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Transmission loss allocation in a deregulated electrical energy market
Electric Power Systems Research 76 (2006) 962–967
Transmission loss allocation in a deregulated electrical energy market S. Abdelkader Dammam College of Technology, Dammant 31472, Saudi Arabia Received 1 May 2005; received in revised form 1 August 2005; accepted 13 December 2005 Available online 24 January 2006
Abstract This paper presents a new method for transmission loss allocation in a deregulated electrical power market. The proposed method is based on physical flow through transmission lines. The contributions of individual loads to the line flows are used as basis for allocating transmission losses to different loads. With minimum assumptions, that sound to be reasonable and cannot be rejected, a novel loss allocation formula is derived. The assumptions made are: a number of currents sharing a transmission line distribute themselves over the cross section in the same manner; that distribution causes the minimum possible power loss. Application of the proposed method is straightforward. It requires only a solved power flow and any simple algorithm for power flow tracing. Both active and reactive powers are considered in the loss allocation procedure. Results of application show the accuracy of the proposed method compared with the commonly used procedures. © 2005 Elsevier B.V. All rights reserved. Keyword: Transmission loss allocation
1. Introduction The last decade has witnessed drastic changes in the electric power industry in many parts of the world. The vertically integrated systems have been restructured and unbundled to one or more generation companies, transmission company, and a number of distribution companies. The power market has been deregulated and the government monopoly is substituted by several private sector competitors. The implementation of deregulation in power systems is based on two main different concepts: power pool and bilateral contracts. In power pool, the generating utilities or the independent power producers and customers both bid for selling and buying power at the pool. Power pool conducts different types of auction: day ahead market, hour ahead market, real time market, etc. In bilateral contracts, generating utilities and customers contract each other for selling and buying power. The seller arranges the transportation of the contracted power over the transmission network. In both cases, a transparent method for allocating transmission losses between all of the interested parties in an equitable and fair manner is required.
E-mail address: sobhy [email protected]. 0378-7796/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2005.12.014
Two facts make the allocation of transmission losses between different loads and generators an uneasy job. The first is that electricity is not a stamped commodity and it is not easy to determine which generator or load is responsible for the flow and loss in a given line. The second is that transmission loss is a non-linear function of line flows. Even, if linearization techniques are used to allocate the flow of a given generator to a certain load, the cross terms associated with the quadratic loss functions do not allow assigning directly losses to generators and consumers [1]. Therefore, the existence of a unique procedure for transmission loss allocation is not expected and several different methods were introduced based on different techniques [2–17]. The simplest procedure is the pro rata [2], where total losses are firstly assigned to generators and loads, normally 50% of losses are assigned to each category. Then losses are allocated to individual generators (customers) proportional to their active power generated (consumed) but not on their relative location within the network. This makes remotely located generators or loads to benefit at the expense of all others. Methods based on calculating transmission loss factors (TLF) have been presented in refs. [3,4]. In ref. [3], TLF is probabilistically calculated from load duration curve considering the non-linearity of transmission flow and loss. In ref. [4], transmission loss penalty factors for area energy interchange are
S. Abdelkader / Electric Power Systems Research 76 (2006) 962–967
calculated based upon the Jacobian method, and considering the electrical distances between energy sending and receiving points and the area inter-tie locations. The proportional sharing principle has been used to develop different methods for loss allocation. Refs. [5–7] are examples of these methods, where the results of a converged power flow are used along with a linear proportional sharing procedure to allocate transmission losses between loads and generators. In these methods, the power flow of generators and loads is traced to determine the transmission system usage by each generator and load. Then, transmission losses caused by each generator or load are determined. In bus-oriented methods, Z-bus is used to determine the loss components associated with different buses [8]. Injection shift distribution factor, generalized generation distribution factors, and generalized load distribution factors are derived based on AC power flow and used to allocate transmission losses to system busses [9]. The marginal transmission loss and incremental load flow concepts have been utilized for loss allocation in refs. [10,11]. A radial equivalent network has been derived in ref. [12] and used for allocation of transmission losses. A nice comparison of the most common practical algorithms for loss allocation is presented in ref. [13]. New concepts such as counter flow [14], center of losses [15] were also introduced and used for loss allocation. To account for incompatible transmission prices between individual countries having cross border trades, a unifying tracing-based method of transmission pricing is introduced in ref. [16]. As it may be clear now, the issue is still a matter of wide controversy and there is no a widely accepted technique for allocating transmission losses. It is, in almost all cases, the agreements and contracts between the interested parties that say how losses would be allocated. This is a direct result of the ambiguity in the allocation of the cross terms of the transmission losses. In a trial to derive a method for fair allocation of transmission losses, the works of ref. [1] ended up with proposing three different methods for allocating the cross terms without proofing which one is the correct one. Also, in ref. [17] a method for cross loss proportioning is introduced and proved to be fair for several special cases. In this paper a method for transmission loss allocation is introduced. The proposed method is based on the physical line flows and the actual line sharing between loads. The paper is organized as follows. Following this introduction, the proposed method is presented, where the proposed equation for unbundling line losses between two loads is derived, followed by generalization to multiple loads in a system with multiple busses. Application to a six-bus test system as well as comparison with two of the most commonly used procedures is also presented, followed by conclusions.
of the conductor area is occupied by each load current. The line can then be unbundled to sub lines, each of them carrying one load current, while all physical relations are maintained. The power loss in each sub line is the power loss caused by the load it carries. Doing that ensures accurate allocation of losses to load currents. This makes the loss allocation between interested parties justifiable and more convincing being having a physical basis. The method is based on the assumption that: when two or more currents share a conductor, they will have equal chances to occupy the conductor cross section. This does not mean that the conductor area will be divided equally between the currents. It means that each current will distribute itself over the conductor area in the very same way the other currents do. A direct result of this assumption is that, the effective area each conductor occupies will be proportional to its magnitude. This assumption and its result sound to be reasonable and cannot be disproved. According to the assumption made above, if a DC line feeds two currents, I1 and I2 (Fig. 1). The conductor area will be divided between the two currents such that a1 :a2 = I1 :I2 . Moreover, it is the very nature of the electric currents, when they have a chance, to distribute itself in a way that minimizes power loss. Now, let the area occupied by I1 be a1 , and that occupied by I2 is a2 = a − a1 . The total power loss in the line will be: PLossL =
ρ 2 ρ 2 I1 + I a1 a − a1 2
(1)
It is easy to prove that PLossL is minimum, only when the condition a1 :a2 = I1 :I2 is satisfied. Under such a condition, a1 and a2 are found to be: a1 =
aI1 , I
a2 =
aI2 I
(2)
and the resistance to I1 and I2 will be r1 and r2 , respectively, and they are given by the following equations in terms of the line resistance, r: r1 =
rI , I1
r2 =
rI I2
(3)
It can be easily realized that smaller current faces a larger resistance because of the smaller portion of the conductor it passes through. This may appear unfair and the customer of that load can claim that if he is given an equal chance to use an equal area as the large load his losses will be smaller. The answer to that is simple and clear, that is the transmission line is constructed with a predefined capacity that is to be fully utilized as possible. Also, the large load will pay more for using larger capacity of the line. Therefore, the fairness of this allocation is not questionable at all. After all, this is what physically happens in the line.
2. The proposed method The proposed method allocates transmission losses to loads based on the actual, physical, usage of the line, i.e. how much
963
Fig. 1. DC line feeding two loads.
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S. Abdelkader / Electric Power Systems Research 76 (2006) 962–967
Fig. 2. Unbundling of the DC line into two lines.
The line can now be split into two lines, as shown in Fig. 2. It can be easily find out that the two lines are equivalent in all electrical aspects to the original line. The power loss caused by each current can now be calculated as: PLoss1 = R1 I12 ,
PLoss2 = R2 I22
(4)
PLoss1 = RI1 I,
PLoss2 = RI2 I
(5)
Eq. (5) can be read in many different equivalent ways that can be used for allocating the line loss to the two loads. (1) Splitting the line into two parallel lines, one separate line for each load current as shown in Fig. 2. Resistances of these lines are inversely proportional to the currents flowing in each line. This ensures that: the equivalent resistance of the two lines is equal to the resistance of the original line, the voltage drop across each line is equal to voltage drop across the original line, and the power loss in the two line is equal to the power loss in the original line. This means the two parallel lines are perfectly equivalent in all aspects of the original line and gives an exact unbundling of the transmission losses. (2) Losses caused by each load is equal to the product of that load current and the voltage drop across the line. (3) Losses caused each load is equal to the product of the voltage drop caused by that load current and the total load current. (4) The total losses can be distributed between loads in proportion to their load currents. All of these are true for the case of DC currents, but for AC lines the story is different. This is due to the fact that AC currents are vectors defined by magnitude and phase, and also due to the presence of line reactance. However, the following section describes how to use the same idea for unbundling the line losses. Now considering an AC line feeding two loads (Fig. 3). Unlike the DC case, currents are not defined by its magnitude only, but also by its phase. The net flow current in the line is the vector sum of the two currents. Line impedance is also different because of the presence of the reactance in addition to resistance.
Fig. 4. Vector diagram of load and line currents.
As shown in Fig. 4, taking the direction of the line current as reference, each of the two load currents can be resolved into two components, one in the direction of the line current and the other perpendicular to it. The perpendicular components of the two currents will be equal in magnitude and opposite in direction, and therefore cancel each other. This means that the line current equals the algebraic sum of the components of load currents that are in its direction. In other words, line current equals the algebraic sum of the projections of load currents on its direction. IL = I1L + I2L
(6)
where I1L =
I1 · IL , IL
I2L =
I2 · IL IL
(7)
where (·) is the dot product operator of vectors. If I1 = a1 + jb1 and I2 = a2 + jb2 , then I1 ·I2 = a1 a2 + b1 b2 . Transmission losses can now be unbundled in a way similar to the DC case but using the contributions of I1 and I2 to the line current, I1L and I2L , instead of I1 and I2 . Therefore, losses caused by I1 and I2 can be calculated as follows: PLoss1 = rI1L IL = r(I1 · IL ) PLoss2 = rI2L IL = r(I2 · IL )
(8)
Now, considering voltage of the receiving bus as reference, I1 , I2 and IL will be: P1 − jQ1 , V where
I1 =
PL = P1 + P2 ,
I2 =
P2 − jQ2 , V
IL =
PL − jQL V
QL = Q1 + Q2
The power loss caused by the first load is: PLoss1 =
(P1 PL + Q1 QL )r V2
(9)
and the power loss caused by the second load is: PLoss2 =
(P2 PL + Q2 QL )r V2
(10)
From Eqs. (8) and (9) it is easy to prove that: PLoss1 = Fig. 3. AC line feeding two loads.
P1 PL + Q1 QL PLossL PL2 + Q2L
(11)
S. Abdelkader / Electric Power Systems Research 76 (2006) 962–967
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and PLoss2 =
P2 PL + Q2 QL PLossL PL2 + Q2L
(12)
And it can be proven that, if n different loads shares a transmission line, the transmission loss caused by the ith load can be calculated as follows: PLossi =
Si · SL PLossL |SL |2
(13)
where Si is the complex power of the ith load, SL the total complex power through the line and PLossL is the total active power loss of the line. All P’s and Q’s in Eqs. (11)–(13) must follow the same sign convention to distinguish between load and injection power, and between leading and lagging reactive power. Eq. (13) is our proposal for allocating transmission losses to different loads sharing the same transmission line. 3. Application to power systems All what is needed to apply the proposed method to a multinode power system is the solved power flow. Different methods are available for power flow tracing [7], by which the flow caused by each load in the system branches can be determined. The line flows caused by the system loads are arranged in a matrix from as follows: F = [fi,j ],
(14)
where fi,j is the flow caused by load (i) in branch (j). Then, losses can be allocated to each load using the following equation: [PLossi ] = [fi,j ] · [Gj ] PLj Sj |Sj |2
Table 1 Bus data for the test system Bus no.
V (pu)
PG (MW)
PL (MW)
QL (MVAR)
1 2 3 4 5 6
1.05 1.1 – – – –
– 50 0 0 0 0
0 0 55 0 30 50
0 0 13 0 18 10
Table 2 Line data for the test system Line From
To
1 1 2 2 4
4 6 3 5 6
R (pu)
X (pu)
Ysh (pu)
0.080 0.123 0.723 0.282 0.097
0.370 0.518 1.050 0.640 0.407
0.014
0.015
(15)
where [Gj ] is an NB × 1 vector, and NB is the number of branches. Gj =
Fig. 5. Six-bus test system.
(16)
where PLj is power loss in branch (j) and Sj is the total flow in branch (j).
Table 3 Transformer data for the test system Bus From
To
4 6
3 5
X (pu)
Tap
0.133 0.300
0.90909 0.97560
Or its equivalent form: [PLossi ] = ([fi,j ][G∗j ])
(17)
[PLossi ] is an NL × 1 vector of the power losses allocated to loads, and NL is the number of load busses.
Results of power flow solution are shown in Tables 4 and 5. Table 4 shows bus voltages, while Table 5 lists the power flows at both ends of each line and transformer, as well as the power loss in each line. The directions of power flows are shown by arrowheads on the single line diagram of the system.
4. Test case To show how the new method is applied and to compare with different methods, the system shown in Fig. 5 is used as a test case. Although, it has got only six buses, application of the new method to any system of a real size will be the same as it is applied to that test system. Moreover, the small size of the system makes power flow through it easily and accurately traced, which guarantees accurate allocation of losses. Data for the system are listed in Tables 1–3.
Table 4 Bus voltages of the test system Bus no.
V
δ
1 2 3 4 5 6
1.0500 1.1000 0.9928 0.9219 0.9060 0.9009
0.000 −3.66 −12.89 −9.89 −12.42 −12.28
PG − PL 95.75 50 −55 0 −30 −50
Q G − QL 51.46 21.44 −13 0 −18 −10
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S. Abdelkader / Electric Power Systems Research 76 (2006) 962–967
Table 5 Line flows and power losses
This can be easily explained recalling that pro rata does not consider neither the physical location nor the reactive power of the load. Also, the ITL considers the active power, whereas the Line Sending end Receiving end Losses proposed method considers both active and reactive powers of From To P Q P Q P Q the loads as well as the physical flows in the lines. 1 4 51.25 27.63 −48.72 −18.69 2.52 11.6 The biggest difference takes place in the loss allocated to 1 6 44.49 23.82 −41.65 −11.85 2.84 11.9 bus 5 load. To determine which one of three methods is more 2 3 17.18 0.74 −15.41 1.82 1.76 2.56 accurate, let us refer to the actual flows and power losses in the 2 5 32.81 20.69 −29.30 −12.73 3.50 7.9 system. It is easy to find out that all the power flow through 4 6 9.14 1.46 −9.04 −3.51 0.104 0.43 4 3 39.58 17.23 −39.58 −14.8 0.00 2.4 lines 2–5 goes to bus 5. Therefore, all the power loss in that line, 6 5 0.69 5.37 −0.69 −5.26 0.00 0.1 3.5 MW, has to be allocated to the load at bus 5. This makes the losses allocated by the pro rata and ITL, 2.388 and 2.3 MW, far below the actual value, while the proposed method allocates 3.638 MW to that load, which sounds reasonable considering The branch flows caused by loads are calculated and matrix the losses caused by the flow coming to bus 5 from the other [F] is formed. The lossy branches (1–4, 1–6, 2–3, 2–5, 4–6) and side. Therefore, it can be claimed that the proposed method is the loaded buses (3, 5, 6) only are considered in the listed order. more accurate and its allocation of losses is justifiable. ⎡ ⎤ 39.584 + j17.234 0.0 + j0.0 15.416 − j1.823 0.0 + j0.0 0.0 + j0.0 ⎢ ⎥ 0.568 + j4.143 0.0 + j0.0 29.3 + 12.73 0.123 + j1.227 ⎦ [F ] = ⎣ 0.125 + j0.509 9.0196 + j0.948 41.083 + j7.714 0.0 + j0.0 0.0 + j0.0 8.9171 + j2.286 The vector [G] is calculated and found to be as follows: ⎤ ⎡ 0.04515 + j0.01732 ⎥ ⎢ ⎢ 0.06312 + j0.01797 ⎥ ⎥ ⎢ ⎥ [G] = ⎢ ⎢ 0.11308 − j0.01337 ⎥ ⎥ ⎢ ⎣ 0.10069 + j0.04374 ⎦ 0.00991 + j0.00388 The loss vector allocating the transmission losses to loads is found to be: ⎡ ⎤ 3.853 ⎢ ⎥ [PLoss ] = ⎣ 3.638 ⎦ 3.253 No losses were allocated to generators. It is the practice of many systems to allocated losses to loads only, the Spanish system is an example. It makes no difference if the losses were portioned between generators and loads. For the purpose of comparison, losses were allocated using the most common techniques, incremental transmission loss (ITL) and pro rata. Results of loss allocation by the two methods are listed along with the proposed method in Table 6. The results show considerable difference between the proposed method and the other two, and a little difference between the Pro rata and ITL. Table 6 Transmission loss allocation (MW) Bus no.
Method Pro rata
3 5 6 Total
ITL
Proposed
4.377 2.388 3.979
4.194 2.300 4.250
3.853 3.638 3.253
10.744
10.7449
10.744
5. Conclusions The paper presents a method for allocation of transmission losses based on the physical power flow in the system. The method uses the contributions of the individual load currents as a basis for determining the shares of individual loads in the total line loss. Being based on the actual contribution of each load to the total line current, the proposed loss allocation formula gives, to a good extent, a fair estimate for the loss caused by each load. That is to say, loads causing the line current, and hence the line loss, to increase will be charged, whereas loads causing decrease in line current will be remunerated. In both cases, the amount of charge or remuneration will be fair and justifiable. The method considers both active power and reactive power of loads in the loss allocation procedure. It can be used also for allocating losses between loads on the same bus in a fair manner. It requires a solved power flow and power flow tracing through the system. Therefore, it can be used only for a posteriori loss allocation. However, the approach on which the method is based is being explored to find a method for a priori loss allocation. References [1] A.G. Exposito, J.M.R. Santos, T.G. Garcia, E.R. Velasco, Fair allocation of transmission power losses, IEEE Trans. Power Syst. 15 (February) (2000) 184–188. [2] M. Ilic, F. Galina, L. Fink, Power System Restructuring: Engineering and Economics, Kluwer, Norwell, MA, 1998. [3] T.K. Hann, J.H. Kim, J.K. Park, Calculation of transmission loss factor considering load variation, IEEE Power Eng. Soc. Meeting (July) (2002) 21–25. [4] Q.C. Lu, S.R. Brammer, Transmission loss penalty factors for area energy interchange, IEEE Trans. Power Syst. 11 (August) (1997) 1185–1193.
S. Abdelkader / Electric Power Systems Research 76 (2006) 962–967 [5] J.W. Bailek, Topological generation and load distribution factors for supplement charge allocation in transmission open access, IEEE Trans. Power Syst. 12 (August) (1996) 1387–1392. [6] G. Strbac, D. Kirschen, S. Ahmed, Allocating transmission system usage on the basis of traceable contributions of generators and loads to flows, IEEE Trans. Power Syst. 13 (May) (1998) 527– 534. [7] D. Kirschen, G. Strbac, Tracing active and reactive power between generators and loads using real and imaginary currents, IEEE Trans. Power Syst. 14 (November) (1999) 1312–1319. [8] A.J. Canejo, F.D. Galiana, I. Kockar, Z-buss loss allocation, IEEE Trans. Power Syst. 16 (February) (2001) 105–110. [9] Y.C. Chang, C.N. Lu, Bus-oriented transmission loss allocation, IEE Proc. Gener. Transm. Distrib. 149 (July) (2002) 402– 406. [10] A. Bhuiya, N. Chowhury, Allocation of transmission losses in a deregulated power system network, in: Proceedings of the IEEE Canadian Conference on Elect. Comput. Eng., Edmonton, Canada, May, 1999, pp. 1148–1152.
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[11] A.M. Leite, J.G. Carvalho Costa, Transmission loss allocation. Part I. Single energy market, IEEE Trans. Power Syst. 18 (November) (2003) 1389–1394. [12] A. Canejo, N. Alguacil, G.F. Ruiz, Allocation of the cost of transmission losses using a radial equivalent network, IEEE Trans. Power Syst. 18 (November) (2003) 1353–1358. [13] A.J. Canejo, J.M. Arroyo, N. Alguacil, A.L. Guijarro, Transmission loss allocation: a comparison of different practical algorithms, IEEE Trans. Power Syst. 17 (August) (2002) 571–576. [14] G. Gross, S. Tao, A physical-flow-based approach framework, IEEE Trans. Power Syst. 15 (May (2)) (2000) 631–637. [15] A.M. Leite, J.G. Carvalho Costa, Transmission loss allocation. Part I. Single energy market, IEEE Trans. Power Syst. 18 (November) (2003) 1395–1401. [16] J.W. Bialek, S. Zeimianek, R. Wallace, A methodology for allocating transmission losses due to cross-border trades, IEEE Trans. Power Syst. 19 (August) (2004) 1255–1262. [17] Z. Yu, The proposed CLP method for allocating real power losses of multiple flows, IEEE Power Eng. Rev. (November) (2001) 45–47.
Transmission loss allocation in a deregulated electrical energy market S. Abdelkader Dammam College of Technology, Dammant 31472, Saudi Arabia Received 1 May 2005; received in revised form 1 August 2005; accepted 13 December 2005 Available online 24 January 2006
Abstract This paper presents a new method for transmission loss allocation in a deregulated electrical power market. The proposed method is based on physical flow through transmission lines. The contributions of individual loads to the line flows are used as basis for allocating transmission losses to different loads. With minimum assumptions, that sound to be reasonable and cannot be rejected, a novel loss allocation formula is derived. The assumptions made are: a number of currents sharing a transmission line distribute themselves over the cross section in the same manner; that distribution causes the minimum possible power loss. Application of the proposed method is straightforward. It requires only a solved power flow and any simple algorithm for power flow tracing. Both active and reactive powers are considered in the loss allocation procedure. Results of application show the accuracy of the proposed method compared with the commonly used procedures. © 2005 Elsevier B.V. All rights reserved. Keyword: Transmission loss allocation
1. Introduction The last decade has witnessed drastic changes in the electric power industry in many parts of the world. The vertically integrated systems have been restructured and unbundled to one or more generation companies, transmission company, and a number of distribution companies. The power market has been deregulated and the government monopoly is substituted by several private sector competitors. The implementation of deregulation in power systems is based on two main different concepts: power pool and bilateral contracts. In power pool, the generating utilities or the independent power producers and customers both bid for selling and buying power at the pool. Power pool conducts different types of auction: day ahead market, hour ahead market, real time market, etc. In bilateral contracts, generating utilities and customers contract each other for selling and buying power. The seller arranges the transportation of the contracted power over the transmission network. In both cases, a transparent method for allocating transmission losses between all of the interested parties in an equitable and fair manner is required.
E-mail address: sobhy [email protected]. 0378-7796/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2005.12.014
Two facts make the allocation of transmission losses between different loads and generators an uneasy job. The first is that electricity is not a stamped commodity and it is not easy to determine which generator or load is responsible for the flow and loss in a given line. The second is that transmission loss is a non-linear function of line flows. Even, if linearization techniques are used to allocate the flow of a given generator to a certain load, the cross terms associated with the quadratic loss functions do not allow assigning directly losses to generators and consumers [1]. Therefore, the existence of a unique procedure for transmission loss allocation is not expected and several different methods were introduced based on different techniques [2–17]. The simplest procedure is the pro rata [2], where total losses are firstly assigned to generators and loads, normally 50% of losses are assigned to each category. Then losses are allocated to individual generators (customers) proportional to their active power generated (consumed) but not on their relative location within the network. This makes remotely located generators or loads to benefit at the expense of all others. Methods based on calculating transmission loss factors (TLF) have been presented in refs. [3,4]. In ref. [3], TLF is probabilistically calculated from load duration curve considering the non-linearity of transmission flow and loss. In ref. [4], transmission loss penalty factors for area energy interchange are
S. Abdelkader / Electric Power Systems Research 76 (2006) 962–967
calculated based upon the Jacobian method, and considering the electrical distances between energy sending and receiving points and the area inter-tie locations. The proportional sharing principle has been used to develop different methods for loss allocation. Refs. [5–7] are examples of these methods, where the results of a converged power flow are used along with a linear proportional sharing procedure to allocate transmission losses between loads and generators. In these methods, the power flow of generators and loads is traced to determine the transmission system usage by each generator and load. Then, transmission losses caused by each generator or load are determined. In bus-oriented methods, Z-bus is used to determine the loss components associated with different buses [8]. Injection shift distribution factor, generalized generation distribution factors, and generalized load distribution factors are derived based on AC power flow and used to allocate transmission losses to system busses [9]. The marginal transmission loss and incremental load flow concepts have been utilized for loss allocation in refs. [10,11]. A radial equivalent network has been derived in ref. [12] and used for allocation of transmission losses. A nice comparison of the most common practical algorithms for loss allocation is presented in ref. [13]. New concepts such as counter flow [14], center of losses [15] were also introduced and used for loss allocation. To account for incompatible transmission prices between individual countries having cross border trades, a unifying tracing-based method of transmission pricing is introduced in ref. [16]. As it may be clear now, the issue is still a matter of wide controversy and there is no a widely accepted technique for allocating transmission losses. It is, in almost all cases, the agreements and contracts between the interested parties that say how losses would be allocated. This is a direct result of the ambiguity in the allocation of the cross terms of the transmission losses. In a trial to derive a method for fair allocation of transmission losses, the works of ref. [1] ended up with proposing three different methods for allocating the cross terms without proofing which one is the correct one. Also, in ref. [17] a method for cross loss proportioning is introduced and proved to be fair for several special cases. In this paper a method for transmission loss allocation is introduced. The proposed method is based on the physical line flows and the actual line sharing between loads. The paper is organized as follows. Following this introduction, the proposed method is presented, where the proposed equation for unbundling line losses between two loads is derived, followed by generalization to multiple loads in a system with multiple busses. Application to a six-bus test system as well as comparison with two of the most commonly used procedures is also presented, followed by conclusions.
of the conductor area is occupied by each load current. The line can then be unbundled to sub lines, each of them carrying one load current, while all physical relations are maintained. The power loss in each sub line is the power loss caused by the load it carries. Doing that ensures accurate allocation of losses to load currents. This makes the loss allocation between interested parties justifiable and more convincing being having a physical basis. The method is based on the assumption that: when two or more currents share a conductor, they will have equal chances to occupy the conductor cross section. This does not mean that the conductor area will be divided equally between the currents. It means that each current will distribute itself over the conductor area in the very same way the other currents do. A direct result of this assumption is that, the effective area each conductor occupies will be proportional to its magnitude. This assumption and its result sound to be reasonable and cannot be disproved. According to the assumption made above, if a DC line feeds two currents, I1 and I2 (Fig. 1). The conductor area will be divided between the two currents such that a1 :a2 = I1 :I2 . Moreover, it is the very nature of the electric currents, when they have a chance, to distribute itself in a way that minimizes power loss. Now, let the area occupied by I1 be a1 , and that occupied by I2 is a2 = a − a1 . The total power loss in the line will be: PLossL =
ρ 2 ρ 2 I1 + I a1 a − a1 2
(1)
It is easy to prove that PLossL is minimum, only when the condition a1 :a2 = I1 :I2 is satisfied. Under such a condition, a1 and a2 are found to be: a1 =
aI1 , I
a2 =
aI2 I
(2)
and the resistance to I1 and I2 will be r1 and r2 , respectively, and they are given by the following equations in terms of the line resistance, r: r1 =
rI , I1
r2 =
rI I2
(3)
It can be easily realized that smaller current faces a larger resistance because of the smaller portion of the conductor it passes through. This may appear unfair and the customer of that load can claim that if he is given an equal chance to use an equal area as the large load his losses will be smaller. The answer to that is simple and clear, that is the transmission line is constructed with a predefined capacity that is to be fully utilized as possible. Also, the large load will pay more for using larger capacity of the line. Therefore, the fairness of this allocation is not questionable at all. After all, this is what physically happens in the line.
2. The proposed method The proposed method allocates transmission losses to loads based on the actual, physical, usage of the line, i.e. how much
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Fig. 1. DC line feeding two loads.
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Fig. 2. Unbundling of the DC line into two lines.
The line can now be split into two lines, as shown in Fig. 2. It can be easily find out that the two lines are equivalent in all electrical aspects to the original line. The power loss caused by each current can now be calculated as: PLoss1 = R1 I12 ,
PLoss2 = R2 I22
(4)
PLoss1 = RI1 I,
PLoss2 = RI2 I
(5)
Eq. (5) can be read in many different equivalent ways that can be used for allocating the line loss to the two loads. (1) Splitting the line into two parallel lines, one separate line for each load current as shown in Fig. 2. Resistances of these lines are inversely proportional to the currents flowing in each line. This ensures that: the equivalent resistance of the two lines is equal to the resistance of the original line, the voltage drop across each line is equal to voltage drop across the original line, and the power loss in the two line is equal to the power loss in the original line. This means the two parallel lines are perfectly equivalent in all aspects of the original line and gives an exact unbundling of the transmission losses. (2) Losses caused by each load is equal to the product of that load current and the voltage drop across the line. (3) Losses caused each load is equal to the product of the voltage drop caused by that load current and the total load current. (4) The total losses can be distributed between loads in proportion to their load currents. All of these are true for the case of DC currents, but for AC lines the story is different. This is due to the fact that AC currents are vectors defined by magnitude and phase, and also due to the presence of line reactance. However, the following section describes how to use the same idea for unbundling the line losses. Now considering an AC line feeding two loads (Fig. 3). Unlike the DC case, currents are not defined by its magnitude only, but also by its phase. The net flow current in the line is the vector sum of the two currents. Line impedance is also different because of the presence of the reactance in addition to resistance.
Fig. 4. Vector diagram of load and line currents.
As shown in Fig. 4, taking the direction of the line current as reference, each of the two load currents can be resolved into two components, one in the direction of the line current and the other perpendicular to it. The perpendicular components of the two currents will be equal in magnitude and opposite in direction, and therefore cancel each other. This means that the line current equals the algebraic sum of the components of load currents that are in its direction. In other words, line current equals the algebraic sum of the projections of load currents on its direction. IL = I1L + I2L
(6)
where I1L =
I1 · IL , IL
I2L =
I2 · IL IL
(7)
where (·) is the dot product operator of vectors. If I1 = a1 + jb1 and I2 = a2 + jb2 , then I1 ·I2 = a1 a2 + b1 b2 . Transmission losses can now be unbundled in a way similar to the DC case but using the contributions of I1 and I2 to the line current, I1L and I2L , instead of I1 and I2 . Therefore, losses caused by I1 and I2 can be calculated as follows: PLoss1 = rI1L IL = r(I1 · IL ) PLoss2 = rI2L IL = r(I2 · IL )
(8)
Now, considering voltage of the receiving bus as reference, I1 , I2 and IL will be: P1 − jQ1 , V where
I1 =
PL = P1 + P2 ,
I2 =
P2 − jQ2 , V
IL =
PL − jQL V
QL = Q1 + Q2
The power loss caused by the first load is: PLoss1 =
(P1 PL + Q1 QL )r V2
(9)
and the power loss caused by the second load is: PLoss2 =
(P2 PL + Q2 QL )r V2
(10)
From Eqs. (8) and (9) it is easy to prove that: PLoss1 = Fig. 3. AC line feeding two loads.
P1 PL + Q1 QL PLossL PL2 + Q2L
(11)
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and PLoss2 =
P2 PL + Q2 QL PLossL PL2 + Q2L
(12)
And it can be proven that, if n different loads shares a transmission line, the transmission loss caused by the ith load can be calculated as follows: PLossi =
Si · SL PLossL |SL |2
(13)
where Si is the complex power of the ith load, SL the total complex power through the line and PLossL is the total active power loss of the line. All P’s and Q’s in Eqs. (11)–(13) must follow the same sign convention to distinguish between load and injection power, and between leading and lagging reactive power. Eq. (13) is our proposal for allocating transmission losses to different loads sharing the same transmission line. 3. Application to power systems All what is needed to apply the proposed method to a multinode power system is the solved power flow. Different methods are available for power flow tracing [7], by which the flow caused by each load in the system branches can be determined. The line flows caused by the system loads are arranged in a matrix from as follows: F = [fi,j ],
(14)
where fi,j is the flow caused by load (i) in branch (j). Then, losses can be allocated to each load using the following equation: [PLossi ] = [fi,j ] · [Gj ] PLj Sj |Sj |2
Table 1 Bus data for the test system Bus no.
V (pu)
PG (MW)
PL (MW)
QL (MVAR)
1 2 3 4 5 6
1.05 1.1 – – – –
– 50 0 0 0 0
0 0 55 0 30 50
0 0 13 0 18 10
Table 2 Line data for the test system Line From
To
1 1 2 2 4
4 6 3 5 6
R (pu)
X (pu)
Ysh (pu)
0.080 0.123 0.723 0.282 0.097
0.370 0.518 1.050 0.640 0.407
0.014
0.015
(15)
where [Gj ] is an NB × 1 vector, and NB is the number of branches. Gj =
Fig. 5. Six-bus test system.
(16)
where PLj is power loss in branch (j) and Sj is the total flow in branch (j).
Table 3 Transformer data for the test system Bus From
To
4 6
3 5
X (pu)
Tap
0.133 0.300
0.90909 0.97560
Or its equivalent form: [PLossi ] = ([fi,j ][G∗j ])
(17)
[PLossi ] is an NL × 1 vector of the power losses allocated to loads, and NL is the number of load busses.
Results of power flow solution are shown in Tables 4 and 5. Table 4 shows bus voltages, while Table 5 lists the power flows at both ends of each line and transformer, as well as the power loss in each line. The directions of power flows are shown by arrowheads on the single line diagram of the system.
4. Test case To show how the new method is applied and to compare with different methods, the system shown in Fig. 5 is used as a test case. Although, it has got only six buses, application of the new method to any system of a real size will be the same as it is applied to that test system. Moreover, the small size of the system makes power flow through it easily and accurately traced, which guarantees accurate allocation of losses. Data for the system are listed in Tables 1–3.
Table 4 Bus voltages of the test system Bus no.
V
δ
1 2 3 4 5 6
1.0500 1.1000 0.9928 0.9219 0.9060 0.9009
0.000 −3.66 −12.89 −9.89 −12.42 −12.28
PG − PL 95.75 50 −55 0 −30 −50
Q G − QL 51.46 21.44 −13 0 −18 −10
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Table 5 Line flows and power losses
This can be easily explained recalling that pro rata does not consider neither the physical location nor the reactive power of the load. Also, the ITL considers the active power, whereas the Line Sending end Receiving end Losses proposed method considers both active and reactive powers of From To P Q P Q P Q the loads as well as the physical flows in the lines. 1 4 51.25 27.63 −48.72 −18.69 2.52 11.6 The biggest difference takes place in the loss allocated to 1 6 44.49 23.82 −41.65 −11.85 2.84 11.9 bus 5 load. To determine which one of three methods is more 2 3 17.18 0.74 −15.41 1.82 1.76 2.56 accurate, let us refer to the actual flows and power losses in the 2 5 32.81 20.69 −29.30 −12.73 3.50 7.9 system. It is easy to find out that all the power flow through 4 6 9.14 1.46 −9.04 −3.51 0.104 0.43 4 3 39.58 17.23 −39.58 −14.8 0.00 2.4 lines 2–5 goes to bus 5. Therefore, all the power loss in that line, 6 5 0.69 5.37 −0.69 −5.26 0.00 0.1 3.5 MW, has to be allocated to the load at bus 5. This makes the losses allocated by the pro rata and ITL, 2.388 and 2.3 MW, far below the actual value, while the proposed method allocates 3.638 MW to that load, which sounds reasonable considering The branch flows caused by loads are calculated and matrix the losses caused by the flow coming to bus 5 from the other [F] is formed. The lossy branches (1–4, 1–6, 2–3, 2–5, 4–6) and side. Therefore, it can be claimed that the proposed method is the loaded buses (3, 5, 6) only are considered in the listed order. more accurate and its allocation of losses is justifiable. ⎡ ⎤ 39.584 + j17.234 0.0 + j0.0 15.416 − j1.823 0.0 + j0.0 0.0 + j0.0 ⎢ ⎥ 0.568 + j4.143 0.0 + j0.0 29.3 + 12.73 0.123 + j1.227 ⎦ [F ] = ⎣ 0.125 + j0.509 9.0196 + j0.948 41.083 + j7.714 0.0 + j0.0 0.0 + j0.0 8.9171 + j2.286 The vector [G] is calculated and found to be as follows: ⎤ ⎡ 0.04515 + j0.01732 ⎥ ⎢ ⎢ 0.06312 + j0.01797 ⎥ ⎥ ⎢ ⎥ [G] = ⎢ ⎢ 0.11308 − j0.01337 ⎥ ⎥ ⎢ ⎣ 0.10069 + j0.04374 ⎦ 0.00991 + j0.00388 The loss vector allocating the transmission losses to loads is found to be: ⎡ ⎤ 3.853 ⎢ ⎥ [PLoss ] = ⎣ 3.638 ⎦ 3.253 No losses were allocated to generators. It is the practice of many systems to allocated losses to loads only, the Spanish system is an example. It makes no difference if the losses were portioned between generators and loads. For the purpose of comparison, losses were allocated using the most common techniques, incremental transmission loss (ITL) and pro rata. Results of loss allocation by the two methods are listed along with the proposed method in Table 6. The results show considerable difference between the proposed method and the other two, and a little difference between the Pro rata and ITL. Table 6 Transmission loss allocation (MW) Bus no.
Method Pro rata
3 5 6 Total
ITL
Proposed
4.377 2.388 3.979
4.194 2.300 4.250
3.853 3.638 3.253
10.744
10.7449
10.744
5. Conclusions The paper presents a method for allocation of transmission losses based on the physical power flow in the system. The method uses the contributions of the individual load currents as a basis for determining the shares of individual loads in the total line loss. Being based on the actual contribution of each load to the total line current, the proposed loss allocation formula gives, to a good extent, a fair estimate for the loss caused by each load. That is to say, loads causing the line current, and hence the line loss, to increase will be charged, whereas loads causing decrease in line current will be remunerated. In both cases, the amount of charge or remuneration will be fair and justifiable. The method considers both active power and reactive power of loads in the loss allocation procedure. It can be used also for allocating losses between loads on the same bus in a fair manner. It requires a solved power flow and power flow tracing through the system. Therefore, it can be used only for a posteriori loss allocation. However, the approach on which the method is based is being explored to find a method for a priori loss allocation. References [1] A.G. Exposito, J.M.R. Santos, T.G. Garcia, E.R. Velasco, Fair allocation of transmission power losses, IEEE Trans. Power Syst. 15 (February) (2000) 184–188. [2] M. Ilic, F. Galina, L. Fink, Power System Restructuring: Engineering and Economics, Kluwer, Norwell, MA, 1998. [3] T.K. Hann, J.H. Kim, J.K. Park, Calculation of transmission loss factor considering load variation, IEEE Power Eng. Soc. Meeting (July) (2002) 21–25. [4] Q.C. Lu, S.R. Brammer, Transmission loss penalty factors for area energy interchange, IEEE Trans. Power Syst. 11 (August) (1997) 1185–1193.
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