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Hindawi International Transactions on Electrical Energy Systems Volume 2022, Article ID 8562670, 15 pages https://doi.org/10.1155/2022/8562670 Research Article Cross-Regional Transaction Path Configuration of Renewable Energy Resources by Graph Theory-Based Transmission Cost Allocation Method 1 1 2 1 1 Yin Yao , Dahuan Lu, Wenzhong Gao, Bo Zhou , and Dongdong Li Department of Electrical Engineering, Shanghai University of Electric Power, Shanghai, China Electrical Engineering Department, University of Denver, Denver, CO 80208, USA Correspondence should be addressed to Bo Zhou; firstname.lastname@example.org Received 7 January 2022; Revised 8 March 2022; Accepted 10 March 2022; Published 22 April 2022 Academic Editor: Mahdiyeh Eslami Copyright © 2022 Yin Yao et al. +is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For the transaction path conﬁguration of renewable energy cross-regional consumption, there are several critical problems such as the key node identiﬁcation, the maximum delivery quota, and the transmission cost allocation (TCA). To solve these problems, ﬁrstly, the simpliﬁed graph model of the ultra-high-voltage (UHV) network is constructed, and the network connectivity and the vulnerability of the key nodes are analyzed from the perspective of the system topology. Secondly, the source and sink nodes are set corresponding to the electricity seller and buyer in the power market, and the Edmonds–Karp algorithm is utilized to search for the augmenting path. Also, the maximum transmission quota of the transaction path is achieved eﬀectively and rapidly. Finally, the social welfare is set as the optimization objective, and the optimal allocation of multiple power ﬂows in multiple feasible transaction paths is carried out. +e case study was conducted based on the 17 cross-regional transactions in China including the typical Gansu-Shanghai renewable energy consumption case. Compared to the existing TCA method, the simulation result shows that the proposed method can eﬀectively utilize the transmission potential, decrease the overall transmission cost, and provide proper economic signals. the southeast region . Also, the power market must be 1. Introduction based on an eﬀective and eﬃcient conﬁguration of the cross- +e imbalance of energy source distribution and the dif- regional transaction paths. ferences in regional economic development have caused a +e cross-regional electricity transaction requires the reverse distribution of the energy supply and demand in ultra-high-voltage (UHV) power grid to minimize the line China. Taking the renewable energy consumption problem loss due to the long-distance transmission. According to the in Gansu province as an example, the renewable energy development scheme of the State Grid Corporation of China generation from wind and solar farms is abundant, but the (SGCC), UHV is deﬁned as the transmission technology with an AC voltage level of 1000 kV and above and DC local electricity consumption capacity in the northwest re- gion is limited due to the comparatively low level of eco- voltage level ±800 kV and above . nomic development. Consequently, the excessive electricity At present, the network structure of the UHV grid is from renewable energy breaks the balance between gener- relatively simple. +e SGCC generally organizes and com- ation and consumption and results in the large-scale elec- pletes the cross-regional electricity transactions annually, tricity abandonment. +erefore, it is urgent to optimize the quarterly, and monthly. However, with the gradual increase distribution of resources nationwide with the support of of AC and DC UHV transmission lines, the complexity of power trading policies. +e objective of power market is to the UHV network will also increase correspondingly. +e transmit the excess electricity to the desired region, such as original bilateral relationship between the seller and buyer 2 International Transactions on Electrical Energy Systems operation of the existing power grid. A responsi- will no longer be adaptive to the market strategy. +e op- timal conﬁguration of the cross-regional transaction path bility-based approach was proposed in  to al- locate the cost of the transmission congestion and will become an increasingly prominent problem. In response to the above problem, the majority of losses to the nodes of the network. In , due to the current research in this area utilizes power economy theory generic complexity of the cooperative game theoretic for modeling, simulation, and optimization. For instance, six problems based on marginal pricing, the min-max transaction models, four model architectures, and corre- fairness policy was utilized to solve this NP-hard sponding risk control strategies for cross-regional transac- problem in polynomial time. With the increase of the tion path conﬁguration were proposed in . To set up the renewable energy penetration, the hybrid AD/DC transmission structure was studied in . +e test platform, a management and control index system was developed in  for cross-regional transactions based on LMP-based nodal pricing method was proposed to provide eﬃcient and accurate solution. In , the supply chain, group control, and risk management. For the test platform, hierarchical optimization of the tie line transmission-network expansion problem and the energy source distraction problem were deﬁned as a planning was carried out to achieve automatic planning and ﬂexible scheduling . To consider the cross-regional trilevel optimization problem based on LMP. transmission line loss, a compensation method was devel- In summary, the long-term marginal cost method oped based on the route method and the average network requires the usage of some highly uncertain as- loss allocation method . sumptions, and the calculation is complicated. +e However, there is relatively few research of TCA of the short-term marginal cost method cannot guarantee cross-regional transaction. +e ideal allocation method the balance of revenue and expenditure. If the meets the following requirements. (1) It must contain suf- network construction investments cannot be re- ﬁcient economic information, which can eﬀectively guide covered, the allocation method is not acceptable by the economic operation of the power grid to make full use of the power company. the existing transmission grid resources. (2) +e transmis- (2) For the allocation method based on the amount of sion grid companies require that their annual revenue and usage, the proportion of the total cost is determined expenditures are balanced through transmission fees to according to the usage of the power grid equipment. ensure the normal operations and the long-term develop- +e actual operation of the system is taken into ment of the power grid. (3) +e method is also required to be account. +e classic MW-Mile (MWM) method simple and easy for implementation, and the results can be allocates the transmission cost based on the usage of veriﬁed to meet the fair and open principle of the power the line capacity and line length. Xiao et al.  market. proposed a power tracing-based equivalent bilateral +ere are mainly two types of allocation methods. +e exchange method in which network users are re- ﬁrst type is based on the cost allocation. +e other type is sponsible for not only their induced power ﬂows based on the amount of usage allocation. +e cost-based but also power ﬂows induced by whom they have allocation methods can be further divided into the em- equivalent bilateral exchanges with. In , a new bedded cost methods and the local marginal price (LMP) eﬃcient method for solving the reactive power methods based on microeconomic theory. tracing problem was proposed in a transmission (i) +e embedded cost method is essentially the ac- system. In , the line capacity was replaced by the counting of the transmission cost. +is method fo- maximum line loading for N − 1 security to achieve cuses on oﬀsetting the actual expenditures of the grid a more fair ﬁxed cost allocation in a pool based operation and investment costs. On the basis of the power market. In , the transmission expansion most common post-stamp method, the distribution model was formulated as a multiobjective optimi- of the ﬁxed and operation cost is deﬁned as an zation problem to facilitate the distributed gener- inﬁnite-person cooperative game. A cooperative ation and defer the transmission investment. game approach which provides stable solution in- Considering the quality of the load, in , the tegrated with appropriate penalties or rewards to power factor was introduced in the MWM method. participants was presented in . +e Aumann– In , the transmission capacity was divided into Shapley value was utilized for TCA , the distri- normal condition capacity, capacity for contin- bution loss allocation , and the proﬁt allocation gency, capacity for future use, and invalid capacity. for demand-side resource (DSR) aggregators . +e +e structural TCA scheme can encourage the ef- proposed game theoretic method ensures the equi- ﬁcient use of the transmission network. +e same table allocation and recovery of the total cost. structural method is applied to the optimal plan- However, the price signal does not contain any ning strategy for the distributed energy resources economic information, so it cannot guide the op- (DERs). In , a circuit theory-based TCA timal use of the power grid resources and the long- method was developed considering the orthogonal term development of the power grid. projection. +e Aumann–Shapley value is used to distribute the interaction term between the in- (ii) +e LMP method aims at maximizing economic volved components. beneﬁts and eﬀectively guiding the economic International Transactions on Electrical Energy Systems 3 transaction are limited. Modeling of UHV and DC +ere are various power ﬂow tracing methods that can be used for the transmission embedded cost allocation. +e hybrid grid for the entire system is almost impos- sible. +e graph theoretic method follows the ca- power ﬂow tracing based on proportional sharing and circuit theory requires line impedance for power ﬂow calculation pacity constraint and equilibrium constraint. +e [21–23]. +e power ﬂow tracing based on optimization  selection of slack bus is not necessary, and the modiﬁes the maximum power output of the generator which counter-ﬂow is considered in the augmenting stage. contributes to line congestion. But in power market, the So, the graph theory-based network ﬂow method is actual power output of the generator is determined during more suitable for the long-term cross-regional the market clearing stage. As for the power ﬂow tracing transaction path conﬁguration with limited grid information. based on the relative electrical distance concept , this method also requires the line impedance and power ﬂow to (2) +e graph theory-based TCA method can provide decide the relative electrical distance. In addition, the the maximum ﬂow capacity and invalid capacity of generation dispatch makes a great impact on the power ﬂow the cross-regional transaction path under complex solutions, and the residual potential of the network is ig- networks. Besides that, the ﬂow between each nored. +erefore, the MWM method requires complete grid transaction pair is decoupled. +e transmission cost parameter information and cannot reﬂect the actual usage of is allocated by three sections: capacity for normal the power grid by the cross-regional transaction. use, capacity for future use, and invalid capacity. +e Besides the reference to the power economy theory, complete allocation of transmission cost is current research also studies the cross-regional power guaranteed. transmission from the topological structure aspect based on In this paper, the network ﬂow algorithm and the the graph theory. In , a transaction tracing-based loss maximum ﬂow algorithm are introduced in Section 2 for allocation scheme for assigning the network losses incurred connectivity analysis and maximum capacity estimation. In due to the transactions occurring between peers in a dy- Section 3, the above algorithms are applied to the simpliﬁed namic environment was presented. In , a network ﬂow graph of the current UHV transmission grid in China. +e approach was developed for the estimation of the cross- graph theory-based TCA method is developed. In Section 4, regional energy trade volume and the partition start-up through the case study of 17 transactions in China (espe- capacity. +e simulation result is applied to the optimization cially the Gansu-Shanghai transaction path that involves the of the local start-up capacity conﬁguration. In , an renewable energy), the comparative analysis between the optimization algorithm was proposed for the renewable original, optimized, and max-ﬂow scenarios is conducted. energy cross-regional transactions. +e objective is to Section 5 summarizes the advantages of the proposed maximize the social beneﬁt and the total trade volume. +e method and discusses the remaining work for future study. algorithm solves the optimization problem by the ﬁx-path method, the point-arc model, and the arc-path model. +e 2. Graph Theory Network Flow Algorithm network ﬂow algorithm was improved in  to achieve the tracking of each cross-regional transaction path. +e graph 2.1. Graph Deﬁnitions and Terms. When the connectivity theory is also applied to the system reliability study. Zhu and mutual relations of the network are involved in the et al.  proposed the impact analysis of the key nodes engineering mathematical problems, a graph can be deﬁned removal on the vulnerability of the UHV grid. for intuitive and visual analysis to solve the problem. A graph G consists of a vertex set V(G) and an edge set 1.1. Contributionsand Organization. In 2060, the proportion E(G). Each edge associates with two vertices (not necessarily of renewable energy power generation will reach more than diﬀerent vertices). Each node in the power transmission grid 70% according to China’s “30·60” decarbonization goal. To can be considered as a vertex in the graph G. Each trans- facilitate the consumption of renewable power, the cross- mission line can be considered as an edge in the graph G. +e regional power market is in the rapid growth stage. In 2020, direction of the line current from nodes u to v is described as the amount of inter-provincial transactions was 1157.7 the ﬂow f(u, v). +e rated power of the transmission line is billion kWh which increased by 9.5% compared to previous described as the edge capacity c(u, v), as shown in Figure 1. year in China . +erefore, the complete allocation of the Assume that the power transmission grid contains neither total cost is a critical problem. Compared to the LMP-based parallel edges nor loops. +e power transmission grid can be TCA method, the graph theoretic TCA method is modiﬁed described by a node vertex set V(G), a transmission line edge based on the MWM TCA method. It is more suitable for the set E(G), and a line capacity set C(G). In summary, the graph emerging market which contains limited grid parameter of the power transmission grid is a directed simple graph information and requires high market clearing eﬃciency. G(V, E, C). +e proposed method provides the following contributions. (1) +e traditional optimal power ﬂow (OPF) method 2.2. Connectivity of Transaction Paths. +e energy seller in requires complete grid parameters and must con- the market, such as the wind farm in the northwest region of sider the selection of slack bus and the counter-ﬂow. China, is described as the source node s in the graph. +e +e cross-regional power market in China contains energy buyer, such as the big power consumer in the east 23 provinces. +e grid parameters involved in the region of China, is described as the sink point t in the graph. 4 International Transactions on Electrical Energy Systems u c (u,v) v +e max-ﬂow algorithm searches and utilizes the residual power transmission capacity between source and sink nodes. f (u,v) Figure 1: Simpliﬁed graph model of the transmission line. 2.3.2. Augmenting Path of Residual Network. In the residual network, the max-ﬂow algorithm searches for the feasible In the directed simple graph G, all diﬀerent possible ﬂows and forms a new residual network. +is process transaction paths between the source node s and the sink constitutes an iterative loop until no new feasible ﬂow can be node t are described as the transaction path set TP(s, t). found. +e new feasible ﬂow is deﬁned as the augmenting From system reliability aspect, the transaction path set is path p which is a path from the source node s to the sink expected to be a non-empty set even if some nodes or edges node t in the residual network G . +e capacity of the need to be removed from the graph due to facility failures or augmenting path, c (p), is deﬁned in equation (3). c (p) is the f f maintenance. maximum additional ﬂow that can be added along the path. If the vertex subset, S ∈ V(G), makes the graph G-S have more than one branch which means G-S is disconnected or ⎧ ⎨ c (p) � min c (u, v)|<u, v>∈ p , f f (3) has only one vertex, then S is called the separating set or c (p)> 0. vertex cut of G. +e minimum size of S is called the con- nectivity of G and denoted as k(G). If the connectivity of G is +e addition of an augmenting path results in a ﬂow with at least k, then G is k-connected. a larger value. +e augmenting path is the increment of the ﬂow and has the property of augmentation. +e ﬂow of augmenting path f is deﬁned in the following equation: 2.3. -e Maximum Flow of Transaction Path. To evaluate the c (p), <u, v>∈ p, maximum transmission capacity between the seller and ⎧ ⎪ f buyer, the maximum ﬂow between the source node (seller) s f (u, v) � −c (p), <u, v>∈ p, (4) p f and the sink node (buyer) t in the directed and weighted 0, else. graph needs to be solved. Moreover, the max-ﬂow solution can provide the visual tracking of the possible “bottleneck” +e physical meaning of the augmenting path in the edge in the transaction path. +e saturated “bottleneck” edge power grid is the incremental trading of the transaction causes the rest of the edges on the path to no longer able to between seller and buyer. +e cross-regional consumption of accommodate any positive ﬂow increase. renewable energy can utilize the multiterminal DC grid and +e concepts of residual network and augmenting path the traditional AC grid. In the traditional AC grid, the cross- are introduced in the following section. Both of them have regional electricity transmission of other energy sources corresponding physical meanings in cross-regional power must be considered. Introducing the concept of augmenting market. path, diﬀerent feasible ﬂows from various types of energy sources can be coordinated and allocated. 2.3.1. Residual Network of Transmission Grid. For the graph, 3. Graph Theory-Based Transmission Cost G(V, E, C), let f be the feasible ﬂow in G. +e residual network intuitively refers to a network composed of edges Optimization and Allocation Method that can accommodate more ﬂows after accounting for the 3.1. Graph -eory-Based Transmission Cost Optimization feasible ﬂow f. For each edge <u, v> in G, the residual ca- Method of Cross-Regional Transaction. +e decision variable pacity c (u, v) is deﬁned as the additional ﬂow that can pass in network ﬂow optimization is the ﬂow f(u, v) on edge e(u, without exceeding the capacity constraints, c(u, v), after v). In the complex network, multiple transactions between taking into account the capacity occupied by the feasible ﬂow diﬀerent buyers and sellers can be concurrent. +e existing f(u, v). TCA method can only provide solution based on the total c (u, v) � c(u, v) − f(u, v). (1) ﬂow on edge and cannot further subdivide the ﬂow for each transaction. +erefore, the expanded graph theory-based Given a graph, G � (V, E, C), and a feasible ﬂow f, the TCA method is proposed to solve the ﬂow optimization residual network is G (V, E ), where the edge set E is f f f problem of the complex network and the concurrency of multiple transaction components . E � (u, v) ∈ V × V|c (u, v)> 0. (2) f f +e optimization variable is expanded from the original +e physical meaning of the residual network in the two-dimensional variable, f(u, v), to the four-dimensional st power grid is the transmission network composed of lines optimization variable f . +e yield spread parameter, b(u, uv st with the residual capacity. +e cross-regional renewable v), on each edge is also extended to the four-dimensional b . uv energy consumption is based on the priority for the In this way, when multiple transactions go through the same transmission demand within the region. +erefore, the edge e(u, v), they can be distinguished by the parameters (s, conﬁguration of transaction paths for renewable energy t) and decouple the multiple transactions on the same edge. sources must be built based on the residual network where +e objective function is shown below to maximize the social the transmission quota within the region has been reserved. welfare. International Transactions on Electrical Energy Systems 5 st st power ﬂow and ignores the direction of the ﬂows. (2) +e max U � f b , uv uv (5) reverse MWM approach considers the counter-ﬂows and (s,t)∈Z charges the user based on the net ﬂows of each transmission line. (3) +e zero counter-ﬂow (ZCF) MWM approach does −p , u � s, s ∈ S, ⎧ ⎪ not consider the counter-ﬂows. +e equations for three st b � p , v � t, t ∈ T, (6) uv t approaches are shown below. −l p , else, uv s ⎧ ⎪ t,e ⎪ TC (absolute MWM approach), ⎪ e st s.t. 0≤ x ≤ c , ⎪ e, max uv e∈E uv ⎪ (7) (s,t)∈Z ⎪ t,e n n TC � TC (reverse MWM approach), st st t ⎪ F f − f � 0, s ∈ TX, (8) e, max uv vu ⎪ e∈E t�1 t�1 ⎪ st st ⎪ t,e f t f t ⎪ TC , ∀f > 0(ZCF MWM approach), uv uv vu vu e t,e + � t , u, v ∈ TX, (9) F uv e, max e∈E c c uv vu (s,t)∈Z (s,t)∈Z (11) where p is the declared electricity price of node s; p is the s t where F is the counter-ﬂow of line e by buyer t, F is t, e e, max declared electricity price of node t; l is the rate of loss uv the maximum ﬂow of line e, and TC is the cost per unit MW allocation of the edge (u, v); S is the set of all sellers; T is the of line e. set of all buyers; Z is the set of all purchase and sale pairs; and TX is the set of all buyers and sellers. Assume that the node sorting starts from s to t, and the total number of nodes is n. 3.3. Network Flow Algorithm-Based TCA Method. +e “fair” Equations (7)–(9) are the channel transmission capacity TCA method is supposed to provide the proper economic constraint, the node ﬂow balance constraint, and the con- signals to the transactions that involves counter-ﬂows. straint of the forward and reverse utilization hours. According to equations (3) and (4), during the process of Equation (7) indicates that all the transaction ﬂows from augmenting path searching, the counter-ﬂows are inher- diﬀerent transaction pairs (s, t) passing through the edge e(u, ently counted in the max-ﬂow algorithm. So, the graph v) are less than the power ﬂow capacity c of the edge e(u, v). uv theory-based TCA method contains following advantages Equation (8) indicates that for all intermediate nodes, the compared to the classic MWM method. Firstly, there is no inﬂow power ﬂow of each transaction pair (s, t) is equal to slack bus in the simpliﬁed graph. Secondly, it is not nec- the outﬂow power ﬂow. Equation (9) indicates that the essary to charge or pay credit to counter-ﬂows separately. transmission line cannot transmit power in both directions +irdly, the maximum future use of the transaction path is at the same time. On the basis of the transmission capacity, provided in the residual network. Fourthly, the network the forward and reverse utilization hours of the transmission ﬂow algorithm based TCA method is still MW-based not line are restricted. +e sum of the forward and reverse energy-based. As a result, the allocation solution still de- utilization hours should be equal to the total available hours pends on the ﬂow usage not energy usage during a period of t of the transmission line. uv time. +e UHV transmission grid covers multiple provincial regions, and the parameter and length of the transmission 3.2. Usage-Based TCA Method. After the network ﬂow op- line are not available. +erefore, the DC power ﬂow-based timization, the total transmission cost needs to be allocated. MWM method cannot be applied. According to Menger’s +e allocation criterion of the MWM method is the “extend theorem , if the source node s and the sink node t are the of use” of each network facility. As stated in Section 1, the nodes of the graph G and st ∉ E(G), then the minimum size MWM method can fully recover the ﬁxed cost of the of s, t-cut is equal to the maximum number of s, t-paths that transmission network based on the actual usage of the active do not intersect each other in each pair. +e minimum cut K power ﬂow and the line length for each transmission line between the source and sink nodes means that the maximum . +e equation is stated below. number of disjoint paths is also K. Based on the above tc L MW e∈E e e t,e theorem, the basic steps of the proposed method are as TC � TC , (10) tc L MW follows: t∈T e∈E e e t,e (1) Test the connectivity of the transaction path <s, t> where tc is the cost per unit length and MW of line e; L is e e to determine the minimum path cut K(s, t) and the length of line e; MW is the active power ﬂow in line e t, e calculate the transmission costs for K diﬀerent due to buyer t; and TC is the total ﬁxed and operational cost paths. +e total number of edges in each diﬀerent involved in the transaction. path is E . As for the problem of counter-ﬂows, there are three common diﬀerent approaches . (1) +e absolute MWM (2) From k � 1, calculate the transmission costs approach charges the user based on the absolute value of the according to the following equation: 6 International Transactions on Electrical Energy Systems K k f (u, v) Lines Set, E k s,t e k TC[f(s, t)] � TC , f (u, v) ∈ 0, F , e e e, max c e=1 e=2 e=E e�1 e k�1 ... (12) c c c 1 2 k s,t s,t F k 2 ,max where f (u, v) is the feasible ﬂow of line e on path k, 1 ,max e (3) s,t (4) 2 ,AG c is the capacity of line e, and F is the max ﬂow 1 ,AG e (4) e, max 2 ,AG (1) of the transaction pair (s, t) online e. f 1 ,AG (2) 2 ,AG (3) +e iteration ends when k � K, and the transmission 1 ,AG 2 ,AG ... cost of the current transaction path and the trans- s,t 1 ,NC s,t mission cost under the maximum ﬂow condition are 2 ,NC (2) 1 ,AG (1) derived. (1) f k ,AG 2 ,AG +e augmenting procedure for lines set E is illustrated (3) k s,t 1 ,AG k ,NC in Figure 2. Assume that the max-ﬂow solution for E is k ,AG found after four iterations of augmenting path searching. (2) k ,AG +e power ﬂow under normal condition for transaction pair ... (3) s,t s,t s,t s,t k ,AG (s, t) of edge e and e is f and f . F and F 1 2 (4) 1,NC 2,NC 1, max 2, max k ,AG s,t are the max ﬂows of edge e and e . +ey are assigned the k ,max 1 2 (4) (4) -c -c -c values f , f and utilized in equation (12). +ere are 1 2 k 1,AG 2,AG two points that need to be mentioned. (1) +e max ﬂow for one edge is not necessarily the ﬂow with the largest absolute value during augmenting. Under max-ﬂow condition, the Normal Condition Initial Augmenting ﬂow on one edge can be limited by other bottleneck line(s). Under Augmenting (2) +e max ﬂow can be the counter-ﬂow which is opposite Figure 2: TCA under normal and max-ﬂow conditions. to the initial direction. 3.4. Embedded TCA Method for Invalid Capacity. Overall the 4. Application of Network Flow Algorithm in the cost of the used capacity of a transmission facility corre- Configuration of Electricity Transaction Path sponds to the power ﬂow f . Also, the future use cost cor- responds to the unused capacity (f – f ). In addition, the e, max e 4.1. Implementation of Maximum Flow Algorithm. cost of the invalid capacity (c – f ) is allocated to buyers e e, max Referring to the concepts of the connectivity and network by an embedded method (post-stamp method). In this way, a ﬂow algorithms mentioned above, a visual and quantitative market-oriented and complete allocation of the total estimation of the path reliability and maximum transmission transmission cost is accomplished. capacity can be obtained. +e key point of implementation +e post-stamp method is the most common and of the speciﬁc algorithm is to search the augmenting path simple method used by electricity utilities, where an entity eﬃciently. pays a rate equal to a ﬁxed charge per unit of energy +e Edmonds–Karp (EK) algorithm is classiﬁed as the transmitted . +e cost allocated to buyer t for invalid Ford–Fulkerson (FF) method. Its basic steps are the same as capacity, TC , is IC the FF method. +e EK algorithm uses breadth-ﬁrst search k (BFS) as the augmenting path search method. +e BFS c − f s,t e e, max ⎡ ⎣ ⎤ ⎦ TC � TC . (13) method is a basic search method, so the logic of the EK IC e e�1 k�1 algorithm is relatively simple. +is algorithm applies to most power grid analyses based on graph theory . s, t +e total transmission cost TC allocated to the +e steps of applying the Edmonds–Karp algorithm are transaction pair (s, t) is shown in equation (14) and is il- as follows: lustrated in Figure 3. +e total cost involves two main components: the valid (1) Initialize the capacity of all edges in the graph. c<u, s,t v> inherits the changed capacity. c<u, v> is initial- capacity TC[f(s, t)] and the invalid capacity TC . IC ized to zero, and the edge <v, u> is the return edge. s,t s,t TC � TC[f(s, t)] + TC . (14) IC Initialize the maximum stream to zero. (2) Start BFS for an augmenting path p from the source At each time node that requires TCA, the proposed node s to the sink node t in the residual network. method calculates and allocates these two costs to each When the point at the ﬁrst of the array is the end transaction pair in the power market. +is method can node, the augmenting path is found; then, go to step ensure the complete allocation of transmission cost and take (3); if it cannot be found, go to step (5). the capacity for future use and invalid capacity into con- sideration. +e ﬂowchart of the graph theory-based method (3) Find the “bottleneck” edge in the augmented path p. is shown in Figure 4. +e “bottleneck” is the edge with the smallest Line Flow International Transactions on Electrical Energy Systems 7 Power ﬂow (MW) Invalid capacity (IC) Capacity reserved for future use (CF) Valid capacity Capacity used for normal condition (CN) Time Horizon Investigated system snapshot (hour) Power ﬂow under CN Max-ﬂow capacity Max ﬂow for future use Transmission capacity Valid capacity Figure 3: Total TCA for CN, CF, and IC. capacity in the path, record this value X, and add it to curtailment problem, Gansu Province has limited local the maximum ﬂow; go to step (4). consumption capacity in the northwest region. It is actively participating in the inter-provincial market and medium- (4) Subtract X from c<u, v> in the augmenting path and term and long-term transactions to promote the increase of add X to all c<v, u> to form a new residual network. wind and solar power generation and facilitate the decrease Go to step (2). of wind and solar curtailment. Shanghai is considered as the (5) Get the maximum ﬂow of the network and end. electricity buyer, which accounts for more than 50% of the electricity purchase in East China . +e current trans- 4.2. Case Study of Cross-Regional Renewable Energy Con- action path from Gansu to Shanghai is Gansu-Shaanxi- Sichuan-Chongqing-Hubei-Shanghai. sumption in China. +e case study in this paper takes the current UHV transmission grid in China as the reference. +e following case study analyzes the system reliability of +e inter-provincial connection channels are transformed the transaction path and estimates the maximum trans- into the simpliﬁed graph G. Assume that a nationwide cross- mission capacity of the transaction path by network con- regional electricity transaction is organized, in which 12 nectivity analysis and maximum ﬂow algorithm. +e provinces participate as electricity sellers and 9 provinces simulation environment of the proposed model is Matlab. participate as electricity buyers . +e bidding data of provinces participating in the power Firstly, the electricity surplus condition and transaction market are given in Table 1, including the amount of buying/ prices declared by each province are collected in a certain selling electricity and bidding price. +e system grid pa- rameters such as source and sink node information of inter- period of time in advance. +e factors such as network loss and maintenance conditions on the cross-regional trans- provincial tie lines, transmission line upper and lower limits, action channel are taken into account. Secondly, according and rate of loss allocation are given in Table 2. to the network ﬂow algorithm-based transmission cost +e network connectivity simulation result is shown in optimization and allocation method, the cross-regional re- Figure 5. +e graph, G(E, V, C), is a 1-connected graph. +e newable energy transaction path plan is formed. Finally, the cut vertices, Hubei, Shaanxi, and Henan, are marked in red. dispatching department conducts the safety check on the +e graph G can be divided into 4 subgraphs by cut vertices. transaction path plan and generates the power transaction +ese four subgraphs correspond to (1) Northeast China, (2) contracts to provide the evidence for electricity settlement. East China, (3) Central China, and (4) Northwest China in the geographic environment. +e Gansu-Shanghai trans- Take Gansu Province as the electricity seller as an ex- ample. As one of the provinces with the most serious wind action path involves three subgraphs. 8 International Transactions on Electrical Energy Systems Start Variable initialization, parameter settings including Selling/buying electricity, bidding price of 23 provinces Power capacity, network loss rate of 26 transmission lines Optimization of social welfare in power market st st Capacity for e objective function: maxU = ∑ f b uv uv (s,t)∈Z normal Constraints: the channel transmission capacity, the node ﬂow condition balance, the forward and reverse utilization hours Determine the size of the minimum cut: K Set k=1 Capacity f (u,v) Calculate the transmission costs of path k: reserved TC e=1 e for future use k=k+1 NO k>K YES f (u,v) = ∑ Output transmission cost of valid capacity: TC[f (s,t)] ∑ TC k=1 e=1 Determine the invalid capacity: C – f e e,max Invalid capacity K E c –f e e,max s,t TC = TC Calculate the transmission costs of invalid capacity: ∑ ∑ IC e k=1 e=1 s,t s,t Output total transmission cost of transaction pair (s, t): TC = TC[f(s,t)]+TC IC End Figure 4: Flowchart of the proposed transaction path conﬁguration method. Similar to the above example, the network connectivity CNY/MWh. From topological structure view, Shaanxi is the analysis is applied to all 17 transactions in the market. If critical vertex for electricity sellers, such as Gansu, Qinghai, there is more than one path between a pair of seller and and Ningxia. However, there is no clear indication in the buyer, all possible transaction paths are included in the traditional transaction data, like trading volume and bidding connectivity analysis. +e number of cut vertex and detailed price. +e network connectivity analysis is a necessary vertex name is shown in Table 3. supplement to the transmission pricing strategy. +e cut vertex info is summarized in Table 4 to indicate +e maximum transmission capacity of the transaction path is estimated by the EK algorithm. +e maximum ca- the importance of the speciﬁc vertex in the market. Hubei is the cut vertex in eight cross-regional transactions. +e pacity takes the transmission line loss, transmission line trading volume and bidding price are 10224.6 GWh and 354 capacity, and the routine maintenance into consideration, CNY/MWh. Shaanxi is cut vertex in 6 transactions, but the and the transaction period is set as 30 days. Four cases of trading volume and bidding price are 34.5 GWh and 297 max-ﬂow problem by EK algorithm are shown in Figure 6. International Transactions on Electrical Energy Systems 9 Table 1: Bidding data of provinces participating in the power market in China. No. Node name Selling electricity (GWh) Buying electricity (GWh) Bidding price (CNY/MWh) 1 Jingjintang 0 857.6 382 2 Hebei 0 0 395 3 Shanxi 1931 0 315 4 Shandong 2790 0 397 5 Shanghai 0 5989 462 6 Jiangsu 0 4900 436 7 Zhejiang 0 3926.1 458 8 Anhui 1000 0 398 9 Fujian 127.3 0 422 10 Hubei 10224.6 0 354 11 Hunan 0 53 371 12 Henan 0 301.8 358 13 Jiangxi 0 585 391 14 Sichuan 255 0 288 15 Chongqing 227 0 291 16 Heilongjiang 1435 0 400 17 Jilin 1431.41 0 376 18 Liaoning 0 321.2 380 19 Shaanxi 0 34.5 297 20 Gansu 220.2 0 277 21 Qinghai 418.6 0 279 22 Ningxia 117.6 0 268 23 Xinjiang 0 0 250 Total 20177.71 16968.2 Table 2: Power capacity and network loss rate of transmission lines. No. Direction Positive direction capacity (MW) Negative direction capacity (MW) Rate of loss allocation 1 Hebei-Jingjintang 4000 4000 0.01 2 Shandong-Hebei 3800 3800 0.01 3 Shanxi-Hebei 7200 7200 0.01 4 Shanxi-Jingjintang 2460 2460 0.01 5 Anhui-Jiangsu 3500 3500 0.01 6 Fujian-Zhejiang 1800 1800 0.01 7 Zhejiang-Shanghai 2600 2600 0.01 8 Jiangsu-Shanghai 3400 3400 0.02 9 Jiangsu-Zhejiang 4000 4000 0.01 10 Sichuan-Chongqing 2200 2200 0.02 11 Hubei-Jiangxi 1600 1600 0.01 12 Hubei-Hunan 2600 1100 0.02 13 Hubei-Chongqing 3000 2000 0.01 14 Hubei-Henan 3000 3000 0.01 15 Jilin-Heilongjiang 2400 2400 0.01 16 Liaoning-Jilin 1400 1400 0.02 17 Gansu-Ningxia 3800 4100 0.01 18 Gansu-Qinghai 2400 2400 0.01 19 Shaanxi-Gansu 2600 2000 0.02 20 Gansu-Xinjiang 1300 1300 0.01 21 Hubei-Jiangsu 3600 3600 0.01 22 Hubei-Shanghai 3600 3600 0.01 23 Shanxi-Henan 1900 1500 0.02 24 Sichuan-Shaanxi 1160 1360 0.01 25 Shaanxi-Henan 1000 1000 0.02 26 Liaoning-Jingjintang 1500 1500 0.02 +e unit is GWh. +e feasible paths and “bottleneck” line are (Gansu-Shaanxi, Sichuan, Chongqing, Hubei, and Shang- highlighted in green and red. hai), the augmenting path (Gansu, Shaanxi, Henan, Hubei, +e current trading volume of Gansu-Shanghai path is and Shanghai) is added in max-ﬂow algorithm. Without 220.2 GWh. In addition to the original transaction path considering the other transaction pair, the “bottleneck” lines 1 1.3 3.6 3.6 3.5 1.8 1.16 1.6 1.4 7.2 1.5 Jilin 3.8 2.4 Xinjiang Heliongjiang Shandong Hebei Shanxi Jingjintang Henan Liaoning Hunan Jiangxi Chongqing Shaanxi Sichuan Jiangsu Gansu Qinghai Shanghai Anhui Ningxia Fujian Zhejiang 2.4 Hubei 2.46 10 International Transactions on Electrical Energy Systems Henan Hubei Shaanxi Unit: GW Figure 5: Simpliﬁed network connectivity simulation. Table 3: Cut vertices of the transaction path(s). No. Vertices of the transaction path(s) # of cut vertex Detailed cut vertex 1 Anhui-Jiangsu-Shanghai 1 Jiangsu 2 Anhui-Jiangsu-Zhejiang 1 Jiangsu 3 Fujian-Zhejiang 0 N/A 4 Gansu-Shaanxi-Henan 1 Shaanxi Gansu-Shaanxi-Sichuan-Chongqing-Hubei-Shanghai, Gansu-Shaanxi-Henan- 5 3 Shaanxi, Henan, Hubei Hubei-Shanghai 6 Hubei-Shanghai, Hubei-Jiangsu-Shanghai 1 Jiangsu 7 Hubei-Jiangsu-Zhejiang 1 Jiangsu 8 Ningxia-Gansu-Shaanxi-Henan 2 Gansu, Shaanxi Gansu, Shaanxi, Henan, 9 Ningxia-Gansu-Shaanxi-Henan-Hubei-Shanghai 4 Hubei Gansu, Shaanxi, Henan, 10 Qinghai-Gansu-Shaanxi-Henan-Hubei-Hunan 4 Hubei Gansu, Shaanxi, Henan, 11 Qinghai-Gansu-Shaanxi-Henan-Hubei-Jiangxi 4 Hubei 12 Shanxi-Jingjintang 0 N/A 13 Shanxi-Jingjintang-Liaoning 1 Jingjintang 14 Shanxi-Henan-Hubei-Jiangsu-Zhejiang 3 Henan, Jiangsu, Hubei 15 Sichuan-Chongqing-Hubei-Shanghai 1 Hubei 16 Chongqing-Hubei-Jiangxi 1 Hubei 17 Chongqing-Hubei-Shanghai 1 Hubei Table 4: Summary of cut vertices. Cut vertex # of transactions involved Trading volume (GWh) Bidding price (CNY/MWh) 1 Hubei 8 10224.6 354 2 Shaanxi 6 34.5 297 3 Jiangsu 5 4900 436 4 Henan 5 301.8 358 5 Gansu 4 220.2 277 6 Jingjintang 1 857.6 382 refer to Gansu-Shaanxi whose capacity is 1411.2 GWh and 72.06%. +e simulation result shows that there is still a huge Shaanxi-Henan whose capacity is 705.6 GWh. +e maxi- transmission margin, and the potential for cross-regional mum transmission estimation is 1411.2 GWh. +e maxi- consumption of renewable energy is promising. +e detailed mum utilization rate of non-bottleneck lines does not exceed “bottleneck” line and max ﬂow are summarized in Table 5. 2.2 2.6 2.6 3.8 1.9 3.4 743.8 2528.8 743.8 743.7 1338.8 743.7 1487.5 2528.8 Qinghai Ningxia Gansu Shaanxi Shanxi Sichuan Qinghai Ningxia Henan Gansu Chongqing Hubai Jingjintang Shaanxi Shanxi Anhui iaoning Sichuan Hunan Jiangsu Jiangxi Henan Fujian Chongqing Shanghai Hubai Jingjintang Qinghai Anhui Liaoning Ningxia Hunan Gansu Jiangsu Jiangxi Fujian Shanghai Shaanxi Shanxi Qinghai Sichuan Ningxia Henan Gansu Chongqing Hubai Jingjintang Shaanxi Shanxi Anhui Liaoning Sichuan Hunan Jiangsu Henan Jiangxi Fujian Chongqing Shanghai Hubai Anhui Hunan Jiangsu Jiangxi Fujian Shanghai Zhejiang Zhejiang Zhejiang Zhejiang International Transactions on Electrical Energy Systems 11 Anhui-Shanghai Qinghai-Hunan Fujian-Zhejiang Gansu-Shanghai Figure 6: Edmonds–Karp max-ﬂow algorithm for cross-regional transaction paths. Table 5: “Bottleneck” transmission line(s), max ﬂow, and invalid capacity of transactions. Max ﬂow IC No. Vertices of the transaction path(s) Bottleneck path(s) (GWh) (MW) 1 Anhui-Jiangsu-Shanghai Jiangsu-Shanghai 2528.8 4535.5 2 Anhui-Jiangsu-Zhejiang Anhui-Jiangsu 2603.2 2111.1 3 Fujian-Zhejiang N/A 1338.8 1623.2 4 Gansu-Shaanxi-Henan Shaanxi-Henan 743.8 949.4 Gansu-Shaanxi-Sichuan-Chongqing-Hubei-Shanghai, Gansu-Shaanxi-Henan- Gansu-Shaanxi Shaanxi- 5 1487.5 6775.5 Hubei-Shanghai Henan 6 Hubei-Shanghai, Hubei-Jiangsu-Shanghai Jiangsu-Shanghai 5206.3 2424.4 7 Hubei-Jiangsu-Zhejiang Hubei-Jiangsu 2677.5 0 8 Ningxia-Gansu-Shaanxi-Henan Shaanxi-Henan 743.8 2768 Ningxia-Gansu-Shaanxi-Henan-Hubei-Shanghai, Ningxia-Gansu-Shaanxi- Gansu-Shaanxi Shaanxi- 9 1487.5 10712.2 Sichuan-Chongqing-Hubei-Shanghai Henan Qinghai-Gansu-Shaanxi-Henan-Hubei-Hunan, Qinghai-Gansu-Shaanxi- Hubei-Hunan Shaanxi- 10 1487.5 11120.5 Sichuan-Chongqing-Hubei-Hunan Henan Qinghai-Gansu-Shaanxi-Henan-Hubei-Jiangxi, Qinghai-Gansu-Shaanxi- Shaanxi-Henan Hubei- 11 1190 9381.6 Sichuan-Chongqing-Hubei-Jiangxi Jiangxi 12 Shanxi-Jingjintang N/A 1830.2 822.8 13 Shanxi-Jingjintang-Liaoning Jingjintang-Liaoning 1115.6 1876.7 14 Shanxi-Henan-Hubei-Jiangsu-Zhejiang Shanxi-Henan 1413.1 2196.7 15 Sichuan-Chongqing-Hubei-Shanghai Chongqing-Hubei 1487.5 3141.9 16 Chongqing-Hubei-Jiangxi Hubei-Jiangxi 1190 2100.8 17 Chongqing-Hubei-Shanghai Chongqing-Hubei 1487.5 1313.3 In the section from Shaanxi to Hubei, there are two paths fails, the complete transaction path can still be in a non-intersecting paths: Shaanxi-Sichuan-Chongqing- working state. +e invalid capacity of the transmission line Hubei and Shaanxi-Henan-Hubei. +en, the minimum cut is after considering the transmission loss rate and the between Shaanxi and Hubei is equal to 2. When one of the periodic maintenance. 743.8 743.8 743.7 743.7 1487.5 743.7 1487.5 1487.5 743.7 1033 305.83 17.22 12 International Transactions on Electrical Energy Systems Gansu Gansu Gansu 1 1 305.83 305.83 Shaanxi Shaanxi Shaanxi 2 2 288.61 Sichuan Sichuan Sichuan Unit:MW Chongqing Chongqing Henan Chongqing Henan 4 4 Hubei Hubei Hubei Shanghai Shanghai Shanghai Original Optimized Max ﬂow Figure 7: Graph theory-based TCA of Gansu-Shanghai transaction path. After the transaction path analysis, the TCA of Gansu- Equations (5)–(9) are utilized in the optimal TCA Shanghai transaction under normal, optimized, and max- method. +e optimized line ﬂow with the social welfare as ﬂow conditions is shown in Figure 7. +e MWM method is the objective is used as the edge ﬂow for the section from utilized in normal condition. Shaanxi to Hubei. f (u, v) TC[f(Gansu, Shanghai)] � TC � 22342.80 CNY. e�1 (15) Gansu−Shaanxi √√√√√ √√√√√√ √ f (u, v) TC [f(Gansu, Shanghai)] � TC opt e e�1 (16) Shaanxi−Sichuan−Chongqing−Hubei and Shaanxi−Henan−Hubei Hubei−Shanghai √√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√ √√√√√ √√√√√√ √ 4 7 5 f (u, v) f (u, v) f (u, v) e,opt e,opt ⎡ ⎣ ⎤ ⎦ + TC + TC + TC � 19296.60 CNY. e e e c c c e e e e�2 e�6 e�5 +e transaction path conﬁguration method in max-ﬂow the edge ﬂow from Qinghai to Hunan to allocate the future condition uses the line capacity of the residual network as use cost. f (u, v) e,res TC [f(Gansu, Shanghai)] � TC res e e�1 (17) 4 7 5 f (u, v) f (u, v) f (u, v) e,res e,res e,res ⎣ ⎦ ⎡ ⎤ + TC + TC + TC � 120739.65 CNY. e e e c c c e e e e�2 e�6 e�5 By the traditional route conﬁguration method, the ﬂow costs. +e transmission cost for 305.83 MW is 22342.80 on each tie line in a single transaction route is the same. CNY. +e unit cost of transmission is 0.073 CNY/kWh. +e Although the MWM algorithm is simple and straightfor- transmission cost for the same amount after social-welfare ward, the transmission capacity of the remaining lines has optimization is 19296.60 CNY. +e unit cost of transmission not been eﬀectively used, resulting in high total transmission is 0.0631 CNY/kWh. It is clear that the transmission cost can 288.61 17.22 305.83 305.83 305.83 305.83 17.22 1033 752.2 1926.2 297.6 752.2 752.2 752.2 365.6 23.6 207.8 12.4 365.6 365.6 23.6 207.8 23.6 702.4 321.2 International Transactions on Electrical Energy Systems 13 Heilongjiang Xinjiang Ningxia Jingjintang Jilin Gansu Shanxi Hebei Liaoning Shandong Shaanxi Qinghai Henan Sichuan Jiangsu 7.6 219.4 12.4 Shanghai Chongqing Hubei Anhui Zhejiang Hunan Jiangxi Fujian 127.3 Unit:GWh 12.4 2379.8 23.6 7.6 Figure 8: Graph theory-based transmission cost optimization. be reduced by 13.63% by the optimization method. In max- transaction path under the practical network. +e network ﬂow condition, without considering the other transaction ﬂow algorithm-based TCA method can provide theoretical pair, the transmission cost for 2000-MW max ﬂow between support for the cross-regional consumption of renewable Gansu and Shanghai is 120739.65 CNY. +e unit cost of energy through the inter-provincial transmission grid. transmission is 0.0688 CNY/kWh, which is still lower than the unit cost of transmission using traditional path con- Abbreviations ﬁguration method. TCA: Transmission cost allocation After the optimization of social welfare, the optimal UHV: Ultra-high voltage results are presented in Figure 8. It provides the optimal ﬂow SGCC: State Grid Corporation of China for each established transactions in the market. +e cost of AC: Alternating current IC is allocated to all the transaction pair by the stamp DC: Direct current method. +e detailed results of IC are given in Table 5. Take LMP: Local marginal price the Gansu-Shanghai transaction path as an example; the DSRs: Demand-side resources total transmission cost is 0.0931 CNY/kWh by equation (14) MWM: MW-Mile including both valid and invalid capacity usage. DERs: Distributed energy resources In summary, the computational eﬃciency of the network OPF: Optimal power ﬂow ﬂow algorithm based TCA method is comparatively high. So, EK: Edmonds–Karp it can accommodate the dynamic changes in the power FF: Ford–Fulkerson system. Besides that, it can eﬀectively identify the critical BFS: Breadth-ﬁrst search node(s) and bottleneck line(s), maximize the social welfare, G: Graph track the decoupled ﬂow, and provide transmission cost for V(G): Vertex set future use. E(G): Edge set f(u, v): +e current ﬂow from node u to v 5. Conclusion c(u, v): +e edge capacity from node u to v In this paper, the network ﬂow algorithm of graph theory is C(G): Line capacity set introduced into the problem of inter-provincial transaction G(V, E, +e directed simple graph path conﬁguration. From the perspective of topological C): structure, the problem of transmission cost optimization and TP(s, t): +e transaction path set allocation under complex network conditions is studied. k(G): +e connectivity of G +e simulation result reveals that the simpliﬁed graph- s: +e source node (seller) based algorithm can evaluate the reliability of the transaction t: +e sink node (buyer) path and estimate the maximum transmission capacity of the c (u, v): +e residual capacity 752.2 1926.2 702.4 297.6 365.6 857.6 321.2 23.6 12.4 12.4 219.4 365.6 14 International Transactions on Electrical Energy Systems Shapley value,” IEEE Transactions on Power Systems, vol. 32, G (V, E ): +e residual network f f no. 2, pp. 1369–1377, 2017. c (p): +e capacity of the augmenting path  Y. P. Molina, O. R. Saavedra, and H. 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