing the cost of a jointly used facility among participants in a co-operative venture [1]. The Aumann – Shapley (AS) methodology, which is part of the cooperative game theory, is applicable to the problem of sharing the loss among the users of the transmission system. It is based on the integration of the marginal costs to provide the so-called AS values. In order to share the cost corresponding to the active power transmission loss, Refs. [10,11] propose to integrate the difference between the Lagrange multipliers obtained from two OPF solutions for minimum active power generation cost, one considering and the other neglecting the resistance of the transmission lines. In Ref. [9], the active power transmission loss is minimized such that the integration of the corresponding Lagrange multipliers provide nodal factors relating the loss to the power injections. The approaches described as follows, based on conventional and OPF solutions, were selected for the present study. According to Ref. [2], the loss change corresponding to an infinitesimal power demand change is reflected on the set of generation buses responsible for loss supply and/or the set of load buses. It is shown that an assignment a priori of the proportion of the total losses Pl to generation/load buses is
R.S. Salgado et al. / Electrical Power and Energy Systems 26 (2004) 81–90
necessary. This proportion is represented P by the components ri of vector r; which are such that ni¼1 ri ¼ 1; with the product rPl providing the fraction of the loss assigned to each bus. Similarly, the power demand supply is supposed to be shared among the generators according to prespecified distribution factors mi ; satisfying the condition Pn tot tot i¼1 mi ¼ 1: The product mPd (where Pd is the total active power demand and m is a vector with components mi ) gives the parcel of the demand supplied by each ith generating unit. Under these assumptions, the incremental active power balance equation is expressed as dPg 2 d Pd ¼ dPðdÞ (where for the sake of simplicity the bus voltage magnitudes are omitted) or, alternatively m dPtot d þ r dPl 2 dPd ¼ dPðdÞ and the infinitesimal increment of the loss attributed to the jth bus, denoted dLj ; is given by dLj ¼
n X a j 2 ai i¼1
aT r
mi dPdj
ð6Þ
where dPdj is the power demand increment at the jth bus, the n-vector a has components ai ¼ 1 2 ITLi (ITLi is the incremental transmission loss factor of the ith bus) and the index i refers to generation buses. Note that the amount dLj =dPdj can be seen as the sensitivity relationship between the loss and the active power injection at the jth bus. The precise contribution of the jth bus to the power transmission loss is obtained by integrating Eq. (6) over the interval corresponding to demand at bus j; that is Lj ¼
( n ðPdj X aj 2 ai 0
i¼1
aT r
) mi
dPdj
ð7Þ
where all terms have been previously defined. Although not shown in Ref. [2], this allocation strategy should take into account the variation of the active power loss with respect to reactive power changes. Since the active power transmission loss is dependent on the bus voltage magnitude (or reactive power distribution), it is expected that neglecting the reactive power can affect the value of the loss allocation coefficients and consequently result in inaccuracy in the loss recovery. The loss recovery through the computation of AS values, as proposed in Ref. [9], is based on the incremental costs resulting from the inclusion of a new participator in a coalition. In the case of the loss allocation, the Lagrange multipliers are used to this aim. The dependence of these factors on the operating points is overcome by calculating a sequence of marginal indices, resulting from small increments in the power injection, from zero up to the load level for which the loss allocation is desired. This means to integrate the marginal costs, such that the parcel of the active power transmission loss corresponding to the jth bus
85
is given by Lj ¼
ð P dj 0
lpj dPdj þ
ðQdj 0
lqj dQdj
ð8Þ
Instead of performing the explicit integration indicated in Eqs. (7) and (8), successive conventional/OPF solutions are used to provide these indices. For this purpose, the power generation/demand patterns are divided in a number of intervals, which are added cumulatively. For each addition the conventional/optimal power flow equations are solved to provide the voltage angles and magnitudes, the losses and the marginal costs/incremental transmission loss coefficients. This process is repeated until the generation matches the load level for which the loss allocation is required. The summation of these loss sensitivity factors provides the desired integration. The arithmetic mean associated with the small increments represents the exact contribution of each bus to the losses. Note that, for purpose of this work the OPF solution minimizes the active power transmission loss, the Lagrange multipliers lp and lq being expressed in MW/MW and MW/Mvar, respectively. Thus, the loss fraction Lj of Eq. (8) is expressed in MW. The procedure adopted to perform this combination can be summarized in the following steps: † Divide the power demand of each bus into N intervals. For the ith bus, each interval corresponds to Pdi =N and Qdi =N: † Starting with null demand, increase each load interval cumulatively. The load of each bus will be increased by Pdi =N and Qdi =N for each addition. † For each load increment, determine the conventional/OPF solution (active power transmission loss minimization). This provides the incremental transmission loss factors/marginal costs. † For each bus, compute the arithmetic mean of the incremental transmission loss factors/marginal costs (AS values), which are the nodal loss factors. The allocation schemes based on integration of sensitivity relationships are highly accurate but demand a large computational effort, since a conventional/optimal power flow solution is necessary at each integration interval. In order to overcome this drawback, scaling intervals are recommended in Ref. [2] which provide nodal loss factors with good level of accuracy in a small number of intervals. As in the case of the strategies based on the direct use of Lagrange multipliers, these allocation methodologies depend on the formulation used to evaluate the sensitivity relationships (marginal costs), and then allow to select the buses among which the loss is allocated. In the present case, the nature marginally crescent of the active power transmission loss function guarantees the non-negativeness of the nodal loss parcels of each bus.
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The main difference between these approaches is the way the sensitivity relationships are integrated. The premise of co-operative game theory requires the integration of marginal costs obtained from a sequence of optimal power flow solutions. If the conventional Newton –Raphson power flow solution is used to this aim, the power dispatch is not optimal as the case of the OPF, and the result of the loss recovery is prone to inaccuracies.
3. Numerical results In order to illustrate the analysis proposed in this work, the loss allocation methodologies described in Section 2 were applied to a 19-bus network whose transmission line data and diagram are presented in Table 1 and Fig. 1, respectively. This network represents an equivalent of the power system of the South –Southeastern region of Brazil. Reactors with nominal values of 200, 150, 300, 100, 150 and 300 Mvar are located, respectively, at buses 13, 15, 16, 17, 18 and 19. With the aim of better illustrating the analysis presented here, the loss allocation proportional to the load buses active power demand is shown, in addition to the loss sharing obtained through the selected allocation techniques. For the sake of simplicity, the strategies used here to loss allocation purposes are referred to as: † PRP: methodology based on proportionality to the bus power demand; † ZBS: methodology based on bus impedance matrix and power injections; † LMB: methodology based on the use of marginal costs; † ISR: methodology based on the use of integrated
sensitivity relationships; † CGT: methodology based on co-operative game theory. An OPF solution to minimize the active power transmission losses was obtained at the load level for which the loss allocation was desired. Thus, the active and reactive power generations are optimally dispatched to supply the active power load and compensating transmission losses. This operation point and the bus impedance matrix were used to calculate the nodal loss factors of the ZBS method. The Lagrange multipliers corresponding to this solution were adjusted to allocate the losses according to methodology LMB and integrated to provide the loss factors according to the premise of co-operative game theory. For allocation technique ISR, the sensitivity relationships between the transmission loss and the power demand of the load buses obtained from a sequence of power flow solutions, from which the last is the same as that obtained from OPF, were used to compute the loss factors of load buses. The nodal loss factors resulting from the application of ISR and CGT were obtained in 30 integration steps. Except for the application of Zbus methodology, the loss was assumed to be shared only by the load buses. Thus, the OPF problem for minimum loss was formulated such that the active and reactive power balances at load buses are the only equality constraints. For application of the criterion of proportionality with respect to the bus active power demand (PRP methodology), the loss fraction of the ith bus was computed as ! Pdi pli ¼ Pl Ptd where Pdi is the active power demand of the ith bus, Ptd is the total active power demand and Pl is the total active power transmission loss.
Table 1 Transmission system data From
To
R (%)
X (%)
B (%)
From
To
R (%)
X (%)
1 1 1 1 2 2 3 3 4 4 4 5 5 6 6 7 7 7 8
11 11 2 2 3 3 4 4 5 12 12 6 6 7 7 8 8 9 9
3.06 3.06 3.16 3.16 1.72 1.72 4.63 4.63 3.86 1.58 1.58 0.96 0.96 3.25 3.25 1.54 1.54 0.65 1.62
11.65 11.65 16.21 16.21 8.54 8.54 23.78 23.78 19.87 8.24 8.24 10.02 10.02 16.50 16.50 8.52 8.52 8.23 9.32
27.02 27.02 27.84 27.84 14.34 14.34 40.80 40.80 34.00 13.75 13.75 8.40 8.40 28.69 28.69 12.50 12.50
8 9 9 10 11 15 13 14 15 15 16 16 15 17 17 18 18 18
9 10 11 11 15 13 14 15 16 16 1 1 17 18 19 12 12 19
1.62 2.04 4.68 3.53
9.32 10.46 23.95 18.09 1.25 2.01 1.94 1.33 2.73 2.98 1.25 1.25 2.05 3.09 2.05 1.25 1.25 3.09
16.12
0.16 0.15 0.11 0.22 0.24
0.16 0.25 0.16
0.25
B (%) 16.12 18.08 41.38 31.25 245.8 237.0 163.7 334.0 363.0
250.2 377.7 250.2
377.7
R.S. Salgado et al. / Electrical Power and Energy Systems 26 (2004) 81–90
87
The objective of these tests was to verify how the loss parcels were modified as a result of the load change. The cases reported in the following sections illustrate these situations. 3.1. Base case Table 2 presents the results of the loss allocation for the base case. The transmission loss corresponding to the OPF solution is 26.75 MW. Buses 11, 16, 17 and 18 are transfer buses and thus no fraction of transmission losses is allocated to these buses. Strategies PRP, LMB, ISR and CGT assign no transmission losses to the generating buses (1, 3, 6, 14, 15 and 19). The demand of buses 1, 3 and 6 is supplied by the generators located at these buses, whereas there is no load located at buses 14, 15 and 19. Thus, the transmission system is not used which implies null loss factors associated to these loads. From the row indicating Total, is noted that the value of the active power transmission losses corresponding to the power flow solution is recovered with satisfactory accuracy in all cases. From columns 6 and 7 of Table 2, it is observed that results provided by PRP and LMB are very close. In this particular case, the magnitude of Lagrange multipliers related to the active power balance are close to 1.00, whereas those corresponding to the reactive power balance are close to zero. For this reason, the scaling factors used to recover the value of the loss work as a kind of proportionality coefficients. Thus, loss parcels determined according to these techniques tend to reflect predominantly the magnitude of the active power load. Column 8 of Table 2 shows that according to the approach ZBS, 10.48 MW (39%) of the total active power transmission loss (26.75 MW) is assigned to the generators
Fig. 1. The 19-bus system.
Similarly, for the application of the LMB allocation strategy the Lagrange P multipliers P were scaled by the factor Pl =P0l (where P0l ¼ lpi Pdi þ lqi Qdi is the value of the active power transmission loss estimated via Lagrange multipliers). In this case, the loss parcel assigned to bus i is ! ! Pl Pl pl i ¼ l P þ l Q P0l pi di P0l qi di Three load conditions were used to analyze the loss allocation obtained from the different methodologies. In the first, the loss parcel of each bus is evaluated for a base case. In the second, loss fractions for all buses are calculated for a given increase in the active load of selected buses. This procedure is repeated in the third case, with an increase in the reactive power demand of a set of chosen buses. Table 2 Loss allocation for the base case Bus
Pg (MW)
Qg (Mvar)
Pd (MW)
1 2 3 4 5 6 7 8 9 10 12 13 14 15 19
490.56
2105.83
430.50 157.90 138.60 158.50 108.50 98.30 188.00 224.20 243.10 124.00 407.90 902.80
241.70 54.90 2.60 44.40 44.80 75.70 106.00 70.10 23.50 26.00 34.20 92.80
3182.30
501.30
Total
301.06
349.13
271.81
94.32
642.47 997.80 427.88
2134.50 2733.09 2515.22
3209.05
21466.48
Qd (Mvar)
CPU time (s)
PRP (MW)
LMB (MW)
1.68
1.66
1.69 1.15
1.68 1.14
2.00 2.38 2.59 1.32 4.34 9.60
2.03 2.44 2.56 1.34 4.27 9.49
26.75
26.61
0.05
0.06
ZBS (MW) 0.29 0.75 0.61 1.19 0.19 0.88 3.06 4.30 1.14 1.95 1.11 2.59 2.54 2.98 3.18
ISR (MW)
CGT (MW)
1.21
1.30
2.19 0.66
1.83 0.68
3.29 4.67 1.98 2.36 2.53 5.99
4.04 5.35 2.05 2.44 2.47 6.46
26.75
24.88
26.61
0.05
0.55
5.33
88
R.S. Salgado et al. / Electrical Power and Energy Systems 26 (2004) 81–90
and 16.27 MW (61%) is attributed to the load buses. It must be recalled that this allocation strategy does not allow to allocate the loss among the load buses only (or alternatively the generation buses only). Note that the loss fraction assigned to the generators is related to the net bus current (power) injection. The power injections at generation buses 1, 3 and 6 are 60.06 (490.56 – 430.50) MW, 162.46 (301.06 – 138.60) MW and 250.83 (349.13 – 98.30) MW whereas those at generation buses 14, 15 and 19, which have no load, are 642.47, 997.80 and 427.88 MW, respectively. Although the active power generated at buses 1, 3 and 6 is high, the load of these buses reduces the current injected in the transmission grid by these generators. This explains why the loss fractions assigned to buses 1, 3 and 6 are not so high. On the other hand, there is no load at buses 14, 15 and 19 and then higher values of current are injected in the transmission system. With respect to load buses, the demand of bus 13 is considerably greater than those of the other buses but its loss parcel is the third greatest. This can be attributed to the electric distance from this bus to the generators, since the analysis of the network topology reveals that this bus is basically supplied by generation buses 14 and 15. Some similarity between the results provided by CGT and ISR could be expected, since both are based on the principle of the integration of sensitivity coefficients. However, the analysis of the last two columns of Table 2 reveals some differences between the loss parcels resulting from the application of these methods. These differences can be explained by examining the solutions of the network equations used to estimate the sensitivity relationships between the losses and the power injections. In CGT, a sequence of optimal solutions is used, and thus the active and reactive powers are optimally scheduled along the whole integration path, the Lagrange multipliers representing then optimal values of the sensitivities. Conventional power flow solutions are used in the application of ISR approach, and thus the generated reactive power is not optimally dispatched, being scheduled simply to satisfy the power balance. In other words, different sets of equations (and therefore different operation points) are used to compute the sensitivities, which explains the different loss parcels. Interesting differences can be observed between allocations obtained through LMB and CGT methods. For example, from the loss parcel of bus 13 (column 7 of Table 2) it could be inferred a considerable influence of this bus on the transmission loss. However, the integration of the corresponding Lagrange multiplier (instantaneous sensibilities) applied for CGT technique provides a different loss fraction for this bus, which reflects its location in the network. This value represents the actual influence of bus 13 on the loss, since it is measured in terms of all possible coalitions between all agents of the power system, as predicated by co-operative game theory [18]. The opposite case is that of buses 7 and 8, for instance, which are
electrically far from the generators. According to method CGT, a larger loss parcel is attributed to these buses, indicating the real effect of their power load on the total loss. The last row of Table 2 shows the CPU time required by each allocation methodology. These results were obtained in an AMD Athlon XP 1700 Processor, with 256 MB (RAM) and 1466 MHz. The large CPU time required by CGT strategy is due to the number of OPF solutions (30 in the present case) necessary to integrate the Lagrange multipliers. 3.2. Changing the active power load Aiming at assessing the loss allocations with respect to cross subsidy in terms of transmission loss, the active power demand of buses mentioned in Section 3.1 was augmented by 100 MW; that is, buses 2 (from 157.90 to 257.90 MW), 4 (from 158.50 to 258.50 MW), 5 (from 108.5 to 208.5 MW) and 7 (from 188.00 to 288.00 MW). The total active power demand increased from 3182.30 to 3582.30 MW and the transmission losses from 26.75 to 37.30 MW (increment of 10.55 MW). Naturally, changes in the transmission loss allocation are expected, with emphasis in the loss fractions corresponding to buses with modified active power demand. Table 3 shows the optimal power dispatch and the loss allocation for the case of increased active power demand. These results present a considerable variation of the fractions corresponding to the buses whose demand was increased. The total parcel of the transmission losses attributed to these buses can be obtained by summing the values in rows corresponding to buses 2, 4, 5, and 7 (columns 6 –10). The fractions of the loss increase of 10.55 MW allocated for each methodology to these buses are: 6.49 MW (61.52%) for PRP, 6.57 MW (62.27%) for LMB, 7.04 MW (66.73%) for ZBS, 8.98 MW (85.12%) for ISR and 9.57 MW (90.71%) for CGT. These values are obtained by summing (for each methodology) the increase in the loss fraction (with respect to the base case) corresponding to these buses. Therefore, methodologies ISR and CGT assign the major part of the loss increase to the buses with active power demand increase, so as to reduce as much as possible the cross subsidy. Note that, because of the change in the power flows due to the load increase, the loss fraction assigned to every bus is modified, being unavoidable to attribute the whole amount of loss increase to the buses whose demand was modified. The increase in the loss fraction of buses 2, 4, 5 and 7 evaluated through the other techniques (PRP, LMB and ZBS) do not significantly reflect the increase in the load of these buses, which indicates a greater level of cross subsidy. 3.3. Changing the reactive power load Table 4 shows the optimal power dispatch and the loss fractions allocated to the buses for the case in which an increase of 100 MVAr is assigned to the load of buses 2
R.S. Salgado et al. / Electrical Power and Energy Systems 26 (2004) 81–90
89
Table 3 Loss allocation after increasing the active power at selected buses Bus
Pg (MW)
1 2 3 4 5 6 7 8 9 10 12 13 14 15 19
530.97
708.33 996.92 498.45
2137.09 2700.95 2509.51
Total
3619.60
21379.63
396.30
488.64
Qg (Mvar) 297.58 256.08
121.58
Pd (MW)
Qd (Mvar)
430.50 257.90 138.60 258.50 208.50 98.30 288.00 224.20 243.10 124.00 407.90 902.80
241.70 54.90 2.60 44.40 44.80 75.70 106.00 70.10 23.50 26.00 34.20 92.80
3582.30
501.30
PRP (MW)
LMB (MW)
3.31
3.30
3.32 2.68
3.33 2.66
3.70 2.88 3.12 1.59 5.24 11.59
3.79 2.96 3.09 1.61 5.15 11.42
37.44
37.31
(reactive power demand increases from 54.9 to 154.9 MVAr), 4 (from 44.40 to 144.40 MVAr), 5 (from 44.80 to 144.80 MVAr) and 7 (from 106.00 to 206.00 MVAr). The total reactive power demand is 901.3 against 501.3 MVAr corresponding to the base case. The increase in reactive power demand resulted in an increase in the active power transmission loss (from 26.75 to 29.55 MW, increase of 2.8 MW) with a major impact on the buses whose demand has been augmented. Note that, although the reactive power demand was increased in 400 Mvar (79.8% with respect to the base case), the active power loss varies only in 11.3%, emphasizing the second order effect of the reactive power distribution over the active power transmission loss. In this case, the parcels of the increase in the transmission losses attributed to buses 2, 4, 5, and 7 and the percentages of the loss increase of 2.80 MW that they
ZBS (MW) 0.47 2.10 1.06 3.25 1.16 1.74 5.72 4.86 1.26 1.96 1.14 1.89 3.48 3.42 3.94 37.44
ISR (MW)
CGT (MW)
3.13
3.40
4.59 2.07
4.36 2.29
6.54 5.47 2.30 2.48 2.85 6.17
7.37 6.04 2.29 2.48 2.72 6.34
35.58
37.30
represent are: 0.68 MW (24.29%) for PRP, 0.72 MW (25.71%) for LMB, 3.07 MW (109.64%) for ZBS, 2.46 MW (87.86%) for ISR and 3.45 MW (123.21%) for CGT. The results obtained through methods CGT and ZBS show that an amount greater than the loss increase is allocated to the buses with increased demand. There is no considerable change in the Lagrange multipliers corresponding to the buses 2, 4, 5 and 7 for OPF solution at this load level, with respect to those of the base case. However, in case of CGT, in which the reactive power is dispatched to minimum active power losses, the integrated Lagrange multipliers are significantly modified, such that the increment in transmission losses is mainly allocated at the buses with increased reactive power load. The reactive power load change considered here degrades the power factor of buses 2, 4, 5 and 7, and results in a lower voltage magnitude with respect to the base case.
Table 4 Loss allocation after increasing the reactive power at selected buses Bus
Pg (MW)
1 2 3 4 5 6 7 8 9 10 12 13 14 15 19
490.62
Total
302.33
349.16
Qg (Mvar) 269.09 233.42
248.81
644.33 996.93 428.19
2144.13 2626.19 2469.21
3211.55
21036.38
Pd (MW)
Qd (Mvar)
430.50 157.90 138.60 158.50 108.50 98.30 188.00 224.20 243.10 124.00 407.90 902.80
241.70 154.90 2.60 144.40 144.80 75.70 206.00 70.10 23.50 26.00 34.20 92.80
3182.30
901.30
PRP (MW)
LMB (MW)
1.86
1.84
1.86 1.27
1.85 1.26
2.21 2.63 2.86 1.46 4.79 10.61
2.28 2.68 2.81 1.46 4.68 10.38
29.55
29.24
ZBS (MW) 0.22 1.50 0.58 1.73 0.59 1.44 4.44 4.65 1.16 1.96 1.17 2.46 2.43 2.46 2.74 29.55
ISR (MW)
CGT (MW)
1.62
2.09
2.60 1.01
2.14 1.39
4.58 5.06 1.98 2.37 2.54 5.94
5.68 5.54 1.94 2.35 2.28 5.80
27.70
29.25
90
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However, the comparison between columns 6 and 7 of Tables 2 and 4 shows that loss allocations provided by PRP and LMB for these buses do not reflect effectively this reactive power load increase. In case of LMB method, it must be recalled that in case of the performance index considered here in the OPF, the Lagrange multipliers corresponding to the reactive power balance are considerably smaller than those related to the active power balance. Besides, the changes in the former multipliers are not so significant, which is the reason for the absence of the major differences previously mentioned. In case of ZBS and CGT methods, the transmission loss is re-allocated, being penalized the buses more responsible for the decrease in the voltage magnitude level.
4. Conclusions The methods presented in this study provide different results with respect to the recovery of the active power transmission loss. Concerning the theoretical concepts and the numerical results shown in this study, the following must be remarked: † Similar results with respect to the active power transmission loss allocation are obtained by the approaches based on the integration of sensitivity relationships or Lagrange multipliers, in terms of both the loss fractions assigned to the load buses and the recovery of the total active power transmission loss value. † The loss allocation based on the integration of sensitivity relationships depends on the integration path. This approach is based on the conventional Newton – Raphson power flow solution, which does not take into account the optimal distribution of reactive power. For this reason a small accuracy error can be expected in the recovery of the transmission loss value. † The application of methodology based on the bus impedance matrix allows to recover accurately the value of the transmission losses. However, loss fractions are attributed to the generation buses. Thus, although the computational effort demanded by this technique is moderate, there is no flexibility to allocate losses for load buses only (or generations buses only). † The transmission loss distribution factors obtained through the direct use of the Lagrange multipliers require additional adjustments to recover the transmission loss accurately.
Acknowledgements R. Salgado thanks Coordenac¸a˜o de Aperfeic¸oamento de Pessoal de Nı´vel Superior, CAPES/Brazil for financial support. C.F. Moyano thanks for the financial support granted by Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico, CNPq/Brazil.
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