Circuit-theory-based method for transmission fixed cost allocation based on game-theory rationalized sharing of mutual-terms

Saeid POUYAFAR; Mehrdad TARAFDAR HAGH; Kazem ZARE

doi:10.1007/s40565-018-0489-y

Circuit-theory-based method for transmission fixed cost allocation based on game-theory rationalized sharing of mutual-terms

POUYAFAR, Saeid; TARAFDAR HAGH, Mehrdad; ZARE, Kazem 2019-01-17 00:00:00 J. Mod. Power Syst. Clean Energy (2019) 7(6):1507–1522 https://doi.org/10.1007/s40565-018-0489-y Circuit-theory-based method for transmission ﬁxed cost allocation based on game-theory rationalized sharing of mutual-terms 1 1,2 Saeid POUYAFAR , Mehrdad TARAFDAR HAGH , Kazem ZARE Abstract This paper proposes a new method to allocate desired in a fair and efﬁcient cost allocation method. the transmission ﬁxed costs among the network participants Finally, the developed technique has been implemented in a pool-based electricity market. The allocation process successfully on the 2383-bus Polish power system to relies on the circuit laws, utilizes the modiﬁed impedance emphasize that the method is applicable to very large matrix and is performed in two individual steps for the systems. generators and loads. To determine the partial branch power ﬂows due to the participants, the equal sharing Keywords Transmission ﬁxed cost allocation, Circuit principle is used and validated by the Shapley and theory, Equal sharing principle, Game theory Aumann-Shapley values as two preferred game-theoretic solutions. The proposed approach is also applied to deter- mine the generators’ contributions into the loads, and a 1 Introduction new concept, named circuit-theory-based equivalent bilat- eral exchange (EBE), is introduced. Using the proposed Among various issues related to the modern restructured method, fairly stable tariffs are provided for the partici- power systems well addressed most recently in the litera- pants. Cross-subsidies are reduced and a fair competition is ture [1, 2], deregulation and its price-based problems [3, 4] made by the proposed method due to the counter-ﬂows are of utmost importance. One of these problems is trans- being alleviated compared with the well-known Z-bus mission cost allocation (TCA). Several methodologies have method. Numerical results are reported and discussed to been proposed in the literature concerning the problem of validate the proposed cost allocation method. Comparative TCA. Traditionally, the costs were allocated to the users by analysis reveals that the method satisﬁes all conditions the Pro-Rata method. Despite the simplicity, the method disregards the network actual extent of use. Recently, the method is enriched and used to allocate the costs of the CrossCheck date: 25 September 2018 unused capacity of the transmission facilities. The task of Received: 28 February 2018 / Accepted: 26 September 2018 / allocating the transmission costs to the users taking into Published online: 17 January 2019 account the network extent of use, was ﬁrst introduced by The Author(s) 2019 the MW-mile method which is now widely applied in the & Mehrdad TARAFDAR HAGH literature [5]. tarafdar@tabrizu.ac.ir Tracing-based methods [6–8] utilize the concept of Saeid POUYAFAR proportional sharing principle (PSP) to trace the power s.pouyafar@tabrizu.ac.ir ﬂow in the network. Reference [6] proved the existence and Kazem ZARE uniqueness of a solution to the tracing problem. The kazem.zare@tabrizu.ac.ir technique of ﬂow tracing has also been extended and used Department of Electrical and Computer Engineering, as an analytical tool for transmission capacity allocation in University of Tabriz, Tabriz, Iran a highly renewable European electricity system [7]. Ref- erence [8] presented a transmission congestion (TC) Engineering Faculty, Near East University, Mersin 10, 99138 Nicosia, North Cyprus, Turkey 123 1508 Saeid POUYAFAR et al. tracing technique based on PSP. Nodal pricing is another analogy (TA) model [27], have an important advantage approach for TCA which is based on locational marginal over any cost allocation method, as previously described. price (LMP) differences, and is currently developed These methods incorporate the network characteristics worldwide. The proposed marginal pricing approach pro- directly into the allocation process. However, due to the vides the correct economic signals to the network partici- non-linear behavior of the power systems, there is still not pants. However, it is not linked to the actual transmission a unique mathematical solution for the contribution of infrastructure cost, thus, not able to recover the total customers into the transmission facilities under these transmission network cost (TNC) [9]. Reference [10] approaches. The results rely mostly on the principle applied checked this fact in several systems around the world uti- to split the mutual terms, as the main causes of the non- lizing LMP-based TCA method, and demonstrated that the linearity, between the participants. A group of papers use maximum network revenue obtained in these systems was the most common sharing principles, namely, proportional only 25% of the TNC. Some authors tried to solve the issue [29], quadratic [30], and equal sharing [19, 28, 31] to split by altering the LMPs to recover the TNC using the concept the mutual terms, whereas the others [23–27] avoid the of Ramsey pricing [10], and introducing the generation and mutual terms by considering a single variable at a time. In nodal injection penalties into the economic dispatch [11]. [29], it is revealed that proportional and equal sharing Marginal and incremental cost allocation methods, based principles bring about more reasonable results for reactive on the concept of sensitivity indices, are other pricing power allocation, compared with the single variable schemes widely applied in the literature, until recently [12]. division. The main drawback of these methods is their sensitivity to Based on the arguments, this paper presents a new cir- the choice of the slack bus. To overcome this limitation, [5] cuit-theory-based TCA method which is developed based utilized the slack bus independent distribution factors, on the modiﬁed Z-bus model. The proposed technique whereas [13] suggested the concept of optimal distributed applies the equal-sharing principle to split the mutual- slack bus. TCA methods based on some form of equiva- terms, and subsequently to determine the partial branch lents have been extended in [14, 15], in which the equiv- power ﬂows due to the participants. It also uses the Shapley alent bilateral exchange (EBE) has been built through the as well as the Aumann-Shapley values, as two preferred optimization as well as tracing-based approaches, respec- cooperative game solution concepts to validate the sharing tively. As an alternative, the optimization approach has principle applied. Moreover, a new concept, named circuit- been used recently along with the min-max fairness criteria theory-based EBE, is introduced by determining the gen- [16] to trace the real power in the network. The application erators and loads contributions into each other. of artiﬁcial intelligence (AI) to power system becomes The innovative contributions of the paper are: popular to explore, especially in power tracing problems 1) The proposed method is applicable to very large [17]. Effect of the possible interactions of components is systems, since it requires less computational efforts, often not considered in neither optimization nor AI-based and overcomes the limitations of the existing Z-bus methods, due to its additive complexity as well as the and proportional sharing (PS) methods to invert the computation time, subsequently leading to inaccuracy in large-scale sparse matrices. some cases. 2) The proposed cost allocation method is fair and There are also a group of papers, with solid economical efﬁcient and is more likely to be accepted by the foundation, that incorporate the concept of cooperative participants, because it is conﬁrmed by the game game theory [18, 19] into the problem of TCA. Although theoretic solutions, and also reﬂects the order of the method behaves well in terms of fairness and efﬁ- magnitude of generators and loads as well as their ciency, signiﬁcantly high computation time is required, if locations in the grid. applied to a large power system [20]. Recently, [21] pro- 3) The proposed method smooths the trend of the posed a beneﬁt-based TCA scheme. The challenging issue Z-bus method to reﬂect the counter-ﬂows, and, in concerning these methods is to ﬁnd the exact beneﬁt that turn, helps to reduce the cross-subsidies. This each user takes from the transmission facilities. Reference property is truly valuable, as higher counter-ﬂows [22] introduced a new load-following-based method to with excessive rewards result in unfair competition, estimate the transmission costs of each participant during a and make the results change signiﬁcantly when speciﬁed time period before entering the market. different MW-mile pricing schemes are used. The use of circuit theory to the TCA is another pricing 4) Highest tariff stability against temporal variations scheme, widely applied in the literature [19, 23–28]. The are observed by the proposed method, compared circuit-theory-based approaches, including Z-bus model with the Z-bus and PS methods. [23, 24] as well as its modiﬁed forms [26, 28], modiﬁed nodal equation (MNE) model [25], and transformer 123 Circuit-theory-based method for transmission ﬁxed cost allocation based on game-theory… 1509 5) The proposed method is less sensitive to the V ¼ z I ð3Þ k k;n n g g calculation reference side of the lines, compared n¼1 with the Z-bus method. th where z is the element of the k row of the matrix Z . 6) The proposed method works consistently for all k;n mod network conﬁgurations, as it overcomes the singu- According to (3), contribution of the generator at bus n into the voltage of the bus k is: larity problems of the similar methods to build the impedance matrix. V ¼ z I ð4Þ k;n k;n n g g g If bus k is a demand bus, its complex power consumption can be written as: 2 Modiﬁed Z-bus theory S ¼ V i ¼ y V V ð5Þ k k k k k k In this study, due to page limit, only the contributions of Applying (3) into (5), we have: the generators are considered. Therefore, the generators are 0 10 1 N N g g X X treated as current injections and the loads as equivalent @ A@ A S ¼ y z I z I ð6Þ k k;n n k g g k;n n impedances. g g n ¼1 n ¼1 g g For a power system with N buses and N lines, suppose b l that there are N generator buses and N demand buses. g d Similarly, for a given line ab, its complex power ﬂow Once the solution of a converged power ﬂow or state can be written as follows: estimator is obtained, system equivalent injection currents and admittance values will be as follows [23]: S ¼ V I ¼ V y V V þ y V ð7Þ ab a a a;sh ab ab a b a P jQ n n g g I ¼ ð1Þ where S and I are the power and current ﬂow; y and V ab ab ab y are series and shunt admittances of the line ab, a,sh where P and Q are the power injected at the generator n n respectively. Substituting equivalent values of the bus g g bus n ; V is the voltage of the generator bus n . g g g voltages from (3) into (7), we have: 0 1 Likewise, the equivalent admittance for a load bus is: P jQ n n @ A d d S ¼ z I ab a;n n g g y ¼ ð2Þ n ¼1 jj V g 2 3 N N g g X X where P and Q are the power consumed at the demand n n 1 d d 4 5 y z z I þ y z I a;sh ab a;n b;n n a;n n bus n ; V is the voltage of the demand bus n . g g g g g d n d d 2 n ¼1 n ¼1 g g Equivalent admittances of the loads are integrated into ð8Þ the network admittance matrix and the ‘‘network Y-mod matrix’’ is built. 3 Sharing principles of mutual-terms Note that treating the loads as constant impedances rather than current injections and, in turn, integrating the As seen in (6) and (8), the network power equations are load impedances into the network Y-bus matrix, in most made up of self-terms, that is, the contribution of an indi- cases, avoids the singularity problems concerning the vidual component, and mutual-terms, given by the product impedance matrix building process. Note also that due to of two distinct components. Thereby, it is necessary to split the TCA methods being developed based on the solved the mutual terms, to allocate the power equations between power ﬂow of the system, and considering the fact that for the involved components. power system steady-state studies, the actual modeling of In general, a simple form of the problem of dividing a the network loads does not inﬂuence the effectiveness of mutual term between two involved components can be the results [32], among numerous static load models, the expressed as follows: constant power PQ load model as the most appropriate one is considered in the proposed TCA method. f ðx ; x Þ¼ 2x x ¼ a x x þ a x x i j i j i i j j i j ð9Þ Using the network modiﬁed impedance matrix, Z ,to mod s:t: a þ a ¼ 2 i j write the relationship between the bus voltages and the bus where x and x are the involved components of the mutual- i j current injections, voltage of a given bus k is expressed as: term; a and a are the contribution coefﬁcients of the i j components into the mutual-term, respectively. Mutual- terms are divided between the involved components, based 123 1510 Saeid POUYAFAR et al. 8 P on the criteria applied to calculate the contribution coefﬁ- x ¼ cðNÞ Global rationality < i i2N cients of the components. Commonly used sharing princi- P ð15Þ x cðSÞ Group/individual rationality ples of mutual-terms are proportional, quadratic, and equal i2S or 50-50 sharing. where 8S N. Proportional sharing principle assumes that the coefﬁcients The Shapley value is a single point solution to the cost a , a are directly proportional to their relevant components: i j game, which is deﬁned as follows. For a n-player coali- a a i j ¼ ð10Þ tional game with real-valued gain function t(), a unique x x i j imputation of the total gain to each player i, denoted by / Regarding the constraint of (9) we have: (i=1, 2, …, n), is given by the Shapley value: i i 2x i S j!ðÞ n jj S 1 ! i i a ¼ < i / ¼ t S [ fg i t S x þ x i j n! ð11Þ S Nnfg i 2x : a ¼ ð16Þ x þ x i j -i th where S is an arbitrary subset of N with the i player Based on the quadratic sharing principle, however, the -i always excluded (denoted by \{i}); S | is the number of coefﬁcients a , a , are proportional to the square of their i j -i players in subset S . According to (16), the coalitional relevant components: rationality of (15) is not a requirement, so the Shapley a a i j ¼ ð12Þ value does not always belong to the core. However, under 2 2 x x i j the whole network game, in most cases, the coalitional rationality holds for the Shapley value solution, thus, it Again, regarding the constraint of (9) we have: belongs to the core as well. In the context of the Shapley 2x value being a part of the core, it is more preferable than > a ¼ < 2 2 x þ x i j other single point solutions, as it holds the axioms of ð13Þ 2x > symmetry, efﬁciency, additivity, and dummy player a ¼ : j 2 2 x þ x [33]. i j There exists another single point solution, namely, the Equal sharing model, on the other hand, assumes an Aumann-Shapley, which is a natural consequence of the equal and unitary value for the coefﬁcients a , a : i j Shapley value method. It is based on the premise that each a ¼ a ¼ 1 ð14Þ i j agent has to be sub-divided into inﬁnitesimal sub-agents, and the Shapley method is applied to each one as if each The sharing principles cannot be proved sub-agent were an individual [19]. mathematically, however, if any of them is conﬁrmed by Despite the fact that the game-theoretic solutions result a game-theoretic solution belonging to the core of the in fair and efﬁcient results, they are not a common TCA game, which is more likely to bring about sensible results. method, because these solutions on realistic sized problems require too much input data, which makes the handling of the game dimension a challenging issue. For example, the 4 Game-theoretic solutions calculations for a network with n participants require 2 - 1 pieces of input data [20]. However, the game-the- If a solution belongs to the core of the game, it is more oretical solutions may be used as a framework to evaluate likely to be accepted by the participants, because the the results of any other usage-based methods, or to prove problem of cross-subsidy is avoided. However, based on whether a mathematical model is rational. the game played, the core may consist of more than one point or it may be empty. To allocate the transmission costs among the users based 5 Proposed method on the game theory, a cost game has to be deﬁned ﬁrst, given by the pair (N, c), where N = {1, 2, …, n} is the set of In this section, ﬁrst, we use the Shapley as well as the players and c is a function that assigns a real number to Aumann-Shapley values, as two preferred cost game each subgroup (coalition) of N. A solution to the cost game solutions, to conﬁrm the equal sharing principle. Then, we is a cost allocation, i.e., a vector x [ R , where any element apply the equal sharing principle to determine the branch x of vector x is the cost allocated to player i. The solution power ﬂow contributions of the participants. The results are is called to be in the core, if, the following rationality then used to allocate the network costs among the users. requirements hold: 123 Circuit-theory-based method for transmission ﬁxed cost allocation based on game-theory… 1511 5.1 Validating the equal sharing principle 5.2 Allocating demand power of buses We apply the Shapley value to solve the general mutual Applying the equal sharing model to spilt the mutual term division problem, deﬁned by (9). terms of the demand power expression (6) between the The problem to be studied might be described as fol- current injections, the contribution of the generator at bus lows: how to impute 2x x to each of x , and x . We ﬁrst n into the demand power at bus k will be: i j i j g 2 0 1 0 13 compute the gains t(), i.e., the value of 2x x when two i j N N g g X X sources each acts alone and act together. 4 @ A @ A5 S ¼ y z I z I þ z I z I k;n k;n n k;n n g g g g g k k;n n k;n n g g g g n ¼1 n ¼1 g g tðÞ fg i ¼ 2x 0 ¼ 0 < i tðÞ fg j ¼ 0 ð17Þ ð22Þ tðÞ fg i; j ¼ 2x x i j Substituting (3) and (4) into (22), the simpliﬁed form of Using the detailed form of the Shapley value formula S will be: k;n (16), the imputation component of f(x,x ) to source x , i j i denoted by f , is: S ¼ y V V þ V V ð23Þ k;n k;n k g k g k k;n 0!ðÞ 2 0 1 ! f ¼ ½ tðÞ fg i 0 The expression obtained, may be applied to build the proposed circuit-theory-based EBE. The concept requires 1!ðÞ 2 1 1 ! ð18Þ þ ½ tðÞ fg i; j tðÞ fg j further investigation and will be developed in the future studies. ¼ 2x x ¼ x x i j i j 5.3 Allocating power ﬂow of branches Similarly, the imputation component of f(x ,x ) to source i j x is calculated. The results, conﬁrms the equal sharing of Again, we use the equal sharing principle to split the mutual terms, as the shares of x and x on 2x x , are the i j i j mutual-terms of the branch power ﬂow expressions same and each equals x x . i j between the current injections. Subsequently, the contri- We perform the same procedure by the Aumann-Shap- bution of the generator at bus n into the power ﬂow of the ley value to split the mutual term, f(x,x ), between its i j branch ab will be: 8 2 3 involved components, x , x . Based on the Aumann-Shapley i j < g value solution concept, unitary participation (UP) of x into 4 5 S ¼ y z I z z I ab;n a;n n g g g ab a;n b;n n g g g 2 : the f(x ,x ) is: i j n ¼1 Z Z 0 19 1 1 oftðÞx g = UP ¼ dt ¼ 2 tx dt ¼ t x ¼ x j j j ðx !f Þ @ A 0 þðz z ÞI z I a;n n a;n b;n n g g ox g g g t¼0 t¼0 n ¼1 ð19Þ 2 0 1 0 13 N N g g X X 4 @ A @ A5 þ y z I z I þ z I z I where each agent is divided into inﬁnitesimal parts (Dx?0) a;n n a;n n a;sh g g a;n n a;n n g g g g g g n ¼1 n ¼1 g g by inﬁnitesimal steps, t. Likewise, solving the problem for UP of x into the f(x , ð24Þ j i x ), we obtain: Using (3) and (4) to replace the related terms in (24), we Z Z 1 1 oftðÞx 1 have: UP ¼ dt ¼ 2ðÞ tx dt ¼ t x j ¼ x i i i ðx !f Þ j 0 hi ox t¼0 t¼0 S ¼ y V V V þ V V V ab;n ab;n a g ab g a b a;n b;n g g ð20Þ To determine the total participation (TP) of the player þ y V V þ V V a;sh a;n a g a a;n into the f(x,x ), the unitary participation is multiplied by i j ð25Þ the amount of the player: TP ¼ UP x ¼ x x ðÞ x !f ðÞ x !f i j i i i ð21Þ 6 Numerical results TP ¼ UP x ¼ x x j i j ðÞ x !f ðÞ x !f j j In this section, the proposed method is compared with It can be observed that the outcomes conﬁrm the equal two well-known TCA techniques, namely, Z-bus theory sharing of mutual terms. [24], and PS method [6]. Two test systems namely 6-bus 123 1512 Saeid POUYAFAR et al. test system and IEEE 30-bus system are used. In the con- text of the 6-bus test system, a shortened time series study is performed to gain more insights into the behavior of the three TCA methods considered. In practice, a large number of time steps need to be analyzed for consecutive studies of large systems. However, the cyclical nature of load and generation proﬁles suggests that some time steps in the proﬁles represent load and generation scenarios that reap- pear over time, and hence can be simulated only once. In this respect, four time steps are considered in our study, which sufﬁciently represent the system off-peak, transi- tional, normal, and peak conditions. For the IEEE 30-bus Fig. 1 6-bus test system system, system peak condition is considered. Any line tariff in $/h, is set to be 1000 times its series reactance. Table 1 6-bus test system branch data Zero counter ﬂow (ZCF) MW-mile pricing is used to Line r (p.u.) x (p.u.) y/2 (p.u.) charge the participants. Thus, the generators neither pay money nor take any reward for the lines in which their 1-2 0.10 0.20 0.02 associated power ﬂow is in opposite direction. Finally, the 1-4 0.05 0.20 0.02 proposed method is applied to a practical system, namely, 1-5 0.08 0.30 0.03 2383-bus Polish 400, 220 and 110 kV networks during 2-3 0.05 0.25 0.03 winter 1999-2000 peak conditions, to emphasize the 2-4 0.05 0.10 0.01 applicability of the method to very large systems. To know 2-5 0.10 0.30 0.02 which type of electricity market with its regulations best 2-6 0.07 0.20 0.02 suits a certain TCA technique, please refer to [34–36] 3-5 0.12 0.26 0.02 which provide key information about transmission pricing 3-6 0.02 0.10 0.01 experiences across various international jurisdictions. 4-5 0.20 0.40 0.04 5-6 0.10 0.30 0.03 6.1 6-bus test system The 6-bus system as shown in Fig. 1 is considered. The Table 2 Active power of loads, generators and losses corresponding displayed values on Fig. 1, represent the active power to system four time steps ﬂows corresponding to the system peak condition. The Time P P P P G1 G2 G3 system has eleven branches with parameters provided in loss L4 L5 L6 step (MW) (MW) (MW) (MW) (MW) (MW) (MW) Table 1, in which r, x, and y denote the resistance, reac- tance, and shunt admittance of the branches, respectively. 1 2.56 3.39 4.74 6.91 50.00* 42.56 45.00* The time series study is performed for the three TCA 2 45.00 45.00 60.00 70.00 50.00* 58.33 45.06* methods, and the results are depicted in Tables 2, 3, 4, 5 3 45.00 60.00 60.00 70.00 50.00* 74.37 60.37 and 6. Table 2 represents the active power of the loads, 4 45.00 45.00 60.00 70.00 77.22 69.27 70.42 generators and losses corresponding to the system four time steps considered. In Table 2, * signiﬁes the minimum PS method traces the power ﬂows using the PSP which can output capacity constraint of G1. be neither proved nor disproved, whereas the proposed According to Tables 3, 4, 5 and 6, preliminary Kirch- method is based on the circuit theory and utilizes the net- hoff’s circuit laws are satisﬁed under the proposed as well work impedance matrix to determine the partial branch as the Z-bus methods. power ﬂows due to the participants. In the case of the Z-bus For each branch of the system, in each of the four time method, however, the generators and loads are both treated steps, the overall contribution due to the generators equals as nodal currents, and hence, their responsibilities for the power ﬂow of the branch, under the proposed as well as network usage are calculated altogether in a common PS methods. This is because, unlike Z-bus method, the process. proposed and PS methodologies perform the allocation Although the amounts of the contributions differ under process for the generators and loads, independently. Nev- the three methods, the dominant positive ones are the same ertheless, the partial branch power ﬂows allocated to the for all the three methods, and belong to the genera- users differ for these two methods. Discrepancies arise tor(s) relatively close to the sending-end bus of the mainly due to the different underlying principles applied. 123 Circuit-theory-based method for transmission ﬁxed cost allocation based on game-theory… 1513 Table 3 Time series study of three TCA methods on 6-bus test Table 4 Time series study of three TCA methods on 6-bus test system (time step 1) system (time step 2) Line number TCA Power ﬂow contribution (MW) Line number TCA Power ﬂow contribution (MW) method method P G1 G2 G3 P G1 G2 G3 line line 1-2 Proposed 9.96 10.46 - 1.03 0.54 1-2 Proposed 7.89 9.96 - 2.14 0.08 Z-bus 17.95 - 4.44 - 1.53 Z-bus 18.17 - 6.07 - 1.52 PS 9.96 0.00 0.00 PS 7.89 0.00 0.00 1-4 Proposed 22.02 11.76 4.16 6.09 1-4 Proposed 21.35 11.28 4.39 5.69 Z-bus 16.19 0.20 1.06 Z-bus 16.08 0.26 1.06 PS 22.02 0.00 0.00 PS 21.35 0.00 0.00 1-5 Proposed 18.02 9.85 4.41 3.76 1-5 Proposed 20.75 10.20 6.02 4.53 Z-bus 15.86 4.24 0.47 Z-bus 15.75 5.82 0.46 PS 18.02 0.00 0.00 PS 20.75 0.00 0.00 2-3 Proposed 0.25 2.89 3.11 - 5.75 2-3 Proposed 3.03 3.13 5.04 - 5.14 Z-bus 7.12 8.13 - 8.11 Z-bus 7.10 11.22 - 7.87 PS 0.05 0.21 0.00 PS 0.36 2.68 0.00 2-4 Proposed 25.73 3.04 11.07 11.62 2-4 Proposed 29.10 3.30 14.03 11.76 Z-bus - 4.16 10.07 5.99 Z-bus - 4.03 13.77 5.89 PS 4.89 20.90 0.00 PS 3.47 25.66 0.00 2-5 Proposed 10.82 2.69 5.06 3.06 2-5 Proposed 15.33 3.56 7.53 4.23 Z-bus 3.75 7.45 1.40 Z-bus 3.82 10.23 1.29 PS 2.06 8.78 0.00 PS 1.83 13.51 0.00 2-6 Proposed 15.61 6.28 7.45 1.89 2-6 Proposed 18.70 6.46 9.98 2.26 Z-bus 7.46 9.15 - 4.25 Z-bus 7.53 12.52 - 4.23 PS 2.97 12.68 0.00 PS 2.23 16.49 0.00 3-5 Proposed 14.73 0.87 3.13 10.73 3-5 Proposed 18.03 1.62 4.42 11.99 Z-bus 0.60 3.19 14.56 Z-bus 0.60 4.41 14.80 PS 0.02 0.07 14.66 PS 0.14 1.00 16.90 3-6 Proposed 30.49 5.09 6.36 19.05 3-6 Proposed 30.02 4.76 6.58 18.68 Z-bus 0.72 0.13 16.97 Z-bus 0.70 0.22 16.92 PS 0.03 0.14 30.34 PS 0.23 1.67 28.15 4-5 Proposed 1.86 1.30 1.09 - 0.53 4-5 Proposed 4.46 1.91 2.24 0.31 Z-bus 6.18 5.23 1.96 Z-bus 6.22 7.09 1.90 PS 1.07 0.83 0.00 PS 2.24 2.31 0.00 5-6 Proposed - 0.49 1.44 - 0.14 - 1.79 5-6 Proposed - 2.96 0.69 - 0.94 - 2.71 Z-bus 4.35 1.35 - 1.38 Z-bus 4.29 1.80 - 1.30 PS - 0.03 - 0.14 - 0.34 PS - 0.15 - 1.14 - 1.77 Total transmission Proposed 498.70 400.10 406.20 Total transmission Proposed 454.10 474.00 376.90 cost ($/h) cost ($/h) Z-bus 895.70 560.30 347.20 Z-bus 817.90 655.80 305.70 PS 553.10 474.30 277.60 PS 507.90 541.80 255.20 Tariffs ($/MWh) Proposed 9.97 9.40 9.02 Tariffs ($/MWh) Proposed 9.08 8.13 8.37 Z-bus 17.91 13.16 7.72 Z-bus 16.36 11.24 6.78 PS 11.06 11.14 6.17 PS 10.16 9.29 5.66 branches. This fact holds for each time step considered. For sending-end bus of the lines 2-3, 2-4, 2-5, and 2-6, incor- example, under the proposed method, the generator at bus 1 porates most in their associated power ﬂows. The same as the closest one to the sending-end bus of the lines 1-2, principle holds for the lines 3-6, 4-5 and 5-6. This property 1-4, and 1-5, has the highest direct contribution to the may be called the nearby effect of the power networks most power ﬂow of those lines. The generator at bus 2, as the evident in the PS method. 123 1514 Saeid POUYAFAR et al. Table 5 Time series study of three TCA methods on 6-bus test Table 6 Time series study of three TCA methods on 6-bus test system (time step 3) system (time step 4) Line number TCA Power ﬂow contribution (MW) Line number TCA Power ﬂow contribution (MW) method method P G1 G2 G3 P G1 G2 G3 line line 1-2 Proposed 5.87 9.69 - 3.38 - 0.44 1-2 Proposed 15.41 16.27 - 1.69 0.83 Z-bus 18.47 - 7.80 - 2.07 Z-bus 28.14 - 7.27 - 2.35 PS 5.87 0.00 0.00 PS 15.41 0.00 0.00 1-4 Proposed 23.90 11.46 5.57 6.86 1-4 Proposed 33.95 17.70 7.48 8.76 Z-bus 15.93 0.35 1.44 Z-bus 24.80 0.43 1.64 PS 23.90 0.00 0.00 PS 33.95 0.00 0.00 1-5 Proposed 20.23 9.58 6.45 4.20 1-5 Proposed 27.86 14.67 7.67 5.53 Z-bus 15.60 7.44 0.63 Z-bus 24.28 6.84 0.72 PS 20.23 0.00 0.00 PS 27.86 0.00 0.00 2-3 Proposed 1.18 2.56 5.71 - 7.09 2-3 Proposed 0.29 3.99 4.60 - 8.31 Z-bus 7.07 14.22 - 10.80 Z-bus 11.03 12.91 - 12.70 PS 0.09 1.09 0.00 PS 0.05 0.24 0.00 2-4 Proposed 40.39 4.90 19.55 15.94 2-4 Proposed 41.74 4.63 19.85 17.26 Z-bus - 3.83 17.60 8.02 Z-bus - 6.23 16.55 9.37 PS 2.96 37.45 0.00 PS 7.62 34.25 0.00 2-5 Proposed 16.40 3.28 8.80 4.32 2-5 Proposed 17.35 3.95 8.77 4.63 Z-bus 3.90 13.02 1.87 Z-bus 5.94 12.03 2.18 PS 1.20 15.20 0.00 PS 3.17 14.24 0.00 2-6 Proposed 22.24 6.64 12.77 2.83 2-6 Proposed 25.03 9.13 12.78 3.12 Z-bus 7.60 15.99 - 5.69 Z-bus 11.70 14.99 - 6.66 PS 1.63 20.62 0.00 PS 4.57 20.54 0.00 3-5 Proposed 20.87 1.62 5.10 14.15 3-5 Proposed 23.18 1.34 5.77 16.07 Z-bus 0.59 5.58 19.53 Z-bus 0.93 5.06 22.81 PS 0.03 0.37 20.48 PS 0.02 0.08 23.10 3-6 Proposed 40.64 6.22 9.81 24.61 3-6 Proposed 47.50 7.52 11.63 28.34 Z-bus 0.67 0.22 22.77 Z-bus 1.08 0.00 26.55 PS 0.06 0.72 39.89 PS 0.04 0.16 47.32 4-5 Proposed 2.53 1.29 1.85 - 0.62 4-5 Proposed 3.21 1.91 1.92 - 0.62 Z-bus 6.21 9.02 2.61 Z-bus 9.44 8.60 3.01 PS 1.08 1.51 0.00 PS 1.82 1.50 0.00 5-6 Proposed - 1.70 1.08 - 0.37 - 2.41 5-6 Proposed - 0.90 1.98 - 0.37 - 2.52 Z-bus 4.27 2.32 - 1.85 Z-bus 6.55 2.26 - 2.13 PS - 0.05 - 0.60 - 1.12 PS - 0.06 - 0.27 - 0.62 Total Proposed 423.40 501.10 380.50 Total Proposed 475.70 444.70 384.60 transmission transmission Z-bus 738.80 714.40 341.90 Z-bus 885.30 574.80 344.30 cost ($/h) cost ($/h) PS 468.90 564.40 271.60 PS 546.60 481.40 277.00 Tariffs ($/MWh) Proposed 8.47 6.74 6.30 Tariffs ($/MWh) Proposed 6.16 6.42 5.46 Z-bus 14.77 9.61 5.66 Z-bus 11.46 8.30 4.89 PS 9.38 7.59 4.50 PS 7.08 6.95 3.93 To gain deeper insight into the results, the three methods smooths the trend of Z-bus method to reﬂect the counter- are also compared in terms of counter ﬂows. Tables 3, 4, 5 ﬂows. For example, in time step 4 representing the system and 6 shows that both circuit-theory-based methods take peak condition, the contribution of the generator at bus 1 into account the counter-ﬂows, while this feature does not into the power ﬂow of the line 5-6, is 6.55 MW counter- exist for the PS method. The proposed method, however, ﬂow under the Z-bus method, and is 1.99 MW counter- 123 Circuit-theory-based method for transmission ﬁxed cost allocation based on game-theory… 1515 Table 7 6-bus test system cost allocation to buses proposed methods, respectively. Note that, higher counter ﬂows with excessive rewards may result in unfair compe- Line Method Cost allocation to Cost Total tition, and in turn, may distort the power market. generator bus ($/h) allocation ($/h) to load Table 7 depicts, in detail, the allocated transmission use Bus 1 Bus 2 Bus 3 buses 4–6 of system (TUoS) costs and tariffs of the generators based ($/h) on the partial branch power ﬂows obtained in the system 1-2 Proposed 95.14 0 4.86 100 200 peak condition (time step 4 in Table 6). As shown in Table 7, each branch cost is allocated by 50-50 ratio Z-bus 181.73 0 0 18.27 between the generators and loads, under the proposed as PS 100 0 0 100 well as PS methods, in spite of different allocated cost 1-4 Proposed 52.14 22.05 25.81 100 200 distributions among the generators as a result of different Z-bus 129.19 2.23 8.53 60.05 partial branch power ﬂows assigned to the generators under PS 100 0 0 100 these methods, previously addressed in Tables 3, 4, 5 and 1-5 Proposed 78.96 41.28 29.76 150 300 6. In contrast, for Z-bus method it is basically the network Z-bus 193.16 54.41 5.69 46.74 parameters that determine the ratio by which the costs are PS 150 0 0 150 allocated between the generators and loads. For instance, 2-3 Proposed 58.07 66.93 0 125 250 the cost of lines 1-5 and 5-6 both equals 300 $/h. The Z-bus 104.28 122.08 0 23.64 proposed technique, as with the PS methodology, assigns PS 22.74 102.26 0 125 150 $/h (50%) of each line cost to the generators. Under 2-4 Proposed 5.55 23.77 20.68 50 100 the Z-bus method, however, 253.26 $/h of line 1-5 cost is Z-bus 0 27.60 15.63 56.77 allocated to the generators and 46.74 $/h to the loads, and PS 9.10 40.90 0 50 for line 5-6, the share of the generators and loads become 2-5 Proposed 34.15 75.83 40.02 150 300 30.83 $/h and 269.17 $/h, respectively. Z-bus 62.66 126.97 23.00 87.37 Based on the results shown in Table 7, it is conﬁrmed PS 27.29 122.71 0 150 that the proposed method considers the amount, the loca- 2-6 Proposed 36.49 51.06 12.45 100 200 tion and the effective use of the line by the generators in its Z-bus 50.93 65.26 0 83.81 allocation process. For example, the generator at bus 2 uses PS 18.19 81.81 0 100 the line 1-2 less, compared to the generator at bus 1, due to 3-5 Proposed 7.51 32.35 90.14 130 260 the power ﬂow direction of the line 1-2. These properties Z-bus 6.47 35.25 158.79 59.49 hold for the proposed method, irrespective of the pricing PS 0.10 0.43 129.47 130 method applied, that is ZCF, absolute value (AV) and 3-6 Proposed 7.91 12.25 29.84 50 100 classic MW-mile pricing. For example, under the proposed Z-bus 2.14 0 52.53 45.33 method, the network costs of the generators using the ZCF PS 0.04 0.17 49.80 50 pricing are 475.74 $/h, 444.73 $/h and 384.53 $/h, while 4-5 Proposed 99.82 100.18 0 200 400 these values change slightly to 485.86 $/h, 399.09 $/h and Z-bus 154.73 140.96 49.31 55.00 420.04 $/h, if the AV pricing is used. In case of the Z-bus PS 109.65 90.35 0 200 method, however, the results change signiﬁcantly, when 5-6 Proposed 0 19.03 130.97 150 300 different pricing schemes are used. The network costs Z-bus 0 0 30.83 269.17 allocated to the generators are 885.29 $/h, 574.75 $/h and PS 9.51 42.75 97.74 150 344.31 $/h, by the ZCF pricing, where these values become 668.43 $/h, 412.85 $/h and 337.77 $/h, by the AV pricing Network Proposed 475.74 444.73 384.53 1305 2610 method. Discrepancies arise because signiﬁcant counter- cost Z-bus 885.29 574.76 344.31 805.64 ﬂows exist under the Z-bus method. PS 546.62 481.38 277.01 1305 According to Table 7 and considering the TUoS tariffs Tariffs Proposed 6.16 6.42 5.46 of the generators, under each of the three methods, the ($/ Z-bus 11.46 8.30 4.89 generator at bus 3 pays the lowest price, whereas the MWh) PS 7.08 6.95 3.93 generator at bus 1 has to pay the highest price, for the use of the network. However, the TUoS tariffs of the genera- tors at buses 1 and 2 are relatively high according to the ﬂow, under the proposed method. In the same time step, Z-bus method, because 69.13% of the network cost is given the generator at bus 2, there exists a counter-ﬂow imposed to the generators. contribution of the generator into the power ﬂow of line The proposed method is less sensitive to the calculation 1-2, with 7.27 MW and 1.69 MW under the Z-bus and the reference side of the lines compared with the Z-bus 123 1516 Saeid POUYAFAR et al. method, for which the results change signiﬁcantly when the calculation reference side is changed. For example, under the proposed method, when P is considered as reference, 1-2 the power ﬂow contribution and the allocated cost to the generator at bus 1 due to the line 1-2, are 16.27 MW and 95.14 $/h, whereas the values change slightly to 15.93 MW and 95.00 $/h when P is considered as ref- 2-1 erence. When Z-bus is used to calculate the same quanti- ties, the values become 28.14 MW and 181.72 $/h for P 1-2 being considered as reference, whereas they change sig- niﬁcantly to 22.43 MW and 125.69 $/h when P is con- 2-1 sidered as reference. 6.2 IEEE 30-bus system To validate the proposed method, the IEEE 30-bus system illustrated in Fig. 2 is used as a test system. Branch data are provided in [37]. Bus data for the base case are Fig. 2 IEEE 30-bus test system provided in Appendix A Table A1. 6.2.1 Contribution of generators into branch power ﬂows The generators are located at buses 1, 2, 22, 27, 23, 13. Figures 3 and 4 illustrate the contributions of the genera- tors into the network real power ﬂow of the branches applied by the three cost allocation methods including Z-bus theory, PS method represented by Bialek, and the proposed method. According to Figs. 3 and 4, the following contributions are obtained: 1) The overall contribution proﬁles of the generators for the three methods are almost comparable, i.e. the power ﬂow contribution of the generators into the neighboring lines is higher compared to the others calculated by each of the three methods, although they have different principles. The generators at buses 1 and 2 contribute most in lines 1-15 (Fig. 3a and b). Likewise, for the generators connected at bus numbers 22, 27, 23 and 13, their associated lines with dominant power ﬂow contribution of the corresponding gener- ators are: line numbers 25-29 for G22; 26-41 for G27; 22, 30, 32 for G23; 16-18 for G13, according to Figs. 3c, d, 4a and b, respectively. It may be entitled ‘‘the nearby effects of the power networks’’. 2) The curves obtained by the PS theory have several lines with zero contribution of the generators. This is because the PS theory does not consider the counter- ﬂow effects in its calculations despite the fact that the concept of counter-ﬂows is indispensable in power Fig. 3 Contribution of generators G1, G2, G22 and G27 into branch ﬂow related problems. power ﬂows of IEEE 30-bus system Figure 5 shows the real power ﬂow allocation of two high loaded branches 1-2 and 6-8. The results are more 123 Circuit-theory-based method for transmission ﬁxed cost allocation based on game-theory… 1517 locational by the PS method (Fig. 5a) thanks to the pro- portional sharing principle which has no counter-ﬂow effect consideration. They are more intense by the Z-bus method in case of high counter-ﬂow shares of some gen- erators (Fig. 5b), and modest by the proposed method. Discrepancies arise by the fact that Z-bus method uses the current ﬂows while the proposed method applies the power ﬂows to allocate. For branch 6-8, each of the three methods allocate the highest contribution to the generators located at buses 1 and 2. The share of the generators located at buses other than 1 and 2 is completely different for the three methods. The PS allocation scheme is completely based on the power ﬂow directions of the lines. Among the generators located at the buses 22, 27, 23 and 13, only G27 contributes into the power ﬂow of the branch 6-8 because there is a path from the generator at the bus 27 to the sending end of the branch 6-8 (the path is 27-28, 28-6) with the same Fig. 4 Contribution of generators G23 and G13 into branch power direction as the real power ﬂow of the branches. For the ﬂows of IEEE 30-bus system Z-bus methodology, the contribution of the generators into the branch ﬂows is highly dependent on their neighboring loads since the method traces the current ﬂow of the par- ticipants. In the proposed methodology, it is the overall location of the generators with respect to the loads that determines the outcomes, which makes the generator tariffs more stable compared to the other two methods (Table 8). For branch 1-2 (Fig. 5b), the contribution of the gener- ator located at bus 1 is dominant by each of the three methods as G1 is connected to the sending end of the line 1-2. Since the counter-ﬂows are not considered in PS method, the adjacent generator G2 located at bus 2 has no share in the power ﬂow of the line 1-2. For the Z-bus method, the share of generator 1 in power ﬂow of the line 1-2 exceeds the real power ﬂow of the line by an amount almost equal to the generator G2’s signiﬁcant counter-ﬂow contribution into the line 1-2. In case of the proposed method, the counter-ﬂow share of the generator G2 is lower and the G1 has a moderate contribution amount into the power ﬂow of the line 1-2. 6.2.2 Generator buses’ TUoS charges Table 8 lists the transmission per-unit costs allocated to the generator buses for the base case and the cases with individual loads altered. P (b) in MW is the active power of the load connected to the bus b. According to Table 8, for the base case, bus numbers 1, 2 and 22 have TUoS tariffs cheaper than buses 27, 23 and Fig. 5 Partial power ﬂows of branches 6-8 and 1-2 due to generators 13 calculated by all the three methods, because large loads 123 1518 Saeid POUYAFAR et al. are connected at buses 2, 7, 8 and 21. The generators G1 for the Z-bus method and ? 18.1 $/MW (121%) for the PS and G2 tariffs are highly dependent on the large loads method. For the generator 27, the tariff growths are ? 11 $/ connected at the buses 2, 7 and 8. Thus, minor changes take MW (22%) by the proposed method, ? 12 $/MW (24%) by place by varying the loads connected at buses relatively far the Z-bus method and ? 16 $/MW (20%) by the PS from that units. This statement is shown in the Table 8 by method. The tariff decrease of the generator bus 23 is 19% altering the loads connected at buses 21, 12, 30, 23 and 27. for both the proposed and the Z-bus method and 22% for The highest TUoS tariff is assigned to the bus 27 by the PS the PS method. The same scenario takes place in case of method due to its relative distance from the large loads and the load increase at buses 30, 23 and 27. For example, the to bus 13 by the proposed method due to its circuit con- highest tariff variation due to the 10 MW load growth of dition characterized by the voltage and current equations of bus 30 is associated to bus 27 calculated by the PS method, the networks. For Z-bus method, the high tariff buses have which is a 14 $/MW decrease from 85 to 71 $/MW. almost the same prices. The generator connected at bus 22 close to the load bus 21, have the lowest TUoS tariff by the 6.3 2383-bus practical system PS method whereas its tariff is equal to the G1 and G2 charges by the proposed method due to their similar circuit Tables 9, 10 and 11 presents the results obtained by conditions. applying the proposed method to the Polish 2383-bus Among the methods discussed, the proposed method has system. To emphasize that the proposed method is appli- the highest tariff stability against the single load variations. cable to very large systems, the 2383-bus system of Polish If the load on bus 21 with the real power demand of 17.5 400, 220 and 110 kV networks during winter 1999-2000 MW is disconnected from the network, the TUoS charges peak conditions is considered. The system data are given in of the generator buses 22, 27 and 23 located in the MATPOWER user’s manual. The proposed method does neighboring zone of bus 21 will be changed by the three not encounter the singularity problems of the Z-bus method methods. The tariff variation of the generator 22 is ? 13 $/ to build the impedance matrix as well as the PS method to MW (36%) for the proposed method, ? 14 $/MW (54%) build the inverted tracing distribution matrices. An Intel Core i5, 2.3 GHz, 6 GB RAM 64-bit computer is used to run the simulations of this system. MATLAB R2016a reported the elapsed time 18.73 s which is fairly a short Table 8 Comparison of TUoS tariffs of generator buses for some operating points on IEEE 30-bus test system running time. It is noted that in Table 11, APG represents the active power generated; TTPU/TTNU represent the Operating point Method Transmission cost per-unit of generator bus ($/MWh) G1 G2 G22 G27 G23 G13 Table 9 First 5 branches with highest tariffs Branch Power ﬂow (MW) Tariffs ($/h) Base case Proposed 34 34 36 51 48 63 Z-bus 38 37 26 51 52 54 1764-1760 16.54 463.20 PS 33 24 15 85 53 44 1763-1761 38.19 452.50 P (21)=0 Proposed 36 36 49 62 39 66 612-413 63.46 324.00 Z-bus 41 39 40 63 42 51 1945-1845 - 40.60 245.50 PS 33 24 33 101 41 43 1489-1426 - 69.30 237.20 P (12)=0 Proposed 35 36 39 55 50 69 Z-bus 41 39 28 54 54 59 PS 34 24 15 89 54 60 P (30)=20 MW Proposed 32 32 33 47 44 60 Z-bus 37 35 25 46 47 51 Table 10 First 5 branches with highest power ﬂows PS 32 24 15 71 51 43 Branch Power ﬂow (MW) Tariffs ($/h) P (23)=20 MW Proposed 33 32 34 50 36 51 Z-bus 37 35 27 51 34 42 138-67 - 771.20 20.56 PS 33 24 15 91 16 41 32-31 - 681.70 0.10 P (27)=20 MW Proposed 33 35 28 42 36 53 18-15 552.20 42.62 Z-bus 39 37 22 41 39 45 15-165 451.60 24.74 PS 34 24 16 60 41 42 132-131 - 416.50 0.10 123 Circuit-theory-based method for transmission ﬁxed cost allocation based on game-theory… 1519 Table 11 Transmission use of system related values for selected buses in Polish 2383-bus test system by proposed method TUoS charge No. Bus number APG (MW) TTPU (MW) TTNU (MW) TTC ($/h) ($/MWh) 1 18 1908 9241 1342 6530 3.42 2 17 1080 5910 623 4226 3.91 3 31 1000 4924 1336 3928 3.93 4 131 872 3992 1277 4426 5.08 5 67 750 3951 882 3494 4.66 6 16 720 3878 450 2722 3.78 7 127 690 3108 931 3215 4.66 8 63 650 3804 565 2508 3.86 9 176 600 2899 1108 2830 4.72 10 139 600 2616 828 2672 4.45 11 1426 495 3014 706 2320 4.69 12 64 450 2699 395 1724 3.83 13 105 430 2218 745 1533 3.57 14 43 410 2340 479 1682 4.10 15 44 410 2214 558 1683 4.11 16 10 400 1944 775 2384 5.96 17 911 370 2172 493 1998 5.40 18 912 370 2235 490 2108 5.70 19 1416 367 3167 807 1927 5.25 20 111 360 1724 409 1578 4.38 21 2164 4.10 21 15 71 17.29 22 2268 1.80 9 7 23 12.57 23 2328 3 13 7 37 12.28 24 2159 12 137 75 137 11.39 25 132 70 796 195 796 11.40 Branch power flow contribution of generator (MW) No. Bus number 1764–1760 1763–1761 612–413 1945–1845 1489–1426 138–67 32–31 18–15 15–165 132–131 1 18 0.45 2.52 3.69 – 1.96 – 3.71 – 92.43 – 68.26 133.24 90.95 – 16.46 2 17 0.30 2.02 2.89 – 1.07 – 2.14 – 41.81 – 28.13 55.10 39.11 – 6.80 3 31 0.47 1.32 3.57 – 0.86 – 1.84 – 25.05 – 207.80 – 2.61 51.53 – 3.82 4 131 0.55 0.21 0.84 43.14 12.98 5.88 – 2.45 – 0.51 – 8.20 – 257.60 5 67 0.28 0.44 1.17 – 1.22 – 1.02 – 136.6 – 14.64 27.01 17.97 – 21.06 6 16 0.18 1.35 1.80 – 0.73 – 1.47 – 28.34 – 19.64 37.20 26.16 – 4.74 7 127 0.67 0.80 0.55 – 3.96 – 0.14 – 6.38 – 6.99 – 1.27 – 15.81 0.40 8 63 0.15 0.48 1.48 – 0.35 – 3.20 – 20.38 – 13.31 17.66 14.97 – 3.14 9 176 0.51 0.71 0.28 – 1.55 – 0.43 – 8.00 – 16.94 – 22.13 – 59.80 – 1.29 10 139 1.62 3.68 1.05 – 1.16 – 0.43 – 7.96 – 8.50 3.67 – 3.86 – 2.64 11 1426 0.11 0.32 1.13 – 0.21 – 25.47 – 11.99 – 9.90 10.48 10.03 – 1.68 12 64 0.11 0.34 1.06 – 0.24 – 2.91 – 14.15 – 9.11 12.47 10.59 – 2.17 13 105 0.11 0.48 0.99 – 0.39 – 0.79 – 20.30 – 13.41 19.94 16.22 – 3.82 14 43 0.14 0.44 1.43 – 0.29 – 0.81 – 11.86 – 27.96 8.18 15.46 – 1.79 15 44 0.14 0.44 1.43 – 0.29 – 0.81 – 11.85 – 28.04 8.17 15.47 – 1.79 16 10 0.26 0.50 – 1.91 – 0.48 – 0.32 – 5.05 – 15.79 – 2.20 7.15 – 0.73 17 911 0.12 0.22 0.60 – 0.49 – 0.54 – 47.78 – 7.21 12.55 8.56 – 8.43 18 912 0.12 0.22 0.59 12.13 8.29 – 0.47 – 0.53 – 46.64 – 7.04 – 8.19 19 1416 0.08 0.23 0.73 – 0.17 4.27 – 10.22 – 6.89 8.56 7.11 – 1.54 20 111 0.57 1.94 0.83 – 0.59 – 0.33 – 6.66 – 5.50 6.36 2.64 – 1.55 21 2164 0.00 0.01 0.00 – 0.02 0.00 – 0.06 – 0.05 – 0.02 – 0.15 – 0.01 22 2268 0.00 0.00 0.00 – 0.01 0.00 – 0.02 – 0.02 – 0.01 – 0.07 – 0.01 23 2328 0.00 0.00 0.00 – 0.01 0.00 – 0.04 – 0.03 0.01 – 0.08 – 0.01 24 2159 0.00 0.03 0.07 – 0.03 0.05 0.18 0.89 2.26 3.51 0.05 25 132 0.01 0.31 – 0.07 – 0.57 0.12 – 19.47 2.59 ̢ 0.55 1.90 7.44 123 1520 Saeid POUYAFAR et al. sum of partial branch power ﬂows due to a generator in the proposed method, however, smooths the trend of the Z-bus same/opposite direction to branch power ﬂow, respectively. method to reﬂect the counter-ﬂows, and therefore helps to The ﬁrst 20 rows (No. 1-20) of Table 11 show the ﬁrst 20 reduce the cross-subsidies. This property is truly valuable, generators in terms of APG, TTPU and TTC. The last 5 since higher counter-ﬂows with excessive rewards bring rows (No. 21-25) of Table 11 show the ﬁrst 5 generators in about unfair competitions. Furthermore, based on a com- terms of TUoS tariffs. parison on the 6-bus system, it is determined that the results of the cost allocation by the Z-bus method change signiﬁcantly when different MW-mile pricing schemes are 7 Conclusion used, whereas the proposed method provides more stable results. This paper presents a new circuit-theory-based method Tariff stability of the proposed cost allocation method is to the problem of TCA. Unlike majority, if not all, similar also assessed on the IEEE 30-bus system. The results methods, the proposed method attempts to justify the way reveal that the proposed method provides a fairly it treats the non-linear behavior of the power systems to stable tariffs against the temporal load variations, as well as mathematically identify the shares of the participants on the generating dispatch strategies. Moreover, based on the network power quantities. The applied principle to split the results on the 6-bus test system, the proposed method is mutual terms of the power equations, as the causes of non- less sensitive to the calculation reference side of the lines, linearity, is conﬁrmed by the Shapley and Aumann-Shap- compared with the Z-bus method. As another advantage, ley values as the preferred transmission network cost game the proposed method works consistently for all network solutions. Moreover, a new concept, named circuit-theory- conﬁgurations, as it overcomes the singularity problems of based EBE, is introduced. the Z-bus as well as PS methods to invert the large-scale The proposed method is compared with two well-known sparse matrices. Computational performance on the TCA techniques, namely, Z-bus and PS methods. Numer- 2383-bus practical system of Poland indicates that the ical case studies on the 6-bus system and the IEEE 30-bus proposed method is quite fast, so it can well deal with the system show that the proposed method outperforms the problem of TCA in large practical power systems. other two methods. According to the results, the proposed Open Access This article is distributed under the terms of the method is fair and efﬁcient, as it reﬂects the network Creative Commons Attribution 4.0 International License (http:// topology as well as the order of magnitude and the location creativecommons.org/licenses/by/4.0/), which permits unrestricted of the generators in the grid. Although the PS method use, distribution, and reproduction in any medium, provided you give intensiﬁes the locational signals, its principle is only based appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were on logical reasoning and can never be proved. It is also made. shown that the PS principle ignores the counter-ﬂows and, in turn, results in considerable tariff instability. The 123 Circuit-theory-based method for transmission ﬁxed cost allocation based on game-theory… 1521 Appendix A Table A1 Load ﬂow result for IEEE 30-bus test system Bus Voltage Generation Load Magnitude (p.u.) Angle () Real (MW) Reactive (Mvar) Real (MW) Reactive (Mvar) 1 0.9824 0 41.5421 - 5.4364 0 0 2 0.9787 - 0.7630 55.4019 1.6748 21.7 12.7 3 0.9769 - 2.3897 0 0 2.4 1.2 4 0.9764 - 2.8386 0 0 7.6 1.6 5 0.9713 - 2.4864 0 0 0 0 6 0.9723 - 3.2287 0 0 0 0 7 0.9623 - 3.4910 0 0 22.8 10.9 8 0.9611 - 3.6819 0 0 30 30 9 0.9903 - 4.1371 0 0 0 0 10 0.9998 - 4.5998 0 0 5.8 2 11 0.9903 - 4.1371 0 0 0 0 12 1.0174 - 4.4979 0 0 11.2 7.5 13 1.0645 - 3.2980 16.2002 35.9303 0 0 14 1.0066 - 5.0397 0 0 6.2 1.6 15 1.0092 - 4.8140 0 0 8.2 2.5 16 1.0028 - 4.8393 0 0 3.5 1.8 17 0.9955 - 4.8873 0 0 9 5.8 18 0.9933 - 5.4843 0 0 3.2 0.9 19 0.9873 - 5.6882 0 0 9.5 3.4 20 0.9896 - 5.4719 0 0 2.2 0.7 21 1.0093 - 4.6208 0 0 17.5 11.2 22 1.0160 - 4.5030 22.7403 34.1971 0 0 23 1.0256 - 3.7557 16.2670 6.9598 3.2 1.6 24 1.0167 - 3.8852 0 0 8.7 6.7 25 1.0438 - 2.0724 0 0 0 0 26 1.0267 - 2.4760 0 0 3.5 2.3 27 1.0690 - 0.7147 39.9090 31.7544 0 0 28 0.9820 - 3.2152 0 0 0 0 29 1.0500 - 1.8494 0 0 2.4 0.9 30 1.0391 - 2.6429 0 0 10.6 1.9 References [5] Yang Z, Zhong H, Xia Q et al (2016) A structural transmission cost allocation scheme based on capacity usage identiﬁcation. 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His research interests include power market and Energy Syst 61:547–552 security assessment of power systems. [22] Shivaie M, Kiani-Moghaddam M, Ansari M (2018) Transmis- sion-service pricing by incorporating load following and cor- Mehrdad TARAFDAR HAGH is with the Faculty of Electrical and relation factors within a restructured environment. Electr Power Computer Engineering, University of Tabriz, Tabriz, Iran, since 2000, Syst Res 163:538–546 where he is currently a Professor. He is also with Engineering Faculty [23] Teng JH (2005) Power ﬂow and loss allocation for deregulated of Near East University, North Cyprus, Turkey. He has published transmission systems. Int J Electr Power Energy Syst more than 300 papers in power system and power-electronic-related 27(4):327–333 topics. His research interests include power system operation, [24] Conejo AJ, Contreras J, Lima DA et al (2007) Z transmission bus distributed generation, ﬂexible AC transmission systems, and power network cost allocation. IEEE Trans Power Syst 22(1):342–349 quality. [25] Khalid SN, Shareef H, Mustafa MW et al (2012) Evaluation of real power and loss contributions for deregulated environment. Kazem ZARE received the B.Sc. and M.Sc. degrees in electrical Int J Electr Power Energy Syst 38(1):63–71 engineering from University of Tabriz, Tabriz, Iran, in 2000 and [26] Nikoukar J, Haghifam MR, Parastar A (2012) Transmission cost 2003, respectively, and Ph.D. degree from Tarbiat Modares Univer- allocation based on the modiﬁed Z-bus. Int J Electr Power sity, Tehran, Iran, in 2009. He is currently an Associate Professor of Energy Syst 42(1):31–37 the Faculty of Electrical and Computer Engineering, University of [27] Abdelkader SM, Morrow DJ, Conejo AJ (2014) Network usage Tabriz. His research interests include power system economics, determination using a transformer analogy. IET Gener Transm distribution networks, microgrid and energy management. Distrib 8(1):81–90 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Modern Power Systems and Clean Energy Springer Journals http://www.deepdyve.com/lp/springer-journals/circuit-theory-based-method-for-transmission-fixed-cost-allocation-T8PXnUrTeO

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Publisher: Springer Journals
Copyright: Copyright © 2019 by The Author(s)
Subject: Energy; Energy Systems; Renewable and Green Energy; Power Electronics, Electrical Machines and Networks
ISSN: 2196-5625
eISSN: 2196-5420
DOI: 10.1007/s40565-018-0489-y
Publisher site: See Article on Publisher Site

Abstract

J. Mod. Power Syst. Clean Energy (2019) 7(6):1507–1522 https://doi.org/10.1007/s40565-018-0489-y Circuit-theory-based method for transmission ﬁxed cost allocation based on game-theory rationalized sharing of mutual-terms 1 1,2 Saeid POUYAFAR , Mehrdad TARAFDAR HAGH , Kazem ZARE Abstract This paper proposes a new method to allocate desired in a fair and efﬁcient cost allocation method. the transmission ﬁxed costs among the network participants Finally, the developed technique has been implemented in a pool-based electricity market. The allocation process successfully on the 2383-bus Polish power system to relies on the circuit laws, utilizes the modiﬁed impedance emphasize that the method is applicable to very large matrix and is performed in two individual steps for the systems. generators and loads. To determine the partial branch power ﬂows due to the participants, the equal sharing Keywords Transmission ﬁxed cost allocation, Circuit principle is used and validated by the Shapley and theory, Equal sharing principle, Game theory Aumann-Shapley values as two preferred game-theoretic solutions. The proposed approach is also applied to deter- mine the generators’ contributions into the loads, and a 1 Introduction new concept, named circuit-theory-based equivalent bilat- eral exchange (EBE), is introduced. Using the proposed Among various issues related to the modern restructured method, fairly stable tariffs are provided for the partici- power systems well addressed most recently in the litera- pants. Cross-subsidies are reduced and a fair competition is ture [1, 2], deregulation and its price-based problems [3, 4] made by the proposed method due to the counter-ﬂows are of utmost importance. One of these problems is trans- being alleviated compared with the well-known Z-bus mission cost allocation (TCA). Several methodologies have method. Numerical results are reported and discussed to been proposed in the literature concerning the problem of validate the proposed cost allocation method. Comparative TCA. Traditionally, the costs were allocated to the users by analysis reveals that the method satisﬁes all conditions the Pro-Rata method. Despite the simplicity, the method disregards the network actual extent of use. Recently, the method is enriched and used to allocate the costs of the CrossCheck date: 25 September 2018 unused capacity of the transmission facilities. The task of Received: 28 February 2018 / Accepted: 26 September 2018 / allocating the transmission costs to the users taking into Published online: 17 January 2019 account the network extent of use, was ﬁrst introduced by The Author(s) 2019 the MW-mile method which is now widely applied in the & Mehrdad TARAFDAR HAGH literature [5]. tarafdar@tabrizu.ac.ir Tracing-based methods [6–8] utilize the concept of Saeid POUYAFAR proportional sharing principle (PSP) to trace the power s.pouyafar@tabrizu.ac.ir ﬂow in the network. Reference [6] proved the existence and Kazem ZARE uniqueness of a solution to the tracing problem. The kazem.zare@tabrizu.ac.ir technique of ﬂow tracing has also been extended and used Department of Electrical and Computer Engineering, as an analytical tool for transmission capacity allocation in University of Tabriz, Tabriz, Iran a highly renewable European electricity system [7]. Ref- erence [8] presented a transmission congestion (TC) Engineering Faculty, Near East University, Mersin 10, 99138 Nicosia, North Cyprus, Turkey 123 1508 Saeid POUYAFAR et al. tracing technique based on PSP. Nodal pricing is another analogy (TA) model [27], have an important advantage approach for TCA which is based on locational marginal over any cost allocation method, as previously described. price (LMP) differences, and is currently developed These methods incorporate the network characteristics worldwide. The proposed marginal pricing approach pro- directly into the allocation process. However, due to the vides the correct economic signals to the network partici- non-linear behavior of the power systems, there is still not pants. However, it is not linked to the actual transmission a unique mathematical solution for the contribution of infrastructure cost, thus, not able to recover the total customers into the transmission facilities under these transmission network cost (TNC) [9]. Reference [10] approaches. The results rely mostly on the principle applied checked this fact in several systems around the world uti- to split the mutual terms, as the main causes of the non- lizing LMP-based TCA method, and demonstrated that the linearity, between the participants. A group of papers use maximum network revenue obtained in these systems was the most common sharing principles, namely, proportional only 25% of the TNC. Some authors tried to solve the issue [29], quadratic [30], and equal sharing [19, 28, 31] to split by altering the LMPs to recover the TNC using the concept the mutual terms, whereas the others [23–27] avoid the of Ramsey pricing [10], and introducing the generation and mutual terms by considering a single variable at a time. In nodal injection penalties into the economic dispatch [11]. [29], it is revealed that proportional and equal sharing Marginal and incremental cost allocation methods, based principles bring about more reasonable results for reactive on the concept of sensitivity indices, are other pricing power allocation, compared with the single variable schemes widely applied in the literature, until recently [12]. division. The main drawback of these methods is their sensitivity to Based on the arguments, this paper presents a new cir- the choice of the slack bus. To overcome this limitation, [5] cuit-theory-based TCA method which is developed based utilized the slack bus independent distribution factors, on the modiﬁed Z-bus model. The proposed technique whereas [13] suggested the concept of optimal distributed applies the equal-sharing principle to split the mutual- slack bus. TCA methods based on some form of equiva- terms, and subsequently to determine the partial branch lents have been extended in [14, 15], in which the equiv- power ﬂows due to the participants. It also uses the Shapley alent bilateral exchange (EBE) has been built through the as well as the Aumann-Shapley values, as two preferred optimization as well as tracing-based approaches, respec- cooperative game solution concepts to validate the sharing tively. As an alternative, the optimization approach has principle applied. Moreover, a new concept, named circuit- been used recently along with the min-max fairness criteria theory-based EBE, is introduced by determining the gen- [16] to trace the real power in the network. The application erators and loads contributions into each other. of artiﬁcial intelligence (AI) to power system becomes The innovative contributions of the paper are: popular to explore, especially in power tracing problems 1) The proposed method is applicable to very large [17]. Effect of the possible interactions of components is systems, since it requires less computational efforts, often not considered in neither optimization nor AI-based and overcomes the limitations of the existing Z-bus methods, due to its additive complexity as well as the and proportional sharing (PS) methods to invert the computation time, subsequently leading to inaccuracy in large-scale sparse matrices. some cases. 2) The proposed cost allocation method is fair and There are also a group of papers, with solid economical efﬁcient and is more likely to be accepted by the foundation, that incorporate the concept of cooperative participants, because it is conﬁrmed by the game game theory [18, 19] into the problem of TCA. Although theoretic solutions, and also reﬂects the order of the method behaves well in terms of fairness and efﬁ- magnitude of generators and loads as well as their ciency, signiﬁcantly high computation time is required, if locations in the grid. applied to a large power system [20]. Recently, [21] pro- 3) The proposed method smooths the trend of the posed a beneﬁt-based TCA scheme. The challenging issue Z-bus method to reﬂect the counter-ﬂows, and, in concerning these methods is to ﬁnd the exact beneﬁt that turn, helps to reduce the cross-subsidies. This each user takes from the transmission facilities. Reference property is truly valuable, as higher counter-ﬂows [22] introduced a new load-following-based method to with excessive rewards result in unfair competition, estimate the transmission costs of each participant during a and make the results change signiﬁcantly when speciﬁed time period before entering the market. different MW-mile pricing schemes are used. The use of circuit theory to the TCA is another pricing 4) Highest tariff stability against temporal variations scheme, widely applied in the literature [19, 23–28]. The are observed by the proposed method, compared circuit-theory-based approaches, including Z-bus model with the Z-bus and PS methods. [23, 24] as well as its modiﬁed forms [26, 28], modiﬁed nodal equation (MNE) model [25], and transformer 123 Circuit-theory-based method for transmission ﬁxed cost allocation based on game-theory… 1509 5) The proposed method is less sensitive to the V ¼ z I ð3Þ k k;n n g g calculation reference side of the lines, compared n¼1 with the Z-bus method. th where z is the element of the k row of the matrix Z . 6) The proposed method works consistently for all k;n mod network conﬁgurations, as it overcomes the singu- According to (3), contribution of the generator at bus n into the voltage of the bus k is: larity problems of the similar methods to build the impedance matrix. V ¼ z I ð4Þ k;n k;n n g g g If bus k is a demand bus, its complex power consumption can be written as: 2 Modiﬁed Z-bus theory S ¼ V i ¼ y V V ð5Þ k k k k k k In this study, due to page limit, only the contributions of Applying (3) into (5), we have: the generators are considered. Therefore, the generators are 0 10 1 N N g g X X treated as current injections and the loads as equivalent @ A@ A S ¼ y z I z I ð6Þ k k;n n k g g k;n n impedances. g g n ¼1 n ¼1 g g For a power system with N buses and N lines, suppose b l that there are N generator buses and N demand buses. g d Similarly, for a given line ab, its complex power ﬂow Once the solution of a converged power ﬂow or state can be written as follows: estimator is obtained, system equivalent injection currents and admittance values will be as follows [23]: S ¼ V I ¼ V y V V þ y V ð7Þ ab a a a;sh ab ab a b a P jQ n n g g I ¼ ð1Þ where S and I are the power and current ﬂow; y and V ab ab ab y are series and shunt admittances of the line ab, a,sh where P and Q are the power injected at the generator n n respectively. Substituting equivalent values of the bus g g bus n ; V is the voltage of the generator bus n . g g g voltages from (3) into (7), we have: 0 1 Likewise, the equivalent admittance for a load bus is: P jQ n n @ A d d S ¼ z I ab a;n n g g y ¼ ð2Þ n ¼1 jj V g 2 3 N N g g X X where P and Q are the power consumed at the demand n n 1 d d 4 5 y z z I þ y z I a;sh ab a;n b;n n a;n n bus n ; V is the voltage of the demand bus n . g g g g g d n d d 2 n ¼1 n ¼1 g g Equivalent admittances of the loads are integrated into ð8Þ the network admittance matrix and the ‘‘network Y-mod matrix’’ is built. 3 Sharing principles of mutual-terms Note that treating the loads as constant impedances rather than current injections and, in turn, integrating the As seen in (6) and (8), the network power equations are load impedances into the network Y-bus matrix, in most made up of self-terms, that is, the contribution of an indi- cases, avoids the singularity problems concerning the vidual component, and mutual-terms, given by the product impedance matrix building process. Note also that due to of two distinct components. Thereby, it is necessary to split the TCA methods being developed based on the solved the mutual terms, to allocate the power equations between power ﬂow of the system, and considering the fact that for the involved components. power system steady-state studies, the actual modeling of In general, a simple form of the problem of dividing a the network loads does not inﬂuence the effectiveness of mutual term between two involved components can be the results [32], among numerous static load models, the expressed as follows: constant power PQ load model as the most appropriate one is considered in the proposed TCA method. f ðx ; x Þ¼ 2x x ¼ a x x þ a x x i j i j i i j j i j ð9Þ Using the network modiﬁed impedance matrix, Z ,to mod s:t: a þ a ¼ 2 i j write the relationship between the bus voltages and the bus where x and x are the involved components of the mutual- i j current injections, voltage of a given bus k is expressed as: term; a and a are the contribution coefﬁcients of the i j components into the mutual-term, respectively. Mutual- terms are divided between the involved components, based 123 1510 Saeid POUYAFAR et al. 8 P on the criteria applied to calculate the contribution coefﬁ- x ¼ cðNÞ Global rationality < i i2N cients of the components. Commonly used sharing princi- P ð15Þ x cðSÞ Group/individual rationality ples of mutual-terms are proportional, quadratic, and equal i2S or 50-50 sharing. where 8S N. Proportional sharing principle assumes that the coefﬁcients The Shapley value is a single point solution to the cost a , a are directly proportional to their relevant components: i j game, which is deﬁned as follows. For a n-player coali- a a i j ¼ ð10Þ tional game with real-valued gain function t(), a unique x x i j imputation of the total gain to each player i, denoted by / Regarding the constraint of (9) we have: (i=1, 2, …, n), is given by the Shapley value: i i 2x i S j!ðÞ n jj S 1 ! i i a ¼ < i / ¼ t S [ fg i t S x þ x i j n! ð11Þ S Nnfg i 2x : a ¼ ð16Þ x þ x i j -i th where S is an arbitrary subset of N with the i player Based on the quadratic sharing principle, however, the -i always excluded (denoted by \{i}); S | is the number of coefﬁcients a , a , are proportional to the square of their i j -i players in subset S . According to (16), the coalitional relevant components: rationality of (15) is not a requirement, so the Shapley a a i j ¼ ð12Þ value does not always belong to the core. However, under 2 2 x x i j the whole network game, in most cases, the coalitional rationality holds for the Shapley value solution, thus, it Again, regarding the constraint of (9) we have: belongs to the core as well. In the context of the Shapley 2x value being a part of the core, it is more preferable than > a ¼ < 2 2 x þ x i j other single point solutions, as it holds the axioms of ð13Þ 2x > symmetry, efﬁciency, additivity, and dummy player a ¼ : j 2 2 x þ x [33]. i j There exists another single point solution, namely, the Equal sharing model, on the other hand, assumes an Aumann-Shapley, which is a natural consequence of the equal and unitary value for the coefﬁcients a , a : i j Shapley value method. It is based on the premise that each a ¼ a ¼ 1 ð14Þ i j agent has to be sub-divided into inﬁnitesimal sub-agents, and the Shapley method is applied to each one as if each The sharing principles cannot be proved sub-agent were an individual [19]. mathematically, however, if any of them is conﬁrmed by Despite the fact that the game-theoretic solutions result a game-theoretic solution belonging to the core of the in fair and efﬁcient results, they are not a common TCA game, which is more likely to bring about sensible results. method, because these solutions on realistic sized problems require too much input data, which makes the handling of the game dimension a challenging issue. For example, the 4 Game-theoretic solutions calculations for a network with n participants require 2 - 1 pieces of input data [20]. However, the game-the- If a solution belongs to the core of the game, it is more oretical solutions may be used as a framework to evaluate likely to be accepted by the participants, because the the results of any other usage-based methods, or to prove problem of cross-subsidy is avoided. However, based on whether a mathematical model is rational. the game played, the core may consist of more than one point or it may be empty. To allocate the transmission costs among the users based 5 Proposed method on the game theory, a cost game has to be deﬁned ﬁrst, given by the pair (N, c), where N = {1, 2, …, n} is the set of In this section, ﬁrst, we use the Shapley as well as the players and c is a function that assigns a real number to Aumann-Shapley values, as two preferred cost game each subgroup (coalition) of N. A solution to the cost game solutions, to conﬁrm the equal sharing principle. Then, we is a cost allocation, i.e., a vector x [ R , where any element apply the equal sharing principle to determine the branch x of vector x is the cost allocated to player i. The solution power ﬂow contributions of the participants. The results are is called to be in the core, if, the following rationality then used to allocate the network costs among the users. requirements hold: 123 Circuit-theory-based method for transmission ﬁxed cost allocation based on game-theory… 1511 5.1 Validating the equal sharing principle 5.2 Allocating demand power of buses We apply the Shapley value to solve the general mutual Applying the equal sharing model to spilt the mutual term division problem, deﬁned by (9). terms of the demand power expression (6) between the The problem to be studied might be described as fol- current injections, the contribution of the generator at bus lows: how to impute 2x x to each of x , and x . We ﬁrst n into the demand power at bus k will be: i j i j g 2 0 1 0 13 compute the gains t(), i.e., the value of 2x x when two i j N N g g X X sources each acts alone and act together. 4 @ A @ A5 S ¼ y z I z I þ z I z I k;n k;n n k;n n g g g g g k k;n n k;n n g g g g n ¼1 n ¼1 g g tðÞ fg i ¼ 2x 0 ¼ 0 < i tðÞ fg j ¼ 0 ð17Þ ð22Þ tðÞ fg i; j ¼ 2x x i j Substituting (3) and (4) into (22), the simpliﬁed form of Using the detailed form of the Shapley value formula S will be: k;n (16), the imputation component of f(x,x ) to source x , i j i denoted by f , is: S ¼ y V V þ V V ð23Þ k;n k;n k g k g k k;n 0!ðÞ 2 0 1 ! f ¼ ½ tðÞ fg i 0 The expression obtained, may be applied to build the proposed circuit-theory-based EBE. The concept requires 1!ðÞ 2 1 1 ! ð18Þ þ ½ tðÞ fg i; j tðÞ fg j further investigation and will be developed in the future studies. ¼ 2x x ¼ x x i j i j 5.3 Allocating power ﬂow of branches Similarly, the imputation component of f(x ,x ) to source i j x is calculated. The results, conﬁrms the equal sharing of Again, we use the equal sharing principle to split the mutual terms, as the shares of x and x on 2x x , are the i j i j mutual-terms of the branch power ﬂow expressions same and each equals x x . i j between the current injections. Subsequently, the contri- We perform the same procedure by the Aumann-Shap- bution of the generator at bus n into the power ﬂow of the ley value to split the mutual term, f(x,x ), between its i j branch ab will be: 8 2 3 involved components, x , x . Based on the Aumann-Shapley i j < g value solution concept, unitary participation (UP) of x into 4 5 S ¼ y z I z z I ab;n a;n n g g g ab a;n b;n n g g g 2 : the f(x ,x ) is: i j n ¼1 Z Z 0 19 1 1 oftðÞx g = UP ¼ dt ¼ 2 tx dt ¼ t x ¼ x j j j ðx !f Þ @ A 0 þðz z ÞI z I a;n n a;n b;n n g g ox g g g t¼0 t¼0 n ¼1 ð19Þ 2 0 1 0 13 N N g g X X 4 @ A @ A5 þ y z I z I þ z I z I where each agent is divided into inﬁnitesimal parts (Dx?0) a;n n a;n n a;sh g g a;n n a;n n g g g g g g n ¼1 n ¼1 g g by inﬁnitesimal steps, t. Likewise, solving the problem for UP of x into the f(x , ð24Þ j i x ), we obtain: Using (3) and (4) to replace the related terms in (24), we Z Z 1 1 oftðÞx 1 have: UP ¼ dt ¼ 2ðÞ tx dt ¼ t x j ¼ x i i i ðx !f Þ j 0 hi ox t¼0 t¼0 S ¼ y V V V þ V V V ab;n ab;n a g ab g a b a;n b;n g g ð20Þ To determine the total participation (TP) of the player þ y V V þ V V a;sh a;n a g a a;n into the f(x,x ), the unitary participation is multiplied by i j ð25Þ the amount of the player: TP ¼ UP x ¼ x x ðÞ x !f ðÞ x !f i j i i i ð21Þ 6 Numerical results TP ¼ UP x ¼ x x j i j ðÞ x !f ðÞ x !f j j In this section, the proposed method is compared with It can be observed that the outcomes conﬁrm the equal two well-known TCA techniques, namely, Z-bus theory sharing of mutual terms. [24], and PS method [6]. Two test systems namely 6-bus 123 1512 Saeid POUYAFAR et al. test system and IEEE 30-bus system are used. In the con- text of the 6-bus test system, a shortened time series study is performed to gain more insights into the behavior of the three TCA methods considered. In practice, a large number of time steps need to be analyzed for consecutive studies of large systems. However, the cyclical nature of load and generation proﬁles suggests that some time steps in the proﬁles represent load and generation scenarios that reap- pear over time, and hence can be simulated only once. In this respect, four time steps are considered in our study, which sufﬁciently represent the system off-peak, transi- tional, normal, and peak conditions. For the IEEE 30-bus Fig. 1 6-bus test system system, system peak condition is considered. Any line tariff in $/h, is set to be 1000 times its series reactance. Table 1 6-bus test system branch data Zero counter ﬂow (ZCF) MW-mile pricing is used to Line r (p.u.) x (p.u.) y/2 (p.u.) charge the participants. Thus, the generators neither pay money nor take any reward for the lines in which their 1-2 0.10 0.20 0.02 associated power ﬂow is in opposite direction. Finally, the 1-4 0.05 0.20 0.02 proposed method is applied to a practical system, namely, 1-5 0.08 0.30 0.03 2383-bus Polish 400, 220 and 110 kV networks during 2-3 0.05 0.25 0.03 winter 1999-2000 peak conditions, to emphasize the 2-4 0.05 0.10 0.01 applicability of the method to very large systems. To know 2-5 0.10 0.30 0.02 which type of electricity market with its regulations best 2-6 0.07 0.20 0.02 suits a certain TCA technique, please refer to [34–36] 3-5 0.12 0.26 0.02 which provide key information about transmission pricing 3-6 0.02 0.10 0.01 experiences across various international jurisdictions. 4-5 0.20 0.40 0.04 5-6 0.10 0.30 0.03 6.1 6-bus test system The 6-bus system as shown in Fig. 1 is considered. The Table 2 Active power of loads, generators and losses corresponding displayed values on Fig. 1, represent the active power to system four time steps ﬂows corresponding to the system peak condition. The Time P P P P G1 G2 G3 system has eleven branches with parameters provided in loss L4 L5 L6 step (MW) (MW) (MW) (MW) (MW) (MW) (MW) Table 1, in which r, x, and y denote the resistance, reac- tance, and shunt admittance of the branches, respectively. 1 2.56 3.39 4.74 6.91 50.00* 42.56 45.00* The time series study is performed for the three TCA 2 45.00 45.00 60.00 70.00 50.00* 58.33 45.06* methods, and the results are depicted in Tables 2, 3, 4, 5 3 45.00 60.00 60.00 70.00 50.00* 74.37 60.37 and 6. Table 2 represents the active power of the loads, 4 45.00 45.00 60.00 70.00 77.22 69.27 70.42 generators and losses corresponding to the system four time steps considered. In Table 2, * signiﬁes the minimum PS method traces the power ﬂows using the PSP which can output capacity constraint of G1. be neither proved nor disproved, whereas the proposed According to Tables 3, 4, 5 and 6, preliminary Kirch- method is based on the circuit theory and utilizes the net- hoff’s circuit laws are satisﬁed under the proposed as well work impedance matrix to determine the partial branch as the Z-bus methods. power ﬂows due to the participants. In the case of the Z-bus For each branch of the system, in each of the four time method, however, the generators and loads are both treated steps, the overall contribution due to the generators equals as nodal currents, and hence, their responsibilities for the power ﬂow of the branch, under the proposed as well as network usage are calculated altogether in a common PS methods. This is because, unlike Z-bus method, the process. proposed and PS methodologies perform the allocation Although the amounts of the contributions differ under process for the generators and loads, independently. Nev- the three methods, the dominant positive ones are the same ertheless, the partial branch power ﬂows allocated to the for all the three methods, and belong to the genera- users differ for these two methods. Discrepancies arise tor(s) relatively close to the sending-end bus of the mainly due to the different underlying principles applied. 123 Circuit-theory-based method for transmission ﬁxed cost allocation based on game-theory… 1513 Table 3 Time series study of three TCA methods on 6-bus test Table 4 Time series study of three TCA methods on 6-bus test system (time step 1) system (time step 2) Line number TCA Power ﬂow contribution (MW) Line number TCA Power ﬂow contribution (MW) method method P G1 G2 G3 P G1 G2 G3 line line 1-2 Proposed 9.96 10.46 - 1.03 0.54 1-2 Proposed 7.89 9.96 - 2.14 0.08 Z-bus 17.95 - 4.44 - 1.53 Z-bus 18.17 - 6.07 - 1.52 PS 9.96 0.00 0.00 PS 7.89 0.00 0.00 1-4 Proposed 22.02 11.76 4.16 6.09 1-4 Proposed 21.35 11.28 4.39 5.69 Z-bus 16.19 0.20 1.06 Z-bus 16.08 0.26 1.06 PS 22.02 0.00 0.00 PS 21.35 0.00 0.00 1-5 Proposed 18.02 9.85 4.41 3.76 1-5 Proposed 20.75 10.20 6.02 4.53 Z-bus 15.86 4.24 0.47 Z-bus 15.75 5.82 0.46 PS 18.02 0.00 0.00 PS 20.75 0.00 0.00 2-3 Proposed 0.25 2.89 3.11 - 5.75 2-3 Proposed 3.03 3.13 5.04 - 5.14 Z-bus 7.12 8.13 - 8.11 Z-bus 7.10 11.22 - 7.87 PS 0.05 0.21 0.00 PS 0.36 2.68 0.00 2-4 Proposed 25.73 3.04 11.07 11.62 2-4 Proposed 29.10 3.30 14.03 11.76 Z-bus - 4.16 10.07 5.99 Z-bus - 4.03 13.77 5.89 PS 4.89 20.90 0.00 PS 3.47 25.66 0.00 2-5 Proposed 10.82 2.69 5.06 3.06 2-5 Proposed 15.33 3.56 7.53 4.23 Z-bus 3.75 7.45 1.40 Z-bus 3.82 10.23 1.29 PS 2.06 8.78 0.00 PS 1.83 13.51 0.00 2-6 Proposed 15.61 6.28 7.45 1.89 2-6 Proposed 18.70 6.46 9.98 2.26 Z-bus 7.46 9.15 - 4.25 Z-bus 7.53 12.52 - 4.23 PS 2.97 12.68 0.00 PS 2.23 16.49 0.00 3-5 Proposed 14.73 0.87 3.13 10.73 3-5 Proposed 18.03 1.62 4.42 11.99 Z-bus 0.60 3.19 14.56 Z-bus 0.60 4.41 14.80 PS 0.02 0.07 14.66 PS 0.14 1.00 16.90 3-6 Proposed 30.49 5.09 6.36 19.05 3-6 Proposed 30.02 4.76 6.58 18.68 Z-bus 0.72 0.13 16.97 Z-bus 0.70 0.22 16.92 PS 0.03 0.14 30.34 PS 0.23 1.67 28.15 4-5 Proposed 1.86 1.30 1.09 - 0.53 4-5 Proposed 4.46 1.91 2.24 0.31 Z-bus 6.18 5.23 1.96 Z-bus 6.22 7.09 1.90 PS 1.07 0.83 0.00 PS 2.24 2.31 0.00 5-6 Proposed - 0.49 1.44 - 0.14 - 1.79 5-6 Proposed - 2.96 0.69 - 0.94 - 2.71 Z-bus 4.35 1.35 - 1.38 Z-bus 4.29 1.80 - 1.30 PS - 0.03 - 0.14 - 0.34 PS - 0.15 - 1.14 - 1.77 Total transmission Proposed 498.70 400.10 406.20 Total transmission Proposed 454.10 474.00 376.90 cost ($/h) cost ($/h) Z-bus 895.70 560.30 347.20 Z-bus 817.90 655.80 305.70 PS 553.10 474.30 277.60 PS 507.90 541.80 255.20 Tariffs ($/MWh) Proposed 9.97 9.40 9.02 Tariffs ($/MWh) Proposed 9.08 8.13 8.37 Z-bus 17.91 13.16 7.72 Z-bus 16.36 11.24 6.78 PS 11.06 11.14 6.17 PS 10.16 9.29 5.66 branches. This fact holds for each time step considered. For sending-end bus of the lines 2-3, 2-4, 2-5, and 2-6, incor- example, under the proposed method, the generator at bus 1 porates most in their associated power ﬂows. The same as the closest one to the sending-end bus of the lines 1-2, principle holds for the lines 3-6, 4-5 and 5-6. This property 1-4, and 1-5, has the highest direct contribution to the may be called the nearby effect of the power networks most power ﬂow of those lines. The generator at bus 2, as the evident in the PS method. 123 1514 Saeid POUYAFAR et al. Table 5 Time series study of three TCA methods on 6-bus test Table 6 Time series study of three TCA methods on 6-bus test system (time step 3) system (time step 4) Line number TCA Power ﬂow contribution (MW) Line number TCA Power ﬂow contribution (MW) method method P G1 G2 G3 P G1 G2 G3 line line 1-2 Proposed 5.87 9.69 - 3.38 - 0.44 1-2 Proposed 15.41 16.27 - 1.69 0.83 Z-bus 18.47 - 7.80 - 2.07 Z-bus 28.14 - 7.27 - 2.35 PS 5.87 0.00 0.00 PS 15.41 0.00 0.00 1-4 Proposed 23.90 11.46 5.57 6.86 1-4 Proposed 33.95 17.70 7.48 8.76 Z-bus 15.93 0.35 1.44 Z-bus 24.80 0.43 1.64 PS 23.90 0.00 0.00 PS 33.95 0.00 0.00 1-5 Proposed 20.23 9.58 6.45 4.20 1-5 Proposed 27.86 14.67 7.67 5.53 Z-bus 15.60 7.44 0.63 Z-bus 24.28 6.84 0.72 PS 20.23 0.00 0.00 PS 27.86 0.00 0.00 2-3 Proposed 1.18 2.56 5.71 - 7.09 2-3 Proposed 0.29 3.99 4.60 - 8.31 Z-bus 7.07 14.22 - 10.80 Z-bus 11.03 12.91 - 12.70 PS 0.09 1.09 0.00 PS 0.05 0.24 0.00 2-4 Proposed 40.39 4.90 19.55 15.94 2-4 Proposed 41.74 4.63 19.85 17.26 Z-bus - 3.83 17.60 8.02 Z-bus - 6.23 16.55 9.37 PS 2.96 37.45 0.00 PS 7.62 34.25 0.00 2-5 Proposed 16.40 3.28 8.80 4.32 2-5 Proposed 17.35 3.95 8.77 4.63 Z-bus 3.90 13.02 1.87 Z-bus 5.94 12.03 2.18 PS 1.20 15.20 0.00 PS 3.17 14.24 0.00 2-6 Proposed 22.24 6.64 12.77 2.83 2-6 Proposed 25.03 9.13 12.78 3.12 Z-bus 7.60 15.99 - 5.69 Z-bus 11.70 14.99 - 6.66 PS 1.63 20.62 0.00 PS 4.57 20.54 0.00 3-5 Proposed 20.87 1.62 5.10 14.15 3-5 Proposed 23.18 1.34 5.77 16.07 Z-bus 0.59 5.58 19.53 Z-bus 0.93 5.06 22.81 PS 0.03 0.37 20.48 PS 0.02 0.08 23.10 3-6 Proposed 40.64 6.22 9.81 24.61 3-6 Proposed 47.50 7.52 11.63 28.34 Z-bus 0.67 0.22 22.77 Z-bus 1.08 0.00 26.55 PS 0.06 0.72 39.89 PS 0.04 0.16 47.32 4-5 Proposed 2.53 1.29 1.85 - 0.62 4-5 Proposed 3.21 1.91 1.92 - 0.62 Z-bus 6.21 9.02 2.61 Z-bus 9.44 8.60 3.01 PS 1.08 1.51 0.00 PS 1.82 1.50 0.00 5-6 Proposed - 1.70 1.08 - 0.37 - 2.41 5-6 Proposed - 0.90 1.98 - 0.37 - 2.52 Z-bus 4.27 2.32 - 1.85 Z-bus 6.55 2.26 - 2.13 PS - 0.05 - 0.60 - 1.12 PS - 0.06 - 0.27 - 0.62 Total Proposed 423.40 501.10 380.50 Total Proposed 475.70 444.70 384.60 transmission transmission Z-bus 738.80 714.40 341.90 Z-bus 885.30 574.80 344.30 cost ($/h) cost ($/h) PS 468.90 564.40 271.60 PS 546.60 481.40 277.00 Tariffs ($/MWh) Proposed 8.47 6.74 6.30 Tariffs ($/MWh) Proposed 6.16 6.42 5.46 Z-bus 14.77 9.61 5.66 Z-bus 11.46 8.30 4.89 PS 9.38 7.59 4.50 PS 7.08 6.95 3.93 To gain deeper insight into the results, the three methods smooths the trend of Z-bus method to reﬂect the counter- are also compared in terms of counter ﬂows. Tables 3, 4, 5 ﬂows. For example, in time step 4 representing the system and 6 shows that both circuit-theory-based methods take peak condition, the contribution of the generator at bus 1 into account the counter-ﬂows, while this feature does not into the power ﬂow of the line 5-6, is 6.55 MW counter- exist for the PS method. The proposed method, however, ﬂow under the Z-bus method, and is 1.99 MW counter- 123 Circuit-theory-based method for transmission ﬁxed cost allocation based on game-theory… 1515 Table 7 6-bus test system cost allocation to buses proposed methods, respectively. Note that, higher counter ﬂows with excessive rewards may result in unfair compe- Line Method Cost allocation to Cost Total tition, and in turn, may distort the power market. generator bus ($/h) allocation ($/h) to load Table 7 depicts, in detail, the allocated transmission use Bus 1 Bus 2 Bus 3 buses 4–6 of system (TUoS) costs and tariffs of the generators based ($/h) on the partial branch power ﬂows obtained in the system 1-2 Proposed 95.14 0 4.86 100 200 peak condition (time step 4 in Table 6). As shown in Table 7, each branch cost is allocated by 50-50 ratio Z-bus 181.73 0 0 18.27 between the generators and loads, under the proposed as PS 100 0 0 100 well as PS methods, in spite of different allocated cost 1-4 Proposed 52.14 22.05 25.81 100 200 distributions among the generators as a result of different Z-bus 129.19 2.23 8.53 60.05 partial branch power ﬂows assigned to the generators under PS 100 0 0 100 these methods, previously addressed in Tables 3, 4, 5 and 1-5 Proposed 78.96 41.28 29.76 150 300 6. In contrast, for Z-bus method it is basically the network Z-bus 193.16 54.41 5.69 46.74 parameters that determine the ratio by which the costs are PS 150 0 0 150 allocated between the generators and loads. For instance, 2-3 Proposed 58.07 66.93 0 125 250 the cost of lines 1-5 and 5-6 both equals 300 $/h. The Z-bus 104.28 122.08 0 23.64 proposed technique, as with the PS methodology, assigns PS 22.74 102.26 0 125 150 $/h (50%) of each line cost to the generators. Under 2-4 Proposed 5.55 23.77 20.68 50 100 the Z-bus method, however, 253.26 $/h of line 1-5 cost is Z-bus 0 27.60 15.63 56.77 allocated to the generators and 46.74 $/h to the loads, and PS 9.10 40.90 0 50 for line 5-6, the share of the generators and loads become 2-5 Proposed 34.15 75.83 40.02 150 300 30.83 $/h and 269.17 $/h, respectively. Z-bus 62.66 126.97 23.00 87.37 Based on the results shown in Table 7, it is conﬁrmed PS 27.29 122.71 0 150 that the proposed method considers the amount, the loca- 2-6 Proposed 36.49 51.06 12.45 100 200 tion and the effective use of the line by the generators in its Z-bus 50.93 65.26 0 83.81 allocation process. For example, the generator at bus 2 uses PS 18.19 81.81 0 100 the line 1-2 less, compared to the generator at bus 1, due to 3-5 Proposed 7.51 32.35 90.14 130 260 the power ﬂow direction of the line 1-2. These properties Z-bus 6.47 35.25 158.79 59.49 hold for the proposed method, irrespective of the pricing PS 0.10 0.43 129.47 130 method applied, that is ZCF, absolute value (AV) and 3-6 Proposed 7.91 12.25 29.84 50 100 classic MW-mile pricing. For example, under the proposed Z-bus 2.14 0 52.53 45.33 method, the network costs of the generators using the ZCF PS 0.04 0.17 49.80 50 pricing are 475.74 $/h, 444.73 $/h and 384.53 $/h, while 4-5 Proposed 99.82 100.18 0 200 400 these values change slightly to 485.86 $/h, 399.09 $/h and Z-bus 154.73 140.96 49.31 55.00 420.04 $/h, if the AV pricing is used. In case of the Z-bus PS 109.65 90.35 0 200 method, however, the results change signiﬁcantly, when 5-6 Proposed 0 19.03 130.97 150 300 different pricing schemes are used. The network costs Z-bus 0 0 30.83 269.17 allocated to the generators are 885.29 $/h, 574.75 $/h and PS 9.51 42.75 97.74 150 344.31 $/h, by the ZCF pricing, where these values become 668.43 $/h, 412.85 $/h and 337.77 $/h, by the AV pricing Network Proposed 475.74 444.73 384.53 1305 2610 method. Discrepancies arise because signiﬁcant counter- cost Z-bus 885.29 574.76 344.31 805.64 ﬂows exist under the Z-bus method. PS 546.62 481.38 277.01 1305 According to Table 7 and considering the TUoS tariffs Tariffs Proposed 6.16 6.42 5.46 of the generators, under each of the three methods, the ($/ Z-bus 11.46 8.30 4.89 generator at bus 3 pays the lowest price, whereas the MWh) PS 7.08 6.95 3.93 generator at bus 1 has to pay the highest price, for the use of the network. However, the TUoS tariffs of the genera- tors at buses 1 and 2 are relatively high according to the ﬂow, under the proposed method. In the same time step, Z-bus method, because 69.13% of the network cost is given the generator at bus 2, there exists a counter-ﬂow imposed to the generators. contribution of the generator into the power ﬂow of line The proposed method is less sensitive to the calculation 1-2, with 7.27 MW and 1.69 MW under the Z-bus and the reference side of the lines compared with the Z-bus 123 1516 Saeid POUYAFAR et al. method, for which the results change signiﬁcantly when the calculation reference side is changed. For example, under the proposed method, when P is considered as reference, 1-2 the power ﬂow contribution and the allocated cost to the generator at bus 1 due to the line 1-2, are 16.27 MW and 95.14 $/h, whereas the values change slightly to 15.93 MW and 95.00 $/h when P is considered as ref- 2-1 erence. When Z-bus is used to calculate the same quanti- ties, the values become 28.14 MW and 181.72 $/h for P 1-2 being considered as reference, whereas they change sig- niﬁcantly to 22.43 MW and 125.69 $/h when P is con- 2-1 sidered as reference. 6.2 IEEE 30-bus system To validate the proposed method, the IEEE 30-bus system illustrated in Fig. 2 is used as a test system. Branch data are provided in [37]. Bus data for the base case are Fig. 2 IEEE 30-bus test system provided in Appendix A Table A1. 6.2.1 Contribution of generators into branch power ﬂows The generators are located at buses 1, 2, 22, 27, 23, 13. Figures 3 and 4 illustrate the contributions of the genera- tors into the network real power ﬂow of the branches applied by the three cost allocation methods including Z-bus theory, PS method represented by Bialek, and the proposed method. According to Figs. 3 and 4, the following contributions are obtained: 1) The overall contribution proﬁles of the generators for the three methods are almost comparable, i.e. the power ﬂow contribution of the generators into the neighboring lines is higher compared to the others calculated by each of the three methods, although they have different principles. The generators at buses 1 and 2 contribute most in lines 1-15 (Fig. 3a and b). Likewise, for the generators connected at bus numbers 22, 27, 23 and 13, their associated lines with dominant power ﬂow contribution of the corresponding gener- ators are: line numbers 25-29 for G22; 26-41 for G27; 22, 30, 32 for G23; 16-18 for G13, according to Figs. 3c, d, 4a and b, respectively. It may be entitled ‘‘the nearby effects of the power networks’’. 2) The curves obtained by the PS theory have several lines with zero contribution of the generators. This is because the PS theory does not consider the counter- ﬂow effects in its calculations despite the fact that the concept of counter-ﬂows is indispensable in power Fig. 3 Contribution of generators G1, G2, G22 and G27 into branch ﬂow related problems. power ﬂows of IEEE 30-bus system Figure 5 shows the real power ﬂow allocation of two high loaded branches 1-2 and 6-8. The results are more 123 Circuit-theory-based method for transmission ﬁxed cost allocation based on game-theory… 1517 locational by the PS method (Fig. 5a) thanks to the pro- portional sharing principle which has no counter-ﬂow effect consideration. They are more intense by the Z-bus method in case of high counter-ﬂow shares of some gen- erators (Fig. 5b), and modest by the proposed method. Discrepancies arise by the fact that Z-bus method uses the current ﬂows while the proposed method applies the power ﬂows to allocate. For branch 6-8, each of the three methods allocate the highest contribution to the generators located at buses 1 and 2. The share of the generators located at buses other than 1 and 2 is completely different for the three methods. The PS allocation scheme is completely based on the power ﬂow directions of the lines. Among the generators located at the buses 22, 27, 23 and 13, only G27 contributes into the power ﬂow of the branch 6-8 because there is a path from the generator at the bus 27 to the sending end of the branch 6-8 (the path is 27-28, 28-6) with the same Fig. 4 Contribution of generators G23 and G13 into branch power direction as the real power ﬂow of the branches. For the ﬂows of IEEE 30-bus system Z-bus methodology, the contribution of the generators into the branch ﬂows is highly dependent on their neighboring loads since the method traces the current ﬂow of the par- ticipants. In the proposed methodology, it is the overall location of the generators with respect to the loads that determines the outcomes, which makes the generator tariffs more stable compared to the other two methods (Table 8). For branch 1-2 (Fig. 5b), the contribution of the gener- ator located at bus 1 is dominant by each of the three methods as G1 is connected to the sending end of the line 1-2. Since the counter-ﬂows are not considered in PS method, the adjacent generator G2 located at bus 2 has no share in the power ﬂow of the line 1-2. For the Z-bus method, the share of generator 1 in power ﬂow of the line 1-2 exceeds the real power ﬂow of the line by an amount almost equal to the generator G2’s signiﬁcant counter-ﬂow contribution into the line 1-2. In case of the proposed method, the counter-ﬂow share of the generator G2 is lower and the G1 has a moderate contribution amount into the power ﬂow of the line 1-2. 6.2.2 Generator buses’ TUoS charges Table 8 lists the transmission per-unit costs allocated to the generator buses for the base case and the cases with individual loads altered. P (b) in MW is the active power of the load connected to the bus b. According to Table 8, for the base case, bus numbers 1, 2 and 22 have TUoS tariffs cheaper than buses 27, 23 and Fig. 5 Partial power ﬂows of branches 6-8 and 1-2 due to generators 13 calculated by all the three methods, because large loads 123 1518 Saeid POUYAFAR et al. are connected at buses 2, 7, 8 and 21. The generators G1 for the Z-bus method and ? 18.1 $/MW (121%) for the PS and G2 tariffs are highly dependent on the large loads method. For the generator 27, the tariff growths are ? 11 $/ connected at the buses 2, 7 and 8. Thus, minor changes take MW (22%) by the proposed method, ? 12 $/MW (24%) by place by varying the loads connected at buses relatively far the Z-bus method and ? 16 $/MW (20%) by the PS from that units. This statement is shown in the Table 8 by method. The tariff decrease of the generator bus 23 is 19% altering the loads connected at buses 21, 12, 30, 23 and 27. for both the proposed and the Z-bus method and 22% for The highest TUoS tariff is assigned to the bus 27 by the PS the PS method. The same scenario takes place in case of method due to its relative distance from the large loads and the load increase at buses 30, 23 and 27. For example, the to bus 13 by the proposed method due to its circuit con- highest tariff variation due to the 10 MW load growth of dition characterized by the voltage and current equations of bus 30 is associated to bus 27 calculated by the PS method, the networks. For Z-bus method, the high tariff buses have which is a 14 $/MW decrease from 85 to 71 $/MW. almost the same prices. The generator connected at bus 22 close to the load bus 21, have the lowest TUoS tariff by the 6.3 2383-bus practical system PS method whereas its tariff is equal to the G1 and G2 charges by the proposed method due to their similar circuit Tables 9, 10 and 11 presents the results obtained by conditions. applying the proposed method to the Polish 2383-bus Among the methods discussed, the proposed method has system. To emphasize that the proposed method is appli- the highest tariff stability against the single load variations. cable to very large systems, the 2383-bus system of Polish If the load on bus 21 with the real power demand of 17.5 400, 220 and 110 kV networks during winter 1999-2000 MW is disconnected from the network, the TUoS charges peak conditions is considered. The system data are given in of the generator buses 22, 27 and 23 located in the MATPOWER user’s manual. The proposed method does neighboring zone of bus 21 will be changed by the three not encounter the singularity problems of the Z-bus method methods. The tariff variation of the generator 22 is ? 13 $/ to build the impedance matrix as well as the PS method to MW (36%) for the proposed method, ? 14 $/MW (54%) build the inverted tracing distribution matrices. An Intel Core i5, 2.3 GHz, 6 GB RAM 64-bit computer is used to run the simulations of this system. MATLAB R2016a reported the elapsed time 18.73 s which is fairly a short Table 8 Comparison of TUoS tariffs of generator buses for some operating points on IEEE 30-bus test system running time. It is noted that in Table 11, APG represents the active power generated; TTPU/TTNU represent the Operating point Method Transmission cost per-unit of generator bus ($/MWh) G1 G2 G22 G27 G23 G13 Table 9 First 5 branches with highest tariffs Branch Power ﬂow (MW) Tariffs ($/h) Base case Proposed 34 34 36 51 48 63 Z-bus 38 37 26 51 52 54 1764-1760 16.54 463.20 PS 33 24 15 85 53 44 1763-1761 38.19 452.50 P (21)=0 Proposed 36 36 49 62 39 66 612-413 63.46 324.00 Z-bus 41 39 40 63 42 51 1945-1845 - 40.60 245.50 PS 33 24 33 101 41 43 1489-1426 - 69.30 237.20 P (12)=0 Proposed 35 36 39 55 50 69 Z-bus 41 39 28 54 54 59 PS 34 24 15 89 54 60 P (30)=20 MW Proposed 32 32 33 47 44 60 Z-bus 37 35 25 46 47 51 Table 10 First 5 branches with highest power ﬂows PS 32 24 15 71 51 43 Branch Power ﬂow (MW) Tariffs ($/h) P (23)=20 MW Proposed 33 32 34 50 36 51 Z-bus 37 35 27 51 34 42 138-67 - 771.20 20.56 PS 33 24 15 91 16 41 32-31 - 681.70 0.10 P (27)=20 MW Proposed 33 35 28 42 36 53 18-15 552.20 42.62 Z-bus 39 37 22 41 39 45 15-165 451.60 24.74 PS 34 24 16 60 41 42 132-131 - 416.50 0.10 123 Circuit-theory-based method for transmission ﬁxed cost allocation based on game-theory… 1519 Table 11 Transmission use of system related values for selected buses in Polish 2383-bus test system by proposed method TUoS charge No. Bus number APG (MW) TTPU (MW) TTNU (MW) TTC ($/h) ($/MWh) 1 18 1908 9241 1342 6530 3.42 2 17 1080 5910 623 4226 3.91 3 31 1000 4924 1336 3928 3.93 4 131 872 3992 1277 4426 5.08 5 67 750 3951 882 3494 4.66 6 16 720 3878 450 2722 3.78 7 127 690 3108 931 3215 4.66 8 63 650 3804 565 2508 3.86 9 176 600 2899 1108 2830 4.72 10 139 600 2616 828 2672 4.45 11 1426 495 3014 706 2320 4.69 12 64 450 2699 395 1724 3.83 13 105 430 2218 745 1533 3.57 14 43 410 2340 479 1682 4.10 15 44 410 2214 558 1683 4.11 16 10 400 1944 775 2384 5.96 17 911 370 2172 493 1998 5.40 18 912 370 2235 490 2108 5.70 19 1416 367 3167 807 1927 5.25 20 111 360 1724 409 1578 4.38 21 2164 4.10 21 15 71 17.29 22 2268 1.80 9 7 23 12.57 23 2328 3 13 7 37 12.28 24 2159 12 137 75 137 11.39 25 132 70 796 195 796 11.40 Branch power flow contribution of generator (MW) No. Bus number 1764–1760 1763–1761 612–413 1945–1845 1489–1426 138–67 32–31 18–15 15–165 132–131 1 18 0.45 2.52 3.69 – 1.96 – 3.71 – 92.43 – 68.26 133.24 90.95 – 16.46 2 17 0.30 2.02 2.89 – 1.07 – 2.14 – 41.81 – 28.13 55.10 39.11 – 6.80 3 31 0.47 1.32 3.57 – 0.86 – 1.84 – 25.05 – 207.80 – 2.61 51.53 – 3.82 4 131 0.55 0.21 0.84 43.14 12.98 5.88 – 2.45 – 0.51 – 8.20 – 257.60 5 67 0.28 0.44 1.17 – 1.22 – 1.02 – 136.6 – 14.64 27.01 17.97 – 21.06 6 16 0.18 1.35 1.80 – 0.73 – 1.47 – 28.34 – 19.64 37.20 26.16 – 4.74 7 127 0.67 0.80 0.55 – 3.96 – 0.14 – 6.38 – 6.99 – 1.27 – 15.81 0.40 8 63 0.15 0.48 1.48 – 0.35 – 3.20 – 20.38 – 13.31 17.66 14.97 – 3.14 9 176 0.51 0.71 0.28 – 1.55 – 0.43 – 8.00 – 16.94 – 22.13 – 59.80 – 1.29 10 139 1.62 3.68 1.05 – 1.16 – 0.43 – 7.96 – 8.50 3.67 – 3.86 – 2.64 11 1426 0.11 0.32 1.13 – 0.21 – 25.47 – 11.99 – 9.90 10.48 10.03 – 1.68 12 64 0.11 0.34 1.06 – 0.24 – 2.91 – 14.15 – 9.11 12.47 10.59 – 2.17 13 105 0.11 0.48 0.99 – 0.39 – 0.79 – 20.30 – 13.41 19.94 16.22 – 3.82 14 43 0.14 0.44 1.43 – 0.29 – 0.81 – 11.86 – 27.96 8.18 15.46 – 1.79 15 44 0.14 0.44 1.43 – 0.29 – 0.81 – 11.85 – 28.04 8.17 15.47 – 1.79 16 10 0.26 0.50 – 1.91 – 0.48 – 0.32 – 5.05 – 15.79 – 2.20 7.15 – 0.73 17 911 0.12 0.22 0.60 – 0.49 – 0.54 – 47.78 – 7.21 12.55 8.56 – 8.43 18 912 0.12 0.22 0.59 12.13 8.29 – 0.47 – 0.53 – 46.64 – 7.04 – 8.19 19 1416 0.08 0.23 0.73 – 0.17 4.27 – 10.22 – 6.89 8.56 7.11 – 1.54 20 111 0.57 1.94 0.83 – 0.59 – 0.33 – 6.66 – 5.50 6.36 2.64 – 1.55 21 2164 0.00 0.01 0.00 – 0.02 0.00 – 0.06 – 0.05 – 0.02 – 0.15 – 0.01 22 2268 0.00 0.00 0.00 – 0.01 0.00 – 0.02 – 0.02 – 0.01 – 0.07 – 0.01 23 2328 0.00 0.00 0.00 – 0.01 0.00 – 0.04 – 0.03 0.01 – 0.08 – 0.01 24 2159 0.00 0.03 0.07 – 0.03 0.05 0.18 0.89 2.26 3.51 0.05 25 132 0.01 0.31 – 0.07 – 0.57 0.12 – 19.47 2.59 ̢ 0.55 1.90 7.44 123 1520 Saeid POUYAFAR et al. sum of partial branch power ﬂows due to a generator in the proposed method, however, smooths the trend of the Z-bus same/opposite direction to branch power ﬂow, respectively. method to reﬂect the counter-ﬂows, and therefore helps to The ﬁrst 20 rows (No. 1-20) of Table 11 show the ﬁrst 20 reduce the cross-subsidies. This property is truly valuable, generators in terms of APG, TTPU and TTC. The last 5 since higher counter-ﬂows with excessive rewards bring rows (No. 21-25) of Table 11 show the ﬁrst 5 generators in about unfair competitions. Furthermore, based on a com- terms of TUoS tariffs. parison on the 6-bus system, it is determined that the results of the cost allocation by the Z-bus method change signiﬁcantly when different MW-mile pricing schemes are 7 Conclusion used, whereas the proposed method provides more stable results. This paper presents a new circuit-theory-based method Tariff stability of the proposed cost allocation method is to the problem of TCA. Unlike majority, if not all, similar also assessed on the IEEE 30-bus system. The results methods, the proposed method attempts to justify the way reveal that the proposed method provides a fairly it treats the non-linear behavior of the power systems to stable tariffs against the temporal load variations, as well as mathematically identify the shares of the participants on the generating dispatch strategies. Moreover, based on the network power quantities. The applied principle to split the results on the 6-bus test system, the proposed method is mutual terms of the power equations, as the causes of non- less sensitive to the calculation reference side of the lines, linearity, is conﬁrmed by the Shapley and Aumann-Shap- compared with the Z-bus method. As another advantage, ley values as the preferred transmission network cost game the proposed method works consistently for all network solutions. Moreover, a new concept, named circuit-theory- conﬁgurations, as it overcomes the singularity problems of based EBE, is introduced. the Z-bus as well as PS methods to invert the large-scale The proposed method is compared with two well-known sparse matrices. Computational performance on the TCA techniques, namely, Z-bus and PS methods. Numer- 2383-bus practical system of Poland indicates that the ical case studies on the 6-bus system and the IEEE 30-bus proposed method is quite fast, so it can well deal with the system show that the proposed method outperforms the problem of TCA in large practical power systems. other two methods. According to the results, the proposed Open Access This article is distributed under the terms of the method is fair and efﬁcient, as it reﬂects the network Creative Commons Attribution 4.0 International License (http:// topology as well as the order of magnitude and the location creativecommons.org/licenses/by/4.0/), which permits unrestricted of the generators in the grid. Although the PS method use, distribution, and reproduction in any medium, provided you give intensiﬁes the locational signals, its principle is only based appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were on logical reasoning and can never be proved. It is also made. shown that the PS principle ignores the counter-ﬂows and, in turn, results in considerable tariff instability. The 123 Circuit-theory-based method for transmission ﬁxed cost allocation based on game-theory… 1521 Appendix A Table A1 Load ﬂow result for IEEE 30-bus test system Bus Voltage Generation Load Magnitude (p.u.) Angle () Real (MW) Reactive (Mvar) Real (MW) Reactive (Mvar) 1 0.9824 0 41.5421 - 5.4364 0 0 2 0.9787 - 0.7630 55.4019 1.6748 21.7 12.7 3 0.9769 - 2.3897 0 0 2.4 1.2 4 0.9764 - 2.8386 0 0 7.6 1.6 5 0.9713 - 2.4864 0 0 0 0 6 0.9723 - 3.2287 0 0 0 0 7 0.9623 - 3.4910 0 0 22.8 10.9 8 0.9611 - 3.6819 0 0 30 30 9 0.9903 - 4.1371 0 0 0 0 10 0.9998 - 4.5998 0 0 5.8 2 11 0.9903 - 4.1371 0 0 0 0 12 1.0174 - 4.4979 0 0 11.2 7.5 13 1.0645 - 3.2980 16.2002 35.9303 0 0 14 1.0066 - 5.0397 0 0 6.2 1.6 15 1.0092 - 4.8140 0 0 8.2 2.5 16 1.0028 - 4.8393 0 0 3.5 1.8 17 0.9955 - 4.8873 0 0 9 5.8 18 0.9933 - 5.4843 0 0 3.2 0.9 19 0.9873 - 5.6882 0 0 9.5 3.4 20 0.9896 - 5.4719 0 0 2.2 0.7 21 1.0093 - 4.6208 0 0 17.5 11.2 22 1.0160 - 4.5030 22.7403 34.1971 0 0 23 1.0256 - 3.7557 16.2670 6.9598 3.2 1.6 24 1.0167 - 3.8852 0 0 8.7 6.7 25 1.0438 - 2.0724 0 0 0 0 26 1.0267 - 2.4760 0 0 3.5 2.3 27 1.0690 - 0.7147 39.9090 31.7544 0 0 28 0.9820 - 3.2152 0 0 0 0 29 1.0500 - 1.8494 0 0 2.4 0.9 30 1.0391 - 2.6429 0 0 10.6 1.9 References [5] Yang Z, Zhong H, Xia Q et al (2016) A structural transmission cost allocation scheme based on capacity usage identiﬁcation. 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His research interests include power system economics, determination using a transformer analogy. IET Gener Transm distribution networks, microgrid and energy management. Distrib 8(1):81–90

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Circuit-theory-based method for transmission fixed cost allocation based on game-theory rationalized sharing of mutual-terms

Circuit-theory-based method for transmission fixed cost allocation based on game-theory rationalized sharing of mutual-terms

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Circuit-theory-based method for transmission fixed cost allocation based on game-theory rationalized sharing of mutual-terms

Circuit-theory-based method for transmission fixed cost allocation based on game-theory rationalized sharing of mutual-terms

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