# Cross-Regional Transaction Path Configuration of Renewable Energy Resources by Graph Theory-Based Transmission Cost Allocation Method

Authors: Yin Yao; Dahuan Lu; Wenzhong Gao; Bo Zhou; Dongdong Li
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

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. The 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.


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## Body

## 1. Introduction

The imbalance of energy source distribution and the differences in regional economic development have caused a reverse distribution of the energy supply and demand in China. Taking the renewable energy consumption problem in Gansu province as an example, the renewable energy generation from wind and solar farms is abundant, but the local electricity consumption capacity in the northwest region is limited due to the comparatively low level of economic development. Consequently, the excessive electricity from renewable energy breaks the balance between generation and consumption and results in the large-scale electricity abandonment. Therefore, it is urgent to optimize the distribution of resources nationwide with the support of power trading policies. The objective of power market is to transmit the excess electricity to the desired region, such as the southeast region [1]. Also, the power market must be based on an effective and efficient configuration of the cross-regional transaction paths.The cross-regional electricity transaction requires the ultra-high-voltage (UHV) power grid to minimize the line loss due to the long-distance transmission. According to the development scheme of the State Grid Corporation of China (SGCC), UHV is defined as the transmission technology with an AC voltage level of 1000 kV and above and DC voltage level ±800 kV and above [2].At present, the network structure of the UHV grid is relatively simple. The SGCC generally organizes and completes the cross-regional electricity transactions annually, quarterly, and monthly. However, with the gradual increase of AC and DC UHV transmission lines, the complexity of the UHV network will also increase correspondingly. The original bilateral relationship between the seller and buyer will no longer be adaptive to the market strategy. The optimal configuration of the cross-regional transaction path will become an increasingly prominent problem.In response to the above problem, the majority of current research in this area utilizes power economy theory for modeling, simulation, and optimization. For instance, six transaction models, four model architectures, and corresponding risk control strategies for cross-regional transaction path configuration were proposed in [1]. To set up the test platform, a management and control index system was developed in [3] for cross-regional transactions based on supply chain, group control, and risk management. For the test platform, hierarchical optimization of the tie line planning was carried out to achieve automatic planning and flexible scheduling [4]. To consider the cross-regional transmission line loss, a compensation method was developed based on the route method and the average network loss allocation method [5].However, there is relatively few research of TCA of the cross-regional transaction. The ideal allocation method meets the following requirements. (1) It must contain sufficient economic information, which can effectively guide the economic operation of the power grid to make full use of the existing transmission grid resources. (2) The transmission grid companies require that their annual revenue and expenditures are balanced through transmission fees to ensure the normal operations and the long-term development of the power grid. (3) The method is also required to be simple and easy for implementation, and the results can be verified to meet the fair and open principle of the power market.There are mainly two types of allocation methods. The first type is based on the cost allocation. The other type is based on the amount of usage allocation. The cost-based allocation methods can be further divided into the embedded cost methods and the local marginal price (LMP) methods based on microeconomic theory.(i)
The embedded cost method is essentially the accounting of the transmission cost. This method focuses on offsetting the actual expenditures of the grid operation and investment costs. On the basis of the most common post-stamp method, the distribution of the fixed and operation cost is defined as an infinite-person cooperative game. A cooperative game approach which provides stable solution integrated with appropriate penalties or rewards to participants was presented in [6]. The Aumann–Shapley value was utilized for TCA [7], the distribution loss allocation [8], and the profit allocation for demand-side resource (DSR) aggregators [9]. The proposed game theoretic method ensures the equitable allocation and recovery of the total cost. However, the price signal does not contain any economic information, so it cannot guide the optimal use of the power grid resources and the long-term development of the power grid.(ii)
The LMP method aims at maximizing economic benefits and effectively guiding the economic operation of the existing power grid. A responsibility-based approach was proposed in [10] to allocate the cost of the transmission congestion and losses to the nodes of the network. In [11], due to the generic complexity of the cooperative game theoretic problems based on marginal pricing, the min-max fairness policy was utilized to solve this NP-hard problem in polynomial time. With the increase of the renewable energy penetration, the hybrid AD/DC transmission structure was studied in [12]. The LMP-based nodal pricing method was proposed to provide efficient and accurate solution. In [13], the transmission-network expansion problem and the energy source distraction problem were defined as a trilevel optimization problem based on LMP.In summary, the long-term marginal cost method requires the usage of some highly uncertain assumptions, and the calculation is complicated. The short-term marginal cost method cannot guarantee the balance of revenue and expenditure. If the network construction investments cannot be recovered, the allocation method is not acceptable by the power company.(2)
For the allocation method based on the amount of usage, the proportion of the total cost is determined according to the usage of the power grid equipment. The actual operation of the system is taken into account. The classic MW-Mile (MWM) method allocates the transmission cost based on the usage of the line capacity and line length. Xiao et al. [14] proposed a power tracing-based equivalent bilateral exchange method in which network users are responsible for not only their induced power flows but also power flows induced by whom they have equivalent bilateral exchanges with. In [15], a new efficient method for solving the reactive power tracing problem was proposed in a transmission system. In [16], the line capacity was replaced by the maximum line loading for N − 1 security to achieve a more fair fixed cost allocation in a pool based power market. In [17], the transmission expansion model was formulated as a multiobjective optimization problem to facilitate the distributed generation and defer the transmission investment. Considering the quality of the load, in [18], the power factor was introduced in the MWM method. In [19], the transmission capacity was divided into normal condition capacity, capacity for contingency, capacity for future use, and invalid capacity. The structural TCA scheme can encourage the efficient use of the transmission network. The same structural method is applied to the optimal planning strategy for the distributed energy resources (DERs). In [20], a circuit theory-based TCA method was developed considering the orthogonal projection. The Aumann–Shapley value is used to distribute the interaction term between the involved components.There are various power flow tracing methods that can be used for the transmission embedded cost allocation. The power flow tracing based on proportional sharing and circuit theory requires line impedance for power flow calculation [21–23]. The power flow tracing based on optimization [24] modifies the maximum power output of the generator which contributes to line congestion. But in power market, the actual power output of the generator is determined during the market clearing stage. As for the power flow tracing based on the relative electrical distance concept [25], this method also requires the line impedance and power flow to decide the relative electrical distance. In addition, the generation dispatch makes a great impact on the power flow solutions, and the residual potential of the network is ignored. Therefore, the MWM method requires complete grid parameter information and cannot reflect the actual usage of the power grid by the cross-regional transaction.Besides the reference to the power economy theory, current research also studies the cross-regional power transmission from the topological structure aspect based on the graph theory. In [26], a transaction tracing-based loss allocation scheme for assigning the network losses incurred due to the transactions occurring between peers in a dynamic environment was presented. In [27], a network flow approach was developed for the estimation of the cross-regional energy trade volume and the partition start-up capacity. The simulation result is applied to the optimization of the local start-up capacity configuration. In [28], an optimization algorithm was proposed for the renewable energy cross-regional transactions. The objective is to maximize the social benefit and the total trade volume. The algorithm solves the optimization problem by the fix-path method, the point-arc model, and the arc-path model. The network flow algorithm was improved in [29] to achieve the tracking of each cross-regional transaction path. The graph theory is also applied to the system reliability study. Zhu et al. [30] proposed the impact analysis of the key nodes removal on the vulnerability of the UHV grid.


### 1.1. Contributions and Organization

In 2060, the proportion of renewable energy power generation will reach more than 70% according to China’s “30·60” decarbonization goal. To facilitate the consumption of renewable power, the cross-regional power market is in the rapid growth stage. In 2020, the amount of inter-provincial transactions was 1157.7 billion kWh which increased by 9.5% compared to previous year in China [31]. Therefore, the complete allocation of the total cost is a critical problem. Compared to the LMP-based TCA method, the graph theoretic TCA method is modified based on the MWM TCA method. It is more suitable for the emerging market which contains limited grid parameter information and requires high market clearing efficiency. The proposed method provides the following contributions.(1)
The traditional optimal power flow (OPF) method requires complete grid parameters and must consider the selection of slack bus and the counter-flow. The cross-regional power market in China contains 23 provinces. The grid parameters involved in the transaction are limited. Modeling of UHV and DC hybrid grid for the entire system is almost impossible. The graph theoretic method follows the capacity constraint and equilibrium constraint. The selection of slack bus is not necessary, and the counter-flow is considered in the augmenting stage. So, the graph theory-based network flow method is more suitable for the long-term cross-regional transaction path configuration with limited grid information.(2)
The graph theory-based TCA method can provide the maximum flow capacity and invalid capacity of the cross-regional transaction path under complex networks. Besides that, the flow between each transaction pair is decoupled. The transmission cost is allocated by three sections: capacity for normal use, capacity for future use, and invalid capacity. The complete allocation of transmission cost is guaranteed.In this paper, the network flow algorithm and the maximum flow algorithm are introduced in Section2 for connectivity analysis and maximum capacity estimation. In Section 3, the above algorithms are applied to the simplified graph of the current UHV transmission grid in China. The graph theory-based TCA method is developed. In Section 4, through the case study of 17 transactions in China (especially the Gansu-Shanghai transaction path that involves the renewable energy), the comparative analysis between the original, optimized, and max-flow scenarios is conducted. Section 5 summarizes the advantages of the proposed method and discusses the remaining work for future study.


## 1.1. Contributions and Organization

In 2060, the proportion of renewable energy power generation will reach more than 70% according to China’s “30·60” decarbonization goal. To facilitate the consumption of renewable power, the cross-regional power market is in the rapid growth stage. In 2020, the amount of inter-provincial transactions was 1157.7 billion kWh which increased by 9.5% compared to previous year in China [31]. Therefore, the complete allocation of the total cost is a critical problem. Compared to the LMP-based TCA method, the graph theoretic TCA method is modified based on the MWM TCA method. It is more suitable for the emerging market which contains limited grid parameter information and requires high market clearing efficiency. The proposed method provides the following contributions.(1)
The traditional optimal power flow (OPF) method requires complete grid parameters and must consider the selection of slack bus and the counter-flow. The cross-regional power market in China contains 23 provinces. The grid parameters involved in the transaction are limited. Modeling of UHV and DC hybrid grid for the entire system is almost impossible. The graph theoretic method follows the capacity constraint and equilibrium constraint. The selection of slack bus is not necessary, and the counter-flow is considered in the augmenting stage. So, the graph theory-based network flow method is more suitable for the long-term cross-regional transaction path configuration with limited grid information.(2)
The graph theory-based TCA method can provide the maximum flow capacity and invalid capacity of the cross-regional transaction path under complex networks. Besides that, the flow between each transaction pair is decoupled. The transmission cost is allocated by three sections: capacity for normal use, capacity for future use, and invalid capacity. The complete allocation of transmission cost is guaranteed.In this paper, the network flow algorithm and the maximum flow algorithm are introduced in Section2 for connectivity analysis and maximum capacity estimation. In Section 3, the above algorithms are applied to the simplified graph of the current UHV transmission grid in China. The graph theory-based TCA method is developed. In Section 4, through the case study of 17 transactions in China (especially the Gansu-Shanghai transaction path that involves the renewable energy), the comparative analysis between the original, optimized, and max-flow scenarios is conducted. Section 5 summarizes the advantages of the proposed method and discusses the remaining work for future study.


## 2. Graph Theory Network Flow Algorithm

### 2.1. Graph Definitions and Terms

When the connectivity and mutual relations of the network are involved in the engineering mathematical problems, a graph can be defined for intuitive and visual analysis to solve the problem.A graphG consists of a vertex set V(G) and an edge set E(G). Each edge associates with two vertices (not necessarily different vertices). Each node in the power transmission grid can be considered as a vertex in the graph G. Each transmission line can be considered as an edge in the graph G. The direction of the line current from nodes u to v is described as the flow f(u, v). The rated power of the transmission line is described as the edge capacity c(u, v), as shown in Figure 1. Assume that the power transmission grid contains neither parallel edges nor loops. The power transmission grid can be described by a node vertex set V(G), a transmission line edge set E(G), and a line capacity set C(G). In summary, the graph of the power transmission grid is a directed simple graph G(V, E, C).Figure 1
Simplified graph model of the transmission line.


### 2.2. Connectivity of Transaction Paths

The energy seller in the market, such as the wind farm in the northwest region of China, is described as the source nodes in the graph. The energy buyer, such as the big power consumer in the east region of China, is described as the sink point t in the graph.In the directed simple graphG, all different possible transaction paths between the source node s and the sink node t are described as the transaction path set TP(s, t). From system reliability aspect, the transaction path set is expected to be a non-empty set even if some nodes or edges need to be removed from the graph due to facility failures or maintenance.If the vertex subset,S∈VG, makes the graph G-S have more than one branch which means G-S is disconnected or has only one vertex, then S is called the separating set or vertex cut of G. The minimum size of S is called the connectivity of G and denoted as k(G). If the connectivity of G is at least k, then G is k-connected.


### 2.3. The Maximum Flow of Transaction Path

To evaluate the maximum transmission capacity between the seller and buyer, the maximum flow between the source node (seller)s and the sink node (buyer) t in the directed and weighted graph needs to be solved. Moreover, the max-flow solution can provide the visual tracking of the possible “bottleneck” edge in the transaction path. The saturated “bottleneck” edge causes the rest of the edges on the path to no longer able to accommodate any positive flow increase.The concepts of residual network and augmenting path are introduced in the following section. Both of them have corresponding physical meanings in cross-regional power market.


#### 2.3.1. Residual Network of Transmission Grid

For the graph,G(V, E, C), let f be the feasible flow in G. The residual network intuitively refers to a network composed of edges that can accommodate more flows after accounting for the feasible flow f. For each edge <u, v> in G, the residual capacity cf(u, v) is defined as the additional flow that can pass without exceeding the capacity constraints, c(u, v), after taking into account the capacity occupied by the feasible flow f(u, v).(1)cfu,v=cu,v−fu,v.Given a graph,G = (V, E, C), and a feasible flow f, the residual network is Gf(V, Ef), where the edge set Ef is(2)Ef=u,v∈V×V|cfu,v>0.The physical meaning of the residual network in the power grid is the transmission network composed of lines with the residual capacity. The cross-regional renewable energy consumption is based on the priority for the transmission demand within the region. Therefore, the configuration of transaction paths for renewable energy sources must be built based on the residual network where the transmission quota within the region has been reserved. The max-flow algorithm searches and utilizes the residual power transmission capacity between source and sink nodes.


#### 2.3.2. Augmenting Path of Residual Network

In the residual network, the max-flow algorithm searches for the feasible flows and forms a new residual network. This process constitutes an iterative loop until no new feasible flow can be found. The new feasible flow is defined as the augmenting pathp which is a path from the source node s to the sink node t in the residual network Gf. The capacity of the augmenting path, cf(p), is defined in equation (3). cf(p) is the maximum additional flow that can be added along the path.(3)cfp=mincfu,v|<u,v>∈p,cfp>0.The addition of an augmenting path results in a flow with a larger value. The augmenting path is the increment of the flow and has the property of augmentation. The flow of augmenting pathfp is defined in the following equation:(4)fpu,v=cfp,<u,v>∈p,−cfp,<u,v>∈p,0,else.The physical meaning of the augmenting path in the power grid is the incremental trading of the transaction between seller and buyer. The cross-regional consumption of renewable energy can utilize the multiterminal DC grid and the traditional AC grid. In the traditional AC grid, the cross-regional electricity transmission of other energy sources must be considered. Introducing the concept of augmenting path, different feasible flows from various types of energy sources can be coordinated and allocated.


## 2.1. Graph Definitions and Terms

When the connectivity and mutual relations of the network are involved in the engineering mathematical problems, a graph can be defined for intuitive and visual analysis to solve the problem.A graphG consists of a vertex set V(G) and an edge set E(G). Each edge associates with two vertices (not necessarily different vertices). Each node in the power transmission grid can be considered as a vertex in the graph G. Each transmission line can be considered as an edge in the graph G. The direction of the line current from nodes u to v is described as the flow f(u, v). The rated power of the transmission line is described as the edge capacity c(u, v), as shown in Figure 1. Assume that the power transmission grid contains neither parallel edges nor loops. The power transmission grid can be described by a node vertex set V(G), a transmission line edge set E(G), and a line capacity set C(G). In summary, the graph of the power transmission grid is a directed simple graph G(V, E, C).Figure 1
Simplified graph model of the transmission line.


## 2.2. Connectivity of Transaction Paths

The energy seller in the market, such as the wind farm in the northwest region of China, is described as the source nodes in the graph. The energy buyer, such as the big power consumer in the east region of China, is described as the sink point t in the graph.In the directed simple graphG, all different possible transaction paths between the source node s and the sink node t are described as the transaction path set TP(s, t). From system reliability aspect, the transaction path set is expected to be a non-empty set even if some nodes or edges need to be removed from the graph due to facility failures or maintenance.If the vertex subset,S∈VG, makes the graph G-S have more than one branch which means G-S is disconnected or has only one vertex, then S is called the separating set or vertex cut of G. The minimum size of S is called the connectivity of G and denoted as k(G). If the connectivity of G is at least k, then G is k-connected.


## 2.3. The Maximum Flow of Transaction Path

To evaluate the maximum transmission capacity between the seller and buyer, the maximum flow between the source node (seller)s and the sink node (buyer) t in the directed and weighted graph needs to be solved. Moreover, the max-flow solution can provide the visual tracking of the possible “bottleneck” edge in the transaction path. The saturated “bottleneck” edge causes the rest of the edges on the path to no longer able to accommodate any positive flow increase.The concepts of residual network and augmenting path are introduced in the following section. Both of them have corresponding physical meanings in cross-regional power market.


### 2.3.1. Residual Network of Transmission Grid

For the graph,G(V, E, C), let f be the feasible flow in G. The residual network intuitively refers to a network composed of edges that can accommodate more flows after accounting for the feasible flow f. For each edge <u, v> in G, the residual capacity cf(u, v) is defined as the additional flow that can pass without exceeding the capacity constraints, c(u, v), after taking into account the capacity occupied by the feasible flow f(u, v).(1)cfu,v=cu,v−fu,v.Given a graph,G = (V, E, C), and a feasible flow f, the residual network is Gf(V, Ef), where the edge set Ef is(2)Ef=u,v∈V×V|cfu,v>0.The physical meaning of the residual network in the power grid is the transmission network composed of lines with the residual capacity. The cross-regional renewable energy consumption is based on the priority for the transmission demand within the region. Therefore, the configuration of transaction paths for renewable energy sources must be built based on the residual network where the transmission quota within the region has been reserved. The max-flow algorithm searches and utilizes the residual power transmission capacity between source and sink nodes.


### 2.3.2. Augmenting Path of Residual Network

In the residual network, the max-flow algorithm searches for the feasible flows and forms a new residual network. This process constitutes an iterative loop until no new feasible flow can be found. The new feasible flow is defined as the augmenting pathp which is a path from the source node s to the sink node t in the residual network Gf. The capacity of the augmenting path, cf(p), is defined in equation (3). cf(p) is the maximum additional flow that can be added along the path.(3)cfp=mincfu,v|<u,v>∈p,cfp>0.The addition of an augmenting path results in a flow with a larger value. The augmenting path is the increment of the flow and has the property of augmentation. The flow of augmenting pathfp is defined in the following equation:(4)fpu,v=cfp,<u,v>∈p,−cfp,<u,v>∈p,0,else.The physical meaning of the augmenting path in the power grid is the incremental trading of the transaction between seller and buyer. The cross-regional consumption of renewable energy can utilize the multiterminal DC grid and the traditional AC grid. In the traditional AC grid, the cross-regional electricity transmission of other energy sources must be considered. Introducing the concept of augmenting 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,G(V, E, C), let f be the feasible flow in G. The residual network intuitively refers to a network composed of edges that can accommodate more flows after accounting for the feasible flow f. For each edge <u, v> in G, the residual capacity cf(u, v) is defined as the additional flow that can pass without exceeding the capacity constraints, c(u, v), after taking into account the capacity occupied by the feasible flow f(u, v).(1)cfu,v=cu,v−fu,v.Given a graph,G = (V, E, C), and a feasible flow f, the residual network is Gf(V, Ef), where the edge set Ef is(2)Ef=u,v∈V×V|cfu,v>0.The physical meaning of the residual network in the power grid is the transmission network composed of lines with the residual capacity. The cross-regional renewable energy consumption is based on the priority for the transmission demand within the region. Therefore, the configuration of transaction paths for renewable energy sources must be built based on the residual network where the transmission quota within the region has been reserved. The max-flow algorithm searches and utilizes the residual power transmission capacity between source and sink nodes.


## 2.3.2. Augmenting Path of Residual Network

In the residual network, the max-flow algorithm searches for the feasible flows and forms a new residual network. This process constitutes an iterative loop until no new feasible flow can be found. The new feasible flow is defined as the augmenting pathp which is a path from the source node s to the sink node t in the residual network Gf. The capacity of the augmenting path, cf(p), is defined in equation (3). cf(p) is the maximum additional flow that can be added along the path.(3)cfp=mincfu,v|<u,v>∈p,cfp>0.The addition of an augmenting path results in a flow with a larger value. The augmenting path is the increment of the flow and has the property of augmentation. The flow of augmenting pathfp is defined in the following equation:(4)fpu,v=cfp,<u,v>∈p,−cfp,<u,v>∈p,0,else.The physical meaning of the augmenting path in the power grid is the incremental trading of the transaction between seller and buyer. The cross-regional consumption of renewable energy can utilize the multiterminal DC grid and the traditional AC grid. In the traditional AC grid, the cross-regional electricity transmission of other energy sources must be considered. Introducing the concept of augmenting path, different feasible flows from various types of energy sources can be coordinated and allocated.


## 3. Graph Theory-Based Transmission Cost Optimization and Allocation Method

### 3.1. Graph Theory-Based Transmission Cost Optimization Method of Cross-Regional Transaction

The decision variable in network flow optimization is the flowf(u, v) on edge e(u, v). In the complex network, multiple transactions between different buyers and sellers can be concurrent. The existing TCA method can only provide solution based on the total flow on edge and cannot further subdivide the flow for each transaction. Therefore, the expanded graph theory-based TCA method is proposed to solve the flow optimization problem of the complex network and the concurrency of multiple transaction components [29].The optimization variable is expanded from the original two-dimensional variable,f(u, v), to the four-dimensional optimization variable fuvst. The yield spread parameter, b(u, v), on each edge is also extended to the four-dimensional buvst. In this way, when multiple transactions go through the same edge e(u, v), they can be distinguished by the parameters (s, t) and decouple the multiple transactions on the same edge. The objective function is shown below to maximize the social welfare.(5)maxU=∑s,t∈Zfuvstbuvst,(6)buvst=−ps,u=s,s∈S,pt,v=t,t∈T,−luvps,else,(7)s.t.0≤∑s,t∈Zxuvst≤cuv,(8)∑t=1nfuvst−∑t=1nfvust=0,s∈TX,(9)∑s,t∈Zfuvsttuvcuv+∑s,t∈Zfvusttvucvu=tuv,u,v∈TX,where ps is the declared electricity price of node s; pt is the declared electricity price of node t; luv is the rate of loss allocation of the edge (u, v); S is the set of all sellers; T is the 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. Equations (7)–(9) are the channel transmission capacity constraint, the node flow balance constraint, and the constraint of the forward and reverse utilization hours.Equation (7) indicates that all the transaction flows from different transaction pairs (s, t) passing through the edge e(u, v) are less than the power flow capacity cuv of the edge e(u, v). Equation (8) indicates that for all intermediate nodes, the inflow power flow of each transaction pair (s, t) is equal to the outflow power flow. Equation (9) indicates that the transmission line cannot transmit power in both directions at the same time. On the basis of the transmission capacity, the forward and reverse utilization hours of the transmission line are restricted. The sum of the forward and reverse utilization hours should be equal to the total available hours tuv of the transmission line.


### 3.2. Usage-Based TCA Method

After the network flow optimization, the total transmission cost needs to be allocated. The allocation criterion of the MWM method is the “extend of use” of each network facility. As stated in Section1, the MWM method can fully recover the fixed cost of the transmission network based on the actual usage of the active power flow and the line length for each transmission line [16]. The equation is stated below.(10)TCt=TC∑e∈EtceLeMWt,e∑t∈T∑e∈EtceLeMWt,e,where tce is the cost per unit length and MW of line e; Le is the length of line e; MWt, e is the active power flow in line e due to buyer t; and TC is the total fixed and operational cost involved in the transaction.As for the problem of counter-flows, there are three common different approaches [16]. (1) The absolute MWM approach charges the user based on the absolute value of the power flow and ignores the direction of the flows. (2) The reverse MWM approach considers the counter-flows and charges the user based on the net flows of each transmission line. (3) The zero counter-flow (ZCF) MWM approach does not consider the counter-flows. The equations for three approaches are shown below.(11)TCt=∑e∈ETCeFt,eFe,maxabsolute MWM approach,∑e∈ETCeFt,eFe,maxreverse MWM approach,∑e∈ETCeFt,eFe,max,∀ft,e>0ZCF MWM approach,where Ft, e is the counter-flow of line e by buyer t, Fe, max is the maximum flow of line e, and TCe is the cost per unit MW of line e.


### 3.3. Network Flow Algorithm-Based TCA Method

The “fair” TCA method is supposed to provide the proper economic signals to the transactions that involves counter-flows. According to equations (3) and (4), during the process of augmenting path searching, the counter-flows are inherently counted in the max-flow algorithm. So, the graph theory-based TCA method contains following advantages compared to the classic MWM method. Firstly, there is no slack bus in the simplified graph. Secondly, it is not necessary to charge or pay credit to counter-flows separately. Thirdly, the maximum future use of the transaction path is provided in the residual network. Fourthly, the network flow algorithm based TCA method is still MW-based not energy-based. As a result, the allocation solution still depends on the flow usage not energy usage during a period of time.The UHV transmission grid covers multiple provincial regions, and the parameter and length of the transmission line are not available. Therefore, the DC power flow-based MWM method cannot be applied. According to Menger’s theorem [32], if the source node s and the sink node t are the nodes of the graph G and st∉EG, then the minimum size of s, t-cut is equal to the maximum number of s, t-paths that do not intersect each other in each pair. The minimum cut K between the source and sink nodes means that the maximum number of disjoint paths is also K. Based on the above theorem, the basic steps of the proposed method are as follows:(1)
Test the connectivity of the transaction path <s, t> to determine the minimum path cut K(s, t) and calculate the transmission costs for K different paths. The total number of edges in each different path is Ek.(2)
Fromk = 1, calculate the transmission costs according to the following equation:(12)TCfs,t=∑k=1K∑e=1EkTCefeku,vce,feku,v∈0,Fe,maxs,t,wherefeku,v is the feasible flow of line e on path k, ce is the capacity of line e, and Fe,maxs,t is the max flow of the transaction pair (s, t) online e.(3)
The iteration ends whenk = K, and the transmission cost of the current transaction path and the transmission cost under the maximum flow condition are derived.The augmenting procedure for lines setEk is illustrated in Figure 2. Assume that the max-flow solution for Ek is found after four iterations of augmenting path searching. The power flow under normal condition for transaction pair (s, t) of edge e1 and e2 is f1,NCs,t and f2,NCs,t. F1,maxs,t and F2,maxs,t are the max flows of edge e1 and e2. They are assigned the values f1,AG4, f2,AG4 and utilized in equation (12). There are two points that need to be mentioned. (1) The max flow for one edge is not necessarily the flow with the largest absolute value during augmenting. Under max-flow condition, the flow on one edge can be limited by other bottleneck line(s). (2) The max flow can be the counter-flow which is opposite to the initial direction.Figure 2
TCA under normal and max-flow conditions.


### 3.4. Embedded TCA Method for Invalid Capacity

Overall the cost of the used capacity of a transmission facility corresponds to the power flowfe. Also, the future use cost corresponds to the unused capacity (fe, max – fe). In addition, the cost of the invalid capacity (ce – fe, max) is allocated to buyers by an embedded method (post-stamp method). In this way, a market-oriented and complete allocation of the total transmission cost is accomplished.The post-stamp method is the most common and simple method used by electricity utilities, where an entity pays a rate equal to a fixed charge per unit of energy transmitted [7]. The cost allocated to buyer t for invalid capacity, TCIC, is(13)TCICs,t=∑k=1K∑e=1EkTCece−fe,maxkce.The total transmission cost TCs, t allocated to the transaction pair (s, t) is shown in equation (14) and is illustrated in Figure 3.Figure 3
Total TCA for CN, CF, and IC.The total cost involves two main components: the valid capacity TC[f(s, t)] and the invalid capacity TCICs,t.(14)TCs,t=TCfs,t+TCICs,t.At each time node that requires TCA, the proposed method calculates and allocates these two costs to each transaction pair in the power market. This method can ensure the complete allocation of transmission cost and take the capacity for future use and invalid capacity into consideration. The flowchart of the graph theory-based method is shown in Figure4.Figure 4
Flowchart of the proposed transaction path configuration method.


## 3.1. Graph Theory-Based Transmission Cost Optimization Method of Cross-Regional Transaction

The decision variable in network flow optimization is the flowf(u, v) on edge e(u, v). In the complex network, multiple transactions between different buyers and sellers can be concurrent. The existing TCA method can only provide solution based on the total flow on edge and cannot further subdivide the flow for each transaction. Therefore, the expanded graph theory-based TCA method is proposed to solve the flow optimization problem of the complex network and the concurrency of multiple transaction components [29].The optimization variable is expanded from the original two-dimensional variable,f(u, v), to the four-dimensional optimization variable fuvst. The yield spread parameter, b(u, v), on each edge is also extended to the four-dimensional buvst. In this way, when multiple transactions go through the same edge e(u, v), they can be distinguished by the parameters (s, t) and decouple the multiple transactions on the same edge. The objective function is shown below to maximize the social welfare.(5)maxU=∑s,t∈Zfuvstbuvst,(6)buvst=−ps,u=s,s∈S,pt,v=t,t∈T,−luvps,else,(7)s.t.0≤∑s,t∈Zxuvst≤cuv,(8)∑t=1nfuvst−∑t=1nfvust=0,s∈TX,(9)∑s,t∈Zfuvsttuvcuv+∑s,t∈Zfvusttvucvu=tuv,u,v∈TX,where ps is the declared electricity price of node s; pt is the declared electricity price of node t; luv is the rate of loss allocation of the edge (u, v); S is the set of all sellers; T is the 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. Equations (7)–(9) are the channel transmission capacity constraint, the node flow balance constraint, and the constraint of the forward and reverse utilization hours.Equation (7) indicates that all the transaction flows from different transaction pairs (s, t) passing through the edge e(u, v) are less than the power flow capacity cuv of the edge e(u, v). Equation (8) indicates that for all intermediate nodes, the inflow power flow of each transaction pair (s, t) is equal to the outflow power flow. Equation (9) indicates that the transmission line cannot transmit power in both directions at the same time. On the basis of the transmission capacity, the forward and reverse utilization hours of the transmission line are restricted. The sum of the forward and reverse utilization hours should be equal to the total available hours tuv of the transmission line.


## 3.2. Usage-Based TCA Method

After the network flow optimization, the total transmission cost needs to be allocated. The allocation criterion of the MWM method is the “extend of use” of each network facility. As stated in Section1, the MWM method can fully recover the fixed cost of the transmission network based on the actual usage of the active power flow and the line length for each transmission line [16]. The equation is stated below.(10)TCt=TC∑e∈EtceLeMWt,e∑t∈T∑e∈EtceLeMWt,e,where tce is the cost per unit length and MW of line e; Le is the length of line e; MWt, e is the active power flow in line e due to buyer t; and TC is the total fixed and operational cost involved in the transaction.As for the problem of counter-flows, there are three common different approaches [16]. (1) The absolute MWM approach charges the user based on the absolute value of the power flow and ignores the direction of the flows. (2) The reverse MWM approach considers the counter-flows and charges the user based on the net flows of each transmission line. (3) The zero counter-flow (ZCF) MWM approach does not consider the counter-flows. The equations for three approaches are shown below.(11)TCt=∑e∈ETCeFt,eFe,maxabsolute MWM approach,∑e∈ETCeFt,eFe,maxreverse MWM approach,∑e∈ETCeFt,eFe,max,∀ft,e>0ZCF MWM approach,where Ft, e is the counter-flow of line e by buyer t, Fe, max is the maximum flow of line e, and TCe is the cost per unit MW of line e.


## 3.3. Network Flow Algorithm-Based TCA Method

The “fair” TCA method is supposed to provide the proper economic signals to the transactions that involves counter-flows. According to equations (3) and (4), during the process of augmenting path searching, the counter-flows are inherently counted in the max-flow algorithm. So, the graph theory-based TCA method contains following advantages compared to the classic MWM method. Firstly, there is no slack bus in the simplified graph. Secondly, it is not necessary to charge or pay credit to counter-flows separately. Thirdly, the maximum future use of the transaction path is provided in the residual network. Fourthly, the network flow algorithm based TCA method is still MW-based not energy-based. As a result, the allocation solution still depends on the flow usage not energy usage during a period of time.The UHV transmission grid covers multiple provincial regions, and the parameter and length of the transmission line are not available. Therefore, the DC power flow-based MWM method cannot be applied. According to Menger’s theorem [32], if the source node s and the sink node t are the nodes of the graph G and st∉EG, then the minimum size of s, t-cut is equal to the maximum number of s, t-paths that do not intersect each other in each pair. The minimum cut K between the source and sink nodes means that the maximum number of disjoint paths is also K. Based on the above theorem, the basic steps of the proposed method are as follows:(1)
Test the connectivity of the transaction path <s, t> to determine the minimum path cut K(s, t) and calculate the transmission costs for K different paths. The total number of edges in each different path is Ek.(2)
Fromk = 1, calculate the transmission costs according to the following equation:(12)TCfs,t=∑k=1K∑e=1EkTCefeku,vce,feku,v∈0,Fe,maxs,t,wherefeku,v is the feasible flow of line e on path k, ce is the capacity of line e, and Fe,maxs,t is the max flow of the transaction pair (s, t) online e.(3)
The iteration ends whenk = K, and the transmission cost of the current transaction path and the transmission cost under the maximum flow condition are derived.The augmenting procedure for lines setEk is illustrated in Figure 2. Assume that the max-flow solution for Ek is found after four iterations of augmenting path searching. The power flow under normal condition for transaction pair (s, t) of edge e1 and e2 is f1,NCs,t and f2,NCs,t. F1,maxs,t and F2,maxs,t are the max flows of edge e1 and e2. They are assigned the values f1,AG4, f2,AG4 and utilized in equation (12). There are two points that need to be mentioned. (1) The max flow for one edge is not necessarily the flow with the largest absolute value during augmenting. Under max-flow condition, the flow on one edge can be limited by other bottleneck line(s). (2) The max flow can be the counter-flow which is opposite to the initial direction.Figure 2
TCA under normal and max-flow conditions.


## 3.4. Embedded TCA Method for Invalid Capacity

Overall the cost of the used capacity of a transmission facility corresponds to the power flowfe. Also, the future use cost corresponds to the unused capacity (fe, max – fe). In addition, the cost of the invalid capacity (ce – fe, max) is allocated to buyers by an embedded method (post-stamp method). In this way, a market-oriented and complete allocation of the total transmission cost is accomplished.The post-stamp method is the most common and simple method used by electricity utilities, where an entity pays a rate equal to a fixed charge per unit of energy transmitted [7]. The cost allocated to buyer t for invalid capacity, TCIC, is(13)TCICs,t=∑k=1K∑e=1EkTCece−fe,maxkce.The total transmission cost TCs, t allocated to the transaction pair (s, t) is shown in equation (14) and is illustrated in Figure 3.Figure 3
Total TCA for CN, CF, and IC.The total cost involves two main components: the valid capacity TC[f(s, t)] and the invalid capacity TCICs,t.(14)TCs,t=TCfs,t+TCICs,t.At each time node that requires TCA, the proposed method calculates and allocates these two costs to each transaction pair in the power market. This method can ensure the complete allocation of transmission cost and take the capacity for future use and invalid capacity into consideration. The flowchart of the graph theory-based method is shown in Figure4.Figure 4
Flowchart of the proposed transaction path configuration method.


## 4. Application of Network Flow Algorithm in the Configuration of Electricity Transaction Path

### 4.1. Implementation of Maximum Flow Algorithm

Referring to the concepts of the connectivity and network flow algorithms mentioned above, a visual and quantitative estimation of the path reliability and maximum transmission capacity can be obtained. The key point of implementation of the specific algorithm is to search the augmenting path efficiently.The Edmonds–Karp (EK) algorithm is classified as the Ford–Fulkerson (FF) method. Its basic steps are the same as the FF method. The EK algorithm uses breadth-first search (BFS) as the augmenting path search method. The BFS method is a basic search method, so the logic of the EK algorithm is relatively simple. This algorithm applies to most power grid analyses based on graph theory [32].The steps of applying the Edmonds–Karp algorithm are as follows:(1)
Initialize the capacity of all edges in the graph.c<u, v> inherits the changed capacity. c<u, v> is initialized to zero, and the edge <v, u> is the return edge. Initialize the maximum stream to zero.(2)
Start BFS for an augmenting pathp from the source node s to the sink node t in the residual network. When the point at the first of the array is the end node, the augmenting path is found; then, go to step (3); if it cannot be found, go to step (5).(3)
Find the “bottleneck” edge in the augmented pathp. The “bottleneck” is the edge with the smallest capacity in the path, record this value X, and add it to the maximum flow; go to step (4).(4)
SubtractX from c<u, v> in the augmenting path and add X to all c<v, u> to form a new residual network. Go to step (2).(5)
Get the maximum flow of the network and end.


### 4.2. Case Study of Cross-Regional Renewable Energy Consumption in China

The case study in this paper takes the current UHV transmission grid in China as the reference. The inter-provincial connection channels are transformed into the simplified graphG. Assume that a nationwide cross-regional electricity transaction is organized, in which 12 provinces participate as electricity sellers and 9 provinces participate as electricity buyers [29].Firstly, the electricity surplus condition and transaction prices declared by each province are collected in a certain period of time in advance. The factors such as network loss and maintenance conditions on the cross-regional transaction channel are taken into account. Secondly, according to the network flow algorithm-based transmission cost optimization and allocation method, the cross-regional renewable energy transaction path plan is formed. Finally, the dispatching department conducts the safety check on the transaction path plan and generates the power transaction contracts to provide the evidence for electricity settlement.Take Gansu Province as the electricity seller as an example. As one of the provinces with the most serious wind curtailment problem, Gansu Province has limited local consumption capacity in the northwest region. It is actively participating in the inter-provincial market and medium-term and long-term transactions to promote the increase of wind and solar power generation and facilitate the decrease of wind and solar curtailment. Shanghai is considered as the electricity buyer, which accounts for more than 50% of the electricity purchase in East China [29]. The current transaction path from Gansu to Shanghai is Gansu-Shaanxi-Sichuan-Chongqing-Hubei-Shanghai.The following case study analyzes the system reliability of the transaction path and estimates the maximum transmission capacity of the transaction path by network connectivity analysis and maximum flow algorithm. The simulation environment of the proposed model is Matlab. The bidding data of provinces participating in the power market are given in Table1, including the amount of buying/selling electricity and bidding price. The system grid parameters such as source and sink node information of inter-provincial tie lines, transmission line upper and lower limits, and rate of loss allocation are given in Table 2.Table 1
Bidding data of provinces participating in the power market in China.
No.Node nameSelling electricity (GWh)Buying electricity (GWh)Bidding price (CNY/MWh)1Jingjintang0857.63822Hebei003953Shanxi193103154Shandong279003975Shanghai059894626Jiangsu049004367Zhejiang03926.14588Anhui100003989Fujian127.3042210Hubei10224.6035411Hunan05337112Henan0301.835813Jiangxi058539114Sichuan255028815Chongqing227029116Heilongjiang1435040017Jilin1431.41037618Liaoning0321.238019Shaanxi034.529720Gansu220.2027721Qinghai418.6027922Ningxia117.6026823Xinjiang00250Total20177.7116968.2Table 2
Power capacity and network loss rate of transmission lines.
No.DirectionPositive direction capacity (MW)Negative direction capacity (MW)Rate of loss allocation1Hebei-Jingjintang400040000.012Shandong-Hebei380038000.013Shanxi-Hebei720072000.014Shanxi-Jingjintang246024600.015Anhui-Jiangsu350035000.016Fujian-Zhejiang180018000.017Zhejiang-Shanghai260026000.018Jiangsu-Shanghai340034000.029Jiangsu-Zhejiang400040000.0110Sichuan-Chongqing220022000.0211Hubei-Jiangxi160016000.0112Hubei-Hunan260011000.0213Hubei-Chongqing300020000.0114Hubei-Henan300030000.0115Jilin-Heilongjiang240024000.0116Liaoning-Jilin140014000.0217Gansu-Ningxia380041000.0118Gansu-Qinghai240024000.0119Shaanxi-Gansu260020000.0220Gansu-Xinjiang130013000.0121Hubei-Jiangsu360036000.0122Hubei-Shanghai360036000.0123Shanxi-Henan190015000.0224Sichuan-Shaanxi116013600.0125Shaanxi-Henan100010000.0226Liaoning-Jingjintang150015000.02The network connectivity simulation result is shown in Figure5. The graph, G(E, V, C), is a 1-connected graph. The cut vertices, Hubei, Shaanxi, and Henan, are marked in red. The graph G can be divided into 4 subgraphs by cut vertices. These four subgraphs correspond to (1) Northeast China, (2) East China, (3) Central China, and (4) Northwest China in the geographic environment. The Gansu-Shanghai transaction path involves three subgraphs.Figure 5
Simplified network connectivity simulation.Similar to the above example, the network connectivity analysis is applied to all 17 transactions in the market. If there is more than one path between a pair of seller and buyer, all possible transaction paths are included in the connectivity analysis. The number of cut vertex and detailed vertex name is shown in Table3.Table 3
Cut vertices of the transaction path(s).
No.Vertices of the transaction path(s)# of cut vertexDetailed cut vertex1Anhui-Jiangsu-Shanghai1Jiangsu2Anhui-Jiangsu-Zhejiang1Jiangsu3Fujian-Zhejiang0N/A4Gansu-Shaanxi-Henan1Shaanxi5Gansu-Shaanxi-Sichuan-Chongqing-Hubei-Shanghai, Gansu-Shaanxi-Henan-Hubei-Shanghai3Shaanxi, Henan, Hubei6Hubei-Shanghai, Hubei-Jiangsu-Shanghai1Jiangsu7Hubei-Jiangsu-Zhejiang1Jiangsu8Ningxia-Gansu-Shaanxi-Henan2Gansu, Shaanxi9Ningxia-Gansu-Shaanxi-Henan-Hubei-Shanghai4Gansu, Shaanxi, Henan, Hubei10Qinghai-Gansu-Shaanxi-Henan-Hubei-Hunan4Gansu, Shaanxi, Henan, Hubei11Qinghai-Gansu-Shaanxi-Henan-Hubei-Jiangxi4Gansu, Shaanxi, Henan, Hubei12Shanxi-Jingjintang0N/A13Shanxi-Jingjintang-Liaoning1Jingjintang14Shanxi-Henan-Hubei-Jiangsu-Zhejiang3Henan, Jiangsu, Hubei15Sichuan-Chongqing-Hubei-Shanghai1Hubei16Chongqing-Hubei-Jiangxi1Hubei17Chongqing-Hubei-Shanghai1HubeiThe cut vertex info is summarized in Table4 to indicate the importance of the specific vertex in the market. Hubei is the cut vertex in eight cross-regional transactions. The trading volume and bidding price are 10224.6 GWh and 354 CNY/MWh. Shaanxi is cut vertex in 6 transactions, but the trading volume and bidding price are 34.5 GWh and 297 CNY/MWh. From topological structure view, Shaanxi is the critical vertex for electricity sellers, such as Gansu, Qinghai, and Ningxia. However, there is no clear indication in the traditional transaction data, like trading volume and bidding price. The network connectivity analysis is a necessary supplement to the transmission pricing strategy.Table 4
Summary of cut vertices.
Cut vertex# of transactions involvedTrading volume (GWh)Bidding price (CNY/MWh)1Hubei810224.63542Shaanxi634.52973Jiangsu549004364Henan5301.83585Gansu4220.22776Jingjintang1857.6382The maximum transmission capacity of the transaction path is estimated by the EK algorithm. The maximum capacity takes the transmission line loss, transmission line capacity, and the routine maintenance into consideration, and the transaction period is set as 30 days. Four cases of max-flow problem by EK algorithm are shown in Figure6. The unit is GWh. The feasible paths and “bottleneck” line are highlighted in green and red.Figure 6
Edmonds–Karp max-flow algorithm for cross-regional transaction paths.The current trading volume of Gansu-Shanghai path is 220.2 GWh. In addition to the original transaction path (Gansu-Shaanxi, Sichuan, Chongqing, Hubei, and Shanghai), the augmenting path (Gansu, Shaanxi, Henan, Hubei, and Shanghai) is added in max-flow algorithm. Without considering the other transaction pair, the “bottleneck” lines refer to Gansu-Shaanxi whose capacity is 1411.2 GWh and Shaanxi-Henan whose capacity is 705.6 GWh. The maximum transmission estimation is 1411.2 GWh. The maximum utilization rate of non-bottleneck lines does not exceed 72.06%. The simulation result shows that there is still a huge transmission margin, and the potential for cross-regional consumption of renewable energy is promising. The detailed “bottleneck” line and max flow are summarized in Table5.Table 5
“Bottleneck” transmission line(s), max flow, and invalid capacity of transactions.
No.Vertices of the transaction path(s)Bottleneck path(s)Max flow (GWh)IC (MW)1Anhui-Jiangsu-ShanghaiJiangsu-Shanghai2528.84535.52Anhui-Jiangsu-ZhejiangAnhui-Jiangsu2603.22111.13Fujian-ZhejiangN/A1338.81623.24Gansu-Shaanxi-HenanShaanxi-Henan743.8949.45Gansu-Shaanxi-Sichuan-Chongqing-Hubei-Shanghai, Gansu-Shaanxi-Henan-Hubei-ShanghaiGansu-Shaanxi Shaanxi-Henan1487.56775.56Hubei-Shanghai, Hubei-Jiangsu-ShanghaiJiangsu-Shanghai5206.32424.47Hubei-Jiangsu-ZhejiangHubei-Jiangsu2677.508Ningxia-Gansu-Shaanxi-HenanShaanxi-Henan743.827689Ningxia-Gansu-Shaanxi-Henan-Hubei-Shanghai, Ningxia-Gansu-Shaanxi-Sichuan-Chongqing-Hubei-ShanghaiGansu-Shaanxi Shaanxi-Henan1487.510712.210Qinghai-Gansu-Shaanxi-Henan-Hubei-Hunan, Qinghai-Gansu-Shaanxi-Sichuan-Chongqing-Hubei-HunanHubei-Hunan Shaanxi-Henan1487.511120.511Qinghai-Gansu-Shaanxi-Henan-Hubei-Jiangxi, Qinghai-Gansu-Shaanxi-Sichuan-Chongqing-Hubei-JiangxiShaanxi-Henan Hubei-Jiangxi11909381.612Shanxi-JingjintangN/A1830.2822.813Shanxi-Jingjintang-LiaoningJingjintang-Liaoning1115.61876.714Shanxi-Henan-Hubei-Jiangsu-ZhejiangShanxi-Henan1413.12196.715Sichuan-Chongqing-Hubei-ShanghaiChongqing-Hubei1487.53141.916Chongqing-Hubei-JiangxiHubei-Jiangxi11902100.817Chongqing-Hubei-ShanghaiChongqing-Hubei1487.51313.3In the section from Shaanxi to Hubei, there are two non-intersecting paths: Shaanxi-Sichuan-Chongqing-Hubei and Shaanxi-Henan-Hubei. Then, the minimum cut between Shaanxi and Hubei is equal to 2. When one of the paths fails, the complete transaction path can still be in a working state. The invalid capacity of the transmission line is after considering the transmission loss rate and the periodic maintenance.After the transaction path analysis, the TCA of Gansu-Shanghai transaction under normal, optimized, and max-flow conditions is shown in Figure7. The MWM method is utilized in normal condition.(15)TCfGansu,Shanghai=∑e=15TCefeu,vce=22342.80CNY.Figure 7
Graph theory-based TCA of Gansu-Shanghai transaction path.Equations (5)–(9) are utilized in the optimal TCA method. The optimized line flow with the social welfare as the objective is used as the edge flow for the section from Shaanxi to Hubei.(16)TCoptfGansu,Shanghai=∑e=11TCefeu,vce︷Gansu−Shaanxi+∑e=24TCefe,optu,vce+∑e=67TCefe,optu,vce︷Shaanxi−Sichuan−Chongqing−Hubei and Shaanxi−Henan−Hubei+∑e=55TCefeu,vce︷Hubei−Shanghai=19296.60CNY.The transaction path configuration method in max-flow condition uses the line capacity of the residual network as the edge flow from Qinghai to Hunan to allocate the future use cost.(17)TCresfGansu,Shanghai=∑e=11TCefe,resu,vce+∑e=24TCefe,resu,vce+∑e=67TCefe,resu,vce+∑e=55TCefe,resu,vce=120739.65CNY.By the traditional route configuration method, the flow on each tie line in a single transaction route is the same. Although the MWM algorithm is simple and straightforward, the transmission capacity of the remaining lines has not been effectively used, resulting in high total transmission costs. The transmission cost for 305.83 MW is 22342.80 CNY. The unit cost of transmission is 0.073 CNY/kWh. The transmission cost for the same amount after social-welfare optimization is 19296.60 CNY. The unit cost of transmission is 0.0631 CNY/kWh. It is clear that the transmission cost can be reduced by 13.63% by the optimization method. In max-flow condition, without considering the other transaction pair, the transmission cost for 2000-MW max flow between Gansu and Shanghai is 120739.65 CNY. The unit cost of transmission is 0.0688 CNY/kWh, which is still lower than the unit cost of transmission using traditional path configuration method.After the optimization of social welfare, the optimal results are presented in Figure8. It provides the optimal flow for each established transactions in the market. The cost of IC is allocated to all the transaction pair by the stamp method. The detailed results of IC are given in Table 5. Take the Gansu-Shanghai transaction path as an example; the total transmission cost is 0.0931 CNY/kWh by equation (14) including both valid and invalid capacity usage.Figure 8
Graph theory-based transmission cost optimization.In summary, the computational efficiency of the network flow algorithm based TCA method is comparatively high. So, it can accommodate the dynamic changes in the power system. Besides that, it can effectively identify the critical node(s) and bottleneck line(s), maximize the social welfare, track the decoupled flow, and provide transmission cost for future use.


## 4.1. Implementation of Maximum Flow Algorithm

Referring to the concepts of the connectivity and network flow algorithms mentioned above, a visual and quantitative estimation of the path reliability and maximum transmission capacity can be obtained. The key point of implementation of the specific algorithm is to search the augmenting path efficiently.The Edmonds–Karp (EK) algorithm is classified as the Ford–Fulkerson (FF) method. Its basic steps are the same as the FF method. The EK algorithm uses breadth-first search (BFS) as the augmenting path search method. The BFS method is a basic search method, so the logic of the EK algorithm is relatively simple. This algorithm applies to most power grid analyses based on graph theory [32].The steps of applying the Edmonds–Karp algorithm are as follows:(1)
Initialize the capacity of all edges in the graph.c<u, v> inherits the changed capacity. c<u, v> is initialized to zero, and the edge <v, u> is the return edge. Initialize the maximum stream to zero.(2)
Start BFS for an augmenting pathp from the source node s to the sink node t in the residual network. When the point at the first of the array is the end node, the augmenting path is found; then, go to step (3); if it cannot be found, go to step (5).(3)
Find the “bottleneck” edge in the augmented pathp. The “bottleneck” is the edge with the smallest capacity in the path, record this value X, and add it to the maximum flow; go to step (4).(4)
SubtractX from c<u, v> in the augmenting path and add X to all c<v, u> to form a new residual network. Go to step (2).(5)
Get the maximum flow of the network and end.


## 4.2. Case Study of Cross-Regional Renewable Energy Consumption in China

The case study in this paper takes the current UHV transmission grid in China as the reference. The inter-provincial connection channels are transformed into the simplified graphG. Assume that a nationwide cross-regional electricity transaction is organized, in which 12 provinces participate as electricity sellers and 9 provinces participate as electricity buyers [29].Firstly, the electricity surplus condition and transaction prices declared by each province are collected in a certain period of time in advance. The factors such as network loss and maintenance conditions on the cross-regional transaction channel are taken into account. Secondly, according to the network flow algorithm-based transmission cost optimization and allocation method, the cross-regional renewable energy transaction path plan is formed. Finally, the dispatching department conducts the safety check on the transaction path plan and generates the power transaction contracts to provide the evidence for electricity settlement.Take Gansu Province as the electricity seller as an example. As one of the provinces with the most serious wind curtailment problem, Gansu Province has limited local consumption capacity in the northwest region. It is actively participating in the inter-provincial market and medium-term and long-term transactions to promote the increase of wind and solar power generation and facilitate the decrease of wind and solar curtailment. Shanghai is considered as the electricity buyer, which accounts for more than 50% of the electricity purchase in East China [29]. The current transaction path from Gansu to Shanghai is Gansu-Shaanxi-Sichuan-Chongqing-Hubei-Shanghai.The following case study analyzes the system reliability of the transaction path and estimates the maximum transmission capacity of the transaction path by network connectivity analysis and maximum flow algorithm. The simulation environment of the proposed model is Matlab. The bidding data of provinces participating in the power market are given in Table1, including the amount of buying/selling electricity and bidding price. The system grid parameters such as source and sink node information of inter-provincial tie lines, transmission line upper and lower limits, and rate of loss allocation are given in Table 2.Table 1
Bidding data of provinces participating in the power market in China.
No.Node nameSelling electricity (GWh)Buying electricity (GWh)Bidding price (CNY/MWh)1Jingjintang0857.63822Hebei003953Shanxi193103154Shandong279003975Shanghai059894626Jiangsu049004367Zhejiang03926.14588Anhui100003989Fujian127.3042210Hubei10224.6035411Hunan05337112Henan0301.835813Jiangxi058539114Sichuan255028815Chongqing227029116Heilongjiang1435040017Jilin1431.41037618Liaoning0321.238019Shaanxi034.529720Gansu220.2027721Qinghai418.6027922Ningxia117.6026823Xinjiang00250Total20177.7116968.2Table 2
Power capacity and network loss rate of transmission lines.
No.DirectionPositive direction capacity (MW)Negative direction capacity (MW)Rate of loss allocation1Hebei-Jingjintang400040000.012Shandong-Hebei380038000.013Shanxi-Hebei720072000.014Shanxi-Jingjintang246024600.015Anhui-Jiangsu350035000.016Fujian-Zhejiang180018000.017Zhejiang-Shanghai260026000.018Jiangsu-Shanghai340034000.029Jiangsu-Zhejiang400040000.0110Sichuan-Chongqing220022000.0211Hubei-Jiangxi160016000.0112Hubei-Hunan260011000.0213Hubei-Chongqing300020000.0114Hubei-Henan300030000.0115Jilin-Heilongjiang240024000.0116Liaoning-Jilin140014000.0217Gansu-Ningxia380041000.0118Gansu-Qinghai240024000.0119Shaanxi-Gansu260020000.0220Gansu-Xinjiang130013000.0121Hubei-Jiangsu360036000.0122Hubei-Shanghai360036000.0123Shanxi-Henan190015000.0224Sichuan-Shaanxi116013600.0125Shaanxi-Henan100010000.0226Liaoning-Jingjintang150015000.02The network connectivity simulation result is shown in Figure5. The graph, G(E, V, C), is a 1-connected graph. The cut vertices, Hubei, Shaanxi, and Henan, are marked in red. The graph G can be divided into 4 subgraphs by cut vertices. These four subgraphs correspond to (1) Northeast China, (2) East China, (3) Central China, and (4) Northwest China in the geographic environment. The Gansu-Shanghai transaction path involves three subgraphs.Figure 5
Simplified network connectivity simulation.Similar to the above example, the network connectivity analysis is applied to all 17 transactions in the market. If there is more than one path between a pair of seller and buyer, all possible transaction paths are included in the connectivity analysis. The number of cut vertex and detailed vertex name is shown in Table3.Table 3
Cut vertices of the transaction path(s).
No.Vertices of the transaction path(s)# of cut vertexDetailed cut vertex1Anhui-Jiangsu-Shanghai1Jiangsu2Anhui-Jiangsu-Zhejiang1Jiangsu3Fujian-Zhejiang0N/A4Gansu-Shaanxi-Henan1Shaanxi5Gansu-Shaanxi-Sichuan-Chongqing-Hubei-Shanghai, Gansu-Shaanxi-Henan-Hubei-Shanghai3Shaanxi, Henan, Hubei6Hubei-Shanghai, Hubei-Jiangsu-Shanghai1Jiangsu7Hubei-Jiangsu-Zhejiang1Jiangsu8Ningxia-Gansu-Shaanxi-Henan2Gansu, Shaanxi9Ningxia-Gansu-Shaanxi-Henan-Hubei-Shanghai4Gansu, Shaanxi, Henan, Hubei10Qinghai-Gansu-Shaanxi-Henan-Hubei-Hunan4Gansu, Shaanxi, Henan, Hubei11Qinghai-Gansu-Shaanxi-Henan-Hubei-Jiangxi4Gansu, Shaanxi, Henan, Hubei12Shanxi-Jingjintang0N/A13Shanxi-Jingjintang-Liaoning1Jingjintang14Shanxi-Henan-Hubei-Jiangsu-Zhejiang3Henan, Jiangsu, Hubei15Sichuan-Chongqing-Hubei-Shanghai1Hubei16Chongqing-Hubei-Jiangxi1Hubei17Chongqing-Hubei-Shanghai1HubeiThe cut vertex info is summarized in Table4 to indicate the importance of the specific vertex in the market. Hubei is the cut vertex in eight cross-regional transactions. The trading volume and bidding price are 10224.6 GWh and 354 CNY/MWh. Shaanxi is cut vertex in 6 transactions, but the trading volume and bidding price are 34.5 GWh and 297 CNY/MWh. From topological structure view, Shaanxi is the critical vertex for electricity sellers, such as Gansu, Qinghai, and Ningxia. However, there is no clear indication in the traditional transaction data, like trading volume and bidding price. The network connectivity analysis is a necessary supplement to the transmission pricing strategy.Table 4
Summary of cut vertices.
Cut vertex# of transactions involvedTrading volume (GWh)Bidding price (CNY/MWh)1Hubei810224.63542Shaanxi634.52973Jiangsu549004364Henan5301.83585Gansu4220.22776Jingjintang1857.6382The maximum transmission capacity of the transaction path is estimated by the EK algorithm. The maximum capacity takes the transmission line loss, transmission line capacity, and the routine maintenance into consideration, and the transaction period is set as 30 days. Four cases of max-flow problem by EK algorithm are shown in Figure6. The unit is GWh. The feasible paths and “bottleneck” line are highlighted in green and red.Figure 6
Edmonds–Karp max-flow algorithm for cross-regional transaction paths.The current trading volume of Gansu-Shanghai path is 220.2 GWh. In addition to the original transaction path (Gansu-Shaanxi, Sichuan, Chongqing, Hubei, and Shanghai), the augmenting path (Gansu, Shaanxi, Henan, Hubei, and Shanghai) is added in max-flow algorithm. Without considering the other transaction pair, the “bottleneck” lines refer to Gansu-Shaanxi whose capacity is 1411.2 GWh and Shaanxi-Henan whose capacity is 705.6 GWh. The maximum transmission estimation is 1411.2 GWh. The maximum utilization rate of non-bottleneck lines does not exceed 72.06%. The simulation result shows that there is still a huge transmission margin, and the potential for cross-regional consumption of renewable energy is promising. The detailed “bottleneck” line and max flow are summarized in Table5.Table 5
“Bottleneck” transmission line(s), max flow, and invalid capacity of transactions.
No.Vertices of the transaction path(s)Bottleneck path(s)Max flow (GWh)IC (MW)1Anhui-Jiangsu-ShanghaiJiangsu-Shanghai2528.84535.52Anhui-Jiangsu-ZhejiangAnhui-Jiangsu2603.22111.13Fujian-ZhejiangN/A1338.81623.24Gansu-Shaanxi-HenanShaanxi-Henan743.8949.45Gansu-Shaanxi-Sichuan-Chongqing-Hubei-Shanghai, Gansu-Shaanxi-Henan-Hubei-ShanghaiGansu-Shaanxi Shaanxi-Henan1487.56775.56Hubei-Shanghai, Hubei-Jiangsu-ShanghaiJiangsu-Shanghai5206.32424.47Hubei-Jiangsu-ZhejiangHubei-Jiangsu2677.508Ningxia-Gansu-Shaanxi-HenanShaanxi-Henan743.827689Ningxia-Gansu-Shaanxi-Henan-Hubei-Shanghai, Ningxia-Gansu-Shaanxi-Sichuan-Chongqing-Hubei-ShanghaiGansu-Shaanxi Shaanxi-Henan1487.510712.210Qinghai-Gansu-Shaanxi-Henan-Hubei-Hunan, Qinghai-Gansu-Shaanxi-Sichuan-Chongqing-Hubei-HunanHubei-Hunan Shaanxi-Henan1487.511120.511Qinghai-Gansu-Shaanxi-Henan-Hubei-Jiangxi, Qinghai-Gansu-Shaanxi-Sichuan-Chongqing-Hubei-JiangxiShaanxi-Henan Hubei-Jiangxi11909381.612Shanxi-JingjintangN/A1830.2822.813Shanxi-Jingjintang-LiaoningJingjintang-Liaoning1115.61876.714Shanxi-Henan-Hubei-Jiangsu-ZhejiangShanxi-Henan1413.12196.715Sichuan-Chongqing-Hubei-ShanghaiChongqing-Hubei1487.53141.916Chongqing-Hubei-JiangxiHubei-Jiangxi11902100.817Chongqing-Hubei-ShanghaiChongqing-Hubei1487.51313.3In the section from Shaanxi to Hubei, there are two non-intersecting paths: Shaanxi-Sichuan-Chongqing-Hubei and Shaanxi-Henan-Hubei. Then, the minimum cut between Shaanxi and Hubei is equal to 2. When one of the paths fails, the complete transaction path can still be in a working state. The invalid capacity of the transmission line is after considering the transmission loss rate and the periodic maintenance.After the transaction path analysis, the TCA of Gansu-Shanghai transaction under normal, optimized, and max-flow conditions is shown in Figure7. The MWM method is utilized in normal condition.(15)TCfGansu,Shanghai=∑e=15TCefeu,vce=22342.80CNY.Figure 7
Graph theory-based TCA of Gansu-Shanghai transaction path.Equations (5)–(9) are utilized in the optimal TCA method. The optimized line flow with the social welfare as the objective is used as the edge flow for the section from Shaanxi to Hubei.(16)TCoptfGansu,Shanghai=∑e=11TCefeu,vce︷Gansu−Shaanxi+∑e=24TCefe,optu,vce+∑e=67TCefe,optu,vce︷Shaanxi−Sichuan−Chongqing−Hubei and Shaanxi−Henan−Hubei+∑e=55TCefeu,vce︷Hubei−Shanghai=19296.60CNY.The transaction path configuration method in max-flow condition uses the line capacity of the residual network as the edge flow from Qinghai to Hunan to allocate the future use cost.(17)TCresfGansu,Shanghai=∑e=11TCefe,resu,vce+∑e=24TCefe,resu,vce+∑e=67TCefe,resu,vce+∑e=55TCefe,resu,vce=120739.65CNY.By the traditional route configuration method, the flow on each tie line in a single transaction route is the same. Although the MWM algorithm is simple and straightforward, the transmission capacity of the remaining lines has not been effectively used, resulting in high total transmission costs. The transmission cost for 305.83 MW is 22342.80 CNY. The unit cost of transmission is 0.073 CNY/kWh. The transmission cost for the same amount after social-welfare optimization is 19296.60 CNY. The unit cost of transmission is 0.0631 CNY/kWh. It is clear that the transmission cost can be reduced by 13.63% by the optimization method. In max-flow condition, without considering the other transaction pair, the transmission cost for 2000-MW max flow between Gansu and Shanghai is 120739.65 CNY. The unit cost of transmission is 0.0688 CNY/kWh, which is still lower than the unit cost of transmission using traditional path configuration method.After the optimization of social welfare, the optimal results are presented in Figure8. It provides the optimal flow for each established transactions in the market. The cost of IC is allocated to all the transaction pair by the stamp method. The detailed results of IC are given in Table 5. Take the Gansu-Shanghai transaction path as an example; the total transmission cost is 0.0931 CNY/kWh by equation (14) including both valid and invalid capacity usage.Figure 8
Graph theory-based transmission cost optimization.In summary, the computational efficiency of the network flow algorithm based TCA method is comparatively high. So, it can accommodate the dynamic changes in the power system. Besides that, it can effectively identify the critical node(s) and bottleneck line(s), maximize the social welfare, track the decoupled flow, and provide transmission cost for future use.


## 5. Conclusion

In this paper, the network flow algorithm of graph theory is introduced into the problem of inter-provincial transaction path configuration. From the perspective of topological structure, the problem of transmission cost optimization and allocation under complex network conditions is studied.The simulation result reveals that the simplified graph-based algorithm can evaluate the reliability of the transaction path and estimate the maximum transmission capacity of the transaction path under the practical network. The network flow algorithm-based TCA method can provide theoretical support for the cross-regional consumption of renewable energy through the inter-provincial transmission grid.



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*Source: 8562670-2022-04-22.xml*