Objective transmission cost allocation based on marginal usage of power network in spot market

Electrical Power and Energy Systems 118 (2020) 105799

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Objective transmission cost allocation based on marginal usage of power network in spot market ⁎

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Zhifang Yanga, , Xingyu Leia, Juan Yua, , Jeremy Linb a State Key Laboratory of Power Transmission Equipment & System Security and New Technology, College of Electrical Engineering, Chongqing University, Chongqing 400030, China b Transmission Analytics Company, Austin, TX 78705, USA

A R T I C LE I N FO

A B S T R A C T

Keywords: Locational marginal price (LMP) Marginal usage Network losses Optimal power flow Transmission cost allocation

Transmission cost allocation is important for a liberalized market with open access to transmission facilities. However, it remains controversial how to objectively allocate the transmission costs because there is no unique answer for the mapping between the power injections at buses and the power flows on branches. However, the marginal usage of the power network can be uniquely determined according to the market-clearing solution. In this paper, an objective transmission cost allocation method is proposed based on the marginal usage of the power network. The influence of the loss modeling on the proposed method is discussed. The “extent of use” of transmission facilities is determined based on the economic principles without subjective assumptions, which obeys the fairness requirement of spot markets. The proposed method is computationally efficient, in which only matrix manipulations are involved. Moreover, the proposed approach is compatible with the standards and procedures of the current spot market operation in the U.S., China, and many other countries and holds promise for industrial applications.

1. Introduction 1.1. Research motivations The open access of transmission network facilities is a basic requirement for deregulated power industries with unbundled generation and demand sectors. The electricity prices include energy price and fixed price. The energy price is primarily determined by the marginal pricing method (such as LMP), in which the transmission losses will be accounted for in the loss component of the LMP. Transmission cost allocation is required to obtain the fixed price or transmission price, which covers the fixed investment costs of the transmission facilities. The transmission price is charged to the network users to recover the opportunity costs of the transmission asset investment and the costs associated with the operation and maintenance of transmission facilities [1]. Hence, a well-designed transmission cost allocation method that reflects the “extent of use” of transmission facilities encourages “grid-friendly” behaviors by charging proportionately for users of the power network [2]. Besides, an ideal transmission cost allocation method should not be based on the subjective assumptions according to the fairness requirement of the spot market design. However, it is hard to achieve absolute fairness based on the

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transmission cost allocation method because there is no unique answer for the contributions of power injections at nodes to power flows on the transmission lines. In fact, determining the relationship between the power injections at nodes and power flows on the transmission lines is an “indeterminate problem”. The “supplementary equation” is required for solving this indeterminate problem, which is inevitable. The involvement of the “supplementary equation” in the transmission cost allocation process may cause disagreement among network participants. For example, for the MW-Mile method [3], network users near the slack bus (or distributed slack buses) commonly experience a lower share of transmission costs than those far away from the slack bus. A different selection of the slack bus leads to totally different transmission prices. The concept of “market center” is proposed which is used as a unique reference point to overcome this problem [4,5]. For the power flow tracing method, the “supplementary equation” of the indeterminate problem is the “proportional sharing principle”, which can be neither proved nor denied [6]. In China, the argument of the fairness of the transmission cost allocation method is one of the reasons for the failure of the spot market trial at the Northeast China Power Grid, in which the power flow tracing method was used. Many network participants had doubt about the transmission prices they received. However, it was difficult for market operators to defend the

Corresponding authors. E-mail addresses: [email protected] (Z. Yang), [email protected] (J. Yu).

https://doi.org/10.1016/j.ijepes.2019.105799 Received 18 June 2019; Received in revised form 4 December 2019; Accepted 18 December 2019 0142-0615/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature

N, K, G

Indices

Vectors and matrices

fijf (fijt )

sets of buses, branches, and generators

f ,f tK,0 constant vector of the linearized losses f K,0 PG vector of generator outputs at all generators PD vector of power demand at all buses t f AN × G, AN × K , AN × K incident matrices IN × N , IK × K identity matrices GSDFK × N matrix formed by GSDFs N Kf × N ,NtK × N sensitive matrix between branch flow and voltage angle

active power from Bus i(j) to Bus j(i)

fijloss hg LF mi − k

losses on branch (i,j) function of the production costs of Generator g loss factor contribution factor of users at Bus i for Branch k to the power flows N, K, G number of buses, branches, and generators generator outputs Pg power demand for users at Bus i Pi transmission price for users at Bus i pi voltage angle at Bus i θi λ, π , σ , ρ Lagrangian multipliers

Operation of vector

XN ≥ YN x i ≥ yi , ∀ i ∈ {1, 2, ⋯, N }

factors (GGDFs/GLDFs) is proposed in [9], in which a historical operating scenario is used to avoid the selection of the slack bus. However, subjective assumptions still remain. Extensions of the MW-Mile method are proposed based on the way to handle the unused capacity, the counter flows, and transmission loss [1,2,10–14]. As the GSDF is obtained by the linear DC OPF model, it should be noted that given the fixed slack buses, the average network usage and marginal network usage information provided by the GSDF are the same in a linear model. Hence, the GSDF-based MW-Mile method can be also regarded as a marginal usage-based method. The power flow tracing method is proposed based on the assumption of the “proportional sharing principle” [6,7,15]. The problem of the counter flow can be avoided. Further studies had been performed to improve the power flow tracing method. For example, reactive power was included in [16]. The basis of the power flow tracing method, i.e., the assumption of the “proportional sharing principle”, can be neither proved nor disproved [6,17]. Besides, according to the feedback from the spot market trial of the Northeast China Power Grid using this method, the computational process of the power flow tracing method is complex, making it difficult for network participants to understand the results. Some studies attempted to determine the contribution of the power injections to the power flows according to the power flow equations [17–20]. The obtained network usage changed if different formulations of the power flow equations were used. Similar to the power flow tracing method, the justification of such approaches can be neither proved nor disproved. Recently, the circuit theory was used to allocate the transmission cost of network users depending on the network characteristics [21–23]. The assumption of the “equal sharing principle” was involved to identify the mathematical terms corresponding to the contribution of the power injections to the power flows. The game theory was also introduced to calculate the average usage of the transmission network, but the computation time of these methods will be considerably high for a large market [23–25]. In prior studies that allocate transmission costs based on the “average usage” of the power network, the loads are matched with generators based on certain assumptions to solve the “indeterminate problem”. Then, the contributions of power injections on branch flows are calculated and transmission costs are allocated accordingly. Although many methods have been proposed to reduce the subjective aspects, it still cannot achieve complete fairness.

justification and fairness of the transmission cost allocation method. The fairness of the transmission cost allocation methods cannot be verified and confirmed because there is no definitive mapping between the power injections to the accumulated power flows. One possible solution to resolve this problem is to allocate the transmission costs based on the marginal network usage information, which can be uniquely determined according to the economic principles. It is well acknowledged that the marginal increase of power injections should be provided by the combination of generators with the lowest costs while subjecting to operational constraints. If the transmission costs are allocated based on the marginal usage information, subjective assumptions can be avoided and the obtained transmission price is expected to be recognized by and acceptable to the network participants. Besides, the marginal network usage information provides a clearer price signal as an incentive for inducing the desirable behavior from network participants. The main purpose of this paper is to investigate and propose how to fairly and conveniently allocate transmission costs based on the marginal usage of the transmission network. 1.2. Literature review In earlier transmission cost allocation methods, such as the postagestamp method and contract-path method, the actual usage of the transmission network facilities is not specifically distinguished [8]. To induce the “grid-friendly” behavior from network participants, a few more methods have been proposed to allocate the transmission cost according to the actual usage of the transmission facility for each network user. Existing transmission cost allocation methods fall into two categories: (1) methods based on the “average usage” information and (2) methods based on the “marginal usage” information. The term “average” means that the contribution of the network users to the transmission facilities is counted according to the current power flow snapshot; while the term “marginal” means that the contribution of the network users to the transmission facilities is calculated based on the next incremental usage of the network users. (1) Transmission Cost Allocation Based on “Average Usage” The MW-Mile method was first proposed for wheeling transactions. The costs associated with each transmission facility are allocated to transactions according to their contributions to the power flow, which are determined by the generation shift distribution factors (GSDFs) [3]. In pool-based markets, the application of the MW-Mile method requires an artificially-selected slack bus to respond to the load increase. The MW-Mile method using generalized generation/load distribution

(2) Transmission Cost Allocation Based on “Marginal Usage” In contrast to the the transmission cost allocation methods based on the “average usage” principle, allocating the transmission costs 2

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according to the “marginal usage” of the network provides the following advantages: (1) the marginal usage of the power network can be uniquely determined by the optimal power flow (OPF) solution, and hence, subjective assumptions can be avoided; (2) the marginal network usage reflects the current operating condition of the power network, which provides a better price signal to induce the rational behavior from network participants. As previously mentioned, the MW-Mile method can also be regarded as a marginal usage-based method. Some studies attempted to improve the GSDF using AC sensitivity analysis [26]. However, for the MW-Mile or AC-based methods, subjective assumptions are still involved because the variations of power injections are responded by selected reference buses. Allocating transmission costs based on the marginal network usage also has promising industrial applications. In Chile and Argentina, the marginal usage of the power network, which is measured by the “marginal participation factors (MAPF)” (known as the “areas of influence”), is used to allocate the transmission costs [27,29]. However, a slack bus needs to be selected to respond to the marginal increase of load for simplification [27]. In the PJM market, a certain percentage of the transmission costs are allocated using a factor called “DFAX” [28]. In the DFAX-based method, despite the reflection of the marginal network usage, the sensitivity factors are similar to the GSDF with artificially-selected slack buses. The slack bus selection made these methods subjective. In our proposed approach, however, the marginal usage of the network can be conveniently determined using only matrix manipulations without depending on the selection of the slack bus, so the subjective assumption is avoided.

2. The scheme of the proposed method The section reviews the basic scheme of the usage-based transmission cost allocation methods. The outline of the proposed method is also provided. 2.1. The scheme of usage-based transmission cost allocation In usage-based transmission cost allocation methods, two stages are involved: the first stage computes the contribution factor of network users to the power flows, and the second stage allocates the transmission costs of each branch to network users according to the contribution factor. The share of the transmission cost of Branch k for users at Bus i, i.e. Fi − k , can be calculated by the following equation:

Fi − k = mi − k Pi

Fk ∑ mt − k Pt

(1)

t∈N

For the MW-Mile method, the contribution factor of users at Bus i for Branch k mi − k is the GSDF or GGDF/GLDF [3]; for the power flow tracing method, mi − k refers to the topological load/generation distribution factors [6]. Then, the transmission price of Bus i can be obtained:

∑ Fi − k pi =

k∈K

Pi

=

∑ k∈K

(mi − k

Fk ) ∑ mt − k Pt t∈N

(2)

Based on (1), there are several alternative choices for transmission cost allocation which are open for discussions: 1.3. Contributions (1) Should the transmission costs be allocated to just loads or just generators? The share of the transmission costs for loads and generators is normally a regulated percentage, for which different system operators have varied choices [29]. In [13], loads and generators are considered as individual participants and not as equivalent injection/withdrawal as in conventional load flow analysis. However, there is still a debate over what is the “right percentage” that provides expected incentives. If the transmission costs are only allocated to loads, Pi in (1) refers to the power demandonly. If the transmission costs are allocated to generators and loads based on a fixed proportion, the corresponding share should be separately allocated to generators and loads. (2) How to handle the unused capacity? For the majority of transmission facilities, the power flows are typically smaller than their capacities. The difference refers to the “unused capacity”. There are some alternative methods for handling costs associated with the unused capacity. In [11], the unused capacity was allocated by the postage-stamp method. In [2], the unused capacity that is identified as “invalid” will not be allocated to network users to avoid overinvestment for vertically-integrated power industries. (3) How to handle counter flows? The counter flow refers to the contribution of the power flow by network users that is in the opposite direction of the dominant power flow. Suppose Fi'− k represents the modified share of transmission costs considering different treatments for the counter flow. There are three approaches to deal with the counter flows: giving credit to the counter flow (Fi'− k = Fi − k ), charging zero fees for the counter flow (Fi'− k =0 , if mi − k Pi × fi < 0 ), or charge the counter flow for the same rate (Fi'− k = |Fi − k|).

In both academia and industries, there still remain controversies over the best approach to fairly allocate the transmission costs to network users. Subjective assumptions are inevitably involved in determining the usage of the power network. In this paper, we propose a novel method for objectively allocating the transmission costs based on the marginal usage of the power network. The major contributions of this paper are as follows: (1) A method for determining the marginal usage of the power network is introduced. Based on the analysis of the Karush–Kuhn–Tucker (KKT) condition of the OPF model, a system of linear equations that determines the marginal usage of the power network is obtained. The contributions of the marginal increase of power injections to power flows are obtained using only matrix manipulations. According to the prevalent practice in power industries, OPF models with two formulations of losses are analyzed. The impact of loss modeling shown in [32,33] on the marginal usage of the power network is investigated. The equivalence with the GSDF-based MWMile using the distributed slack power flow model is discussed. (2) An objective method of transmission cost allocation based on the marginal usage of the power network is presented based on a general scheme for the marginal usage-based transmission cost allocation. No subjective assumptions are involved. The proposed method is computationally efficient, easy to implement, and compatible with the current market practices. Case studies using this method on several benchmark systems show the advantages of the proposed method.

There is no “ideal answer” regarding the above questions. Instead, the answers depend on the specific market policy. Different policies only influence second stage. These policies do not influence the generality of the first stage. In this paper, the following choices are made: (1) transmission costs are allocated to loads only (our proposed method can be easily extended to regulatory policy which decides to allocate

In general, this paper presents a transmission cost allocation method in accordance with the fairness requirement of the electricity markets, which provides a new approach to charging the transmission service tariff in an unbiased and equitable manner.

3

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cost to both generators and loads), (2) the costs associated with the unused capacity are treated the same, and (3) the counter flow is charged at the same rate.

(2) Linearizing Losses Using the Loss Factor In major spot markets in the US, the losses are linearized based on the loss factors, which are obtained from the sensitivity analysis of the operating condition of a base case system [32]. For such approach, fijloss is modeled as LFij (θi − θj ) + offsetij . The obtained model is denoted by L2 . MDC Note that the DC OPF model in [32] uses the system power balancing equation. By comparison, the nodal power balancing equations are used in this paper. These two formulations of OPF models are equivalent. The latter one is used here to better illustrate the proposed method.

2.2. The scheme of the proposed method The transmission cost allocation method based on the marginal use can provide a price signal to balance the power flow and allow the optimal use of transmission resources. An illustrative example is provided in Fig. 1. Generator 1 (G1) serves the marginal increase of Load 1 and Load 2. For the marginal usage-based transmission pricing methods, the obtained price signal should induce the users to reduce the power flows on branches by charging more for those who have a larger usage, which could enhance the system operating security and defer the transmission facility investment. The cost allocation for Branch 1 is investigated. If the “average usage” information is used, it may claim that the large percentage of Load 2 is served by Generator 2 (G2), and consequently, the majority of the transmission costs of Branch 1 will be allocated to Load 1. However, it can be easily observed that the increase of Load 1 and Load 2 has the same impact on Branch 1. From this prospective, to guide the users to reduce the future increase of power flows on Branch 1, Load 1 and Load 2 should be charged at the same rate for the usage of Branch 1. In this case, the transmission price provided by the marginal usage-based method could give a signal reflecting the present operating status. On the contrary, the transmission price using the “average usage” gives a distorted price signal because the transmission price for Load 1 will be smaller than that for Load 2. As a marginal usage-based transmission cost allocation method, the proposed method uses the scheme described above. Existing methods for calculating mi − k involve subjective assumptions in one way or another, which makes the transmission allocation methods “subjective”. The major contribution of the proposed method is an objective way for calculating mi − k based on the marginal network usage information that is derived. The usage of the power network is determined by the economic principles, which are well acknowledged by network participants.

(3) Other Approaches In some power systems, such as those in New Zealand and Australia, losses are modeled by a piecewise linear function [33]. This approach can also be represented by the basic model MDC . In this paper, our focus is on the prior two approaches for handling the losses. 3.2. Analysis of the KKT condition To determine the marginal usage of the power network, the KKT condition needs to be analyzed. For better clarification, model MDC is rewritten in a compact formulation as follows:

min hG (PG)

(3)

s.t. f f t t AN×G PG − PD = AN ×K f K + AN×K f K , [λK ]

(4)

f f Kf = N Kf ×N θ N + f K,0 , [πKf]

(5)

f tK = NtK×N θ N + f tK,0, [πKt]

(6)

−

f max K

⩽ fK ⩽

f max K

(7)

PGmin ⩽ PG ⩽ PGmax , [ ρ̲ G , ρ¯G ]

(8)

In (5) and (6), the expression of the coefficient matrices, namely N Kf × N and NtK × N , are modified according to the formulations of losses. f f K,0 and f tK,0 represent the constant term of the linearized losses. More details about the linearized losses can be found in [35,36]. The transmission constraint (7) is discussed. In some studies, fK is used as the average power flow at the receiving and sending end: fK = (f Kf + f tK)/2 . In this paper, we separately constrain the power flows at receiving and sending end as in MATPOWER [37].

3. Determination of the marginal usage This section discusses the methods for determining the marginal usage of the network. In accordance with the current spot market practice, the derivation is based on a general DC OPF model with losses. 3.1. Formulation of the DC OPF model

f t − f max ⩽ f Kf ⩽ f max ⩽ f tK ⩽ f max ¯Kf]; −f max ¯Kt] K K , [ σ̲ K , σ K K , [ σ̲ K , σ

The basic formulation of the OPF model is discussed first. For the purpose of the computational robustness and efficiency, system operators commonly use the DC OPF solver to determine the operation manner for power systems[30]. A general formulation of the DC OPF model, which is denoted by MDC , is presented in Appendix A. Based on the method for handling the losses, DC OPF models that are currently used in power industries can be divided into the following categories:

(9)

In the absence of demand participation in a typical spot market, the maximization of social welfare is similar to the minimization of operational cost of power suppliers. Hence, the load demand elasticities are not considered in MDC , which can be also considered as long as the OPF model is linear. Suppose the Lagrangian function of the OPF model is denoted by L. According to the KKT condition, following equations hold for the optimal solution of model MDC :

(1) Adding Losses to the Loads

∂L/ ∂θ N = (N Kf ×N)T πKf + (NtK×N)T πKt = 0 N

Some power industries, such as those in China, modify the loads according to the estimate of the total losses in the transmission network [31]. This method is easy to implement. However, the major drawback of this approach is that losses are irrelevant with the obtained generator scheduling. The DC OPF model used by these power industries can be obtained by fixing fijloss to zero in (33) and (34), and modifying Pi, D in (32). The

Fig. 1. An illustrative example.

L1 obtained DC OPF model is denoted by MDC

4

(10)

Electrical Power and Energy Systems 118 (2020) 105799

Z. Yang, et al. f f T ∂L/ ∂f Kf = −πKf + (AN ×K) λN − σK = 0 K

(11)

denoted by the following equation:

∂L/ ∂f Kf = −πKt + (AtN×K)T λN − σKt = 0 K

(12)

∂L/ ∂PG = [∂h (PG)/ ∂PG ]G×1 − (AN×G)T λN − ( ρ̲ G − ρ¯G ) = 0G

(13)

⎛ λN ⎞ ⎜ PG ⎟ R (N + G + 2K ) × (N + G + 2K ) f = bN + G + 2K + N(N + G + 2K ) ×N PD ⎜ σK ⎟ ⎜σt ⎟ ⎝ K⎠

where

σKf

f

= σ − _ K

σ¯Kf

σKt

and

t

σ¯Kt

= σ − _ K

in (11) and (12), respectively.

Combining Eqs. (10)–(12), it can be inferred that: f t T f T t MN × N λN + (N K × N) σK + (N K × N) σK = 0 N

where the coefficient matrix for variables is denoted by R (N + G + 2K) × (N + G + 2K) . On the right-hand side of (23), elements that are relevant to the power demand is extracted for the convenience of determining the marginal network usage.

(14)

where f f t t T MN×N = (AN ×K N K×N + AN×K NK×N)

(15) 3.3. Calculating the marginal usage of the power network

Eq. (14) describes the relationship between the LMPs (λN ) and the Lagrangian multipliers of the branch flow limits (σKf and σKt , also referred to as the “congestion costs”). To understand the physical meaning of MN × N , based on the definition (15), the formulation of MN × N is presented as follows [34]:

⎧ ⎪ ⎪ ⎪ MN×N :

MN×N (i, i) =

(i, j ) ∈ K 1

MN×N (i, j ) = −( x − ij

From (23), the marginal usage information can be obtained. For the marginal increase in power demand, the binding statuses of branch flow limits and generator output limits remain the same. Hence, Eq. (23) still holds. From (23), it can be deduced that:

loss 1 ∂fij ) 2 ∂θ

1

∑

(x + ij

PG = LG × (N + G + 2K ) R (N- 1+ G + 2K ) × (N + G + 2K ) [bN + G + 2K + N(N + G + 2K ) ×N PD] (24)

loss 1 ∂fij ) 2 ∂θij

⎨ , (i, j ) ∈ K loss ⎪ 1 1 ∂fij ( , ) ( ) = − + M j i ⎪ N×N 2 ∂θij xij ⎪ 0 , otherwise ⎩

where matrix LG × (N + G + 2K) = [0G × N , IG × G , 0G × 2K] extracts PG among all variables. Hence, it can be obtained that: (16)

(∂PG / ∂PD)G × N = LG × (N + G + 2K ) R (N- 1+ G + 2K ) × (N + G + 2K )N(N + G + 2K ) ×N

th

According to (16), the summation of the i column of MN × N equals zero. Hence, matrix MN × N is singular. In fact, the rank of MN × N is (N − 1) for a stable operating condition [34,35]. If the losses are added to the loads and fijloss = 0 , MN × N becomes the well-acknowledged B matrix in the DC OPF model. According to the complementary slackness condition, the elements in σKf and σKt equal zero when the corresponding branch flow limits are not binding. Hence, following equations can be obtained:

σijf

=

σ̲ ijf

−

σ¯ijf

= 0 , if −

f ijmax

<

fijf

(23)

<

f ijmax

fijf = f ijmax or − f ijmax , otherwise σijt = σ̲ ijt − σ¯ijt = 0 , if − f ijmax < fijt < f ijmax fijt = f ijmax or − f ijmax , otherwise

(25)

Eq. (25) describes the response from the generators for the marginal increase in power demands. The marginal usage of the network can be obtained by taking (25) into (19) and (20):

(∂f Kf/ ∂PD) K×N = DKf ×N [AN×G (∂PG / ∂PD)G × N − IN×N]

(26)

(∂f tK/ ∂PD) K×N = DtK×N [AN×G (∂PG / ∂PD)G × N − IN×N]

(27)

where the formulation of matrices DKf × N and DKf × N is provided in the Appendix. The difference of the absolute values of elements in f Kf and f tK originates from the losses. As one can observe, only matrix manipulations are involved during the derivation process. Because the coefficient matrices are highly sparse, efficient computational performance should be easily achieved.

, (i, j ) ∈ K (17)

, (i, j ) ∈ K (18)

where

fijf = Dijf - N (AN×G PG − PD) + fijf,0

(19)

fijt = Dtij - N (AN×G PG − PD) + fijt,0

(20)

3.4. Discussion on the marginal usage of the power network In this subsection, the comparison with the traditional GSDF methods, the discussion of the influence of losses and the costs of congested lines, and the economic principle of the proposed method are presented.

In (19) and (20), the branch flows are expressed as a function of generator outputs to eliminate voltage angles. The expressions of coefficients are provided in Appendix B. Similarly, using the complementary slackness equations, following equations for ρ¯G and ρ can be obtained:

(1) Comparison with the Traditional GSDF-Based Method

_ G

λi = ∂hG (PG)/ ∂Pg , if Pgmin < Pg < Pgmax Pg = Pgmax or Pg = Pgmin , otherwise

The GSDF-based method is the most common approach for determining the usage of the power network. Using the GSDF, the usage of the power network is similar to (26) and (27):

, g ∈ Gi , i ∈ N (21)

λN , PG , σKf ,

and σKt as Combining (14), (17), (18), and (21), regarding variables, the number of variables, i.e., (N + 2 K + G), equals the number of equations. Because the rank of matrix MN × N is (N − 1), one of the equations is redundant. The following equation that enforces the power balance is used to replace any equation in (14):

∑ Pg = ∑ Pi,D g∈G

i∈N

+

∑ (i, j ) ∈ K

fijloss

(∂fK/ ∂PD) K×N = GSDFK×N [AN×G (∂PG / ∂PD)GSlack × N − IN×N ]

(28)

Comparing (28) with (26) and (27), the most significant difference between the proposed method and the GSDF-based method lies in how to determine the response from the generators. In our proposed method, (∂PG / ∂PD) is determined by (25) with the objective of minimizing the incremental operation costs. However, in the GSDF-based methods, the value of (∂PG / ∂PD) depends on the subjective selection of the slack bus. For example, if the bus connected with generator m is chosen as the slack bus, the formulation of (∂PG / ∂PD)GSlack × N should be:

(22)

Therefore, there are (N + 2 K + G) independent equations described by (14) (removing any row), (17), (18), (21), and (22), which is 5

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(∂PG / ∂PD)GSlack ×N

1 2 1 ⎛0 ⋮ = ⋮⎜ m 1 ⎜ ⋮ ⋮ ⎜ G ⎝0

⋯ N 0 … 0⎞ ⋮ … ⋮ ⎟ 1 … 1 ⎟ ⋮ … ⋮ ⎟ 0 … 0⎠

situation, there is only one marginal generator in the OPF solution. If the slack bus for the GSDF-based MW-Mile method is set to be the marginal generator, the proposed method and the GSDF-based MWMile method is equivalent. (2) hg (Pg ) is nonlinear (normally quadratic). In this situation, the number of marginal generators is usually more than one, which means that multiple generators respond to the increase of loads. Hence, if distributed slack buses are used and the weights are the same as that described by (25), the GSDF-based MW-Mile method is equivalent to the proposed method. From the discussion above, it can L1 be inferred that if model MDC is used and there are no congestions in the network, the proposed method provides an approach to the participation factors of distributed slack buses objectively based on the marginal usage of the power network.. However, if there are congestions or the losses are explicitly modL2 eled (as in MDC ), the response from generators are different for the load increase at different buses. In this situation, the equivalency between the proposed method and the traditional MW-Mile based method does not exist.

(29)

It can be inferred that different choice of the slack bus leads to a completely different evaluation of the usage of the power network. System operators may use distributed slack buses to compensate the drawback of the GSDF. For distributed slack buses, multiple rows in (29) are nonzero. The summation of each column of (∂PG / ∂PD)GSlack ×N equals 1. The obtained network usage depends on the combinations and weights of the distributed slack buses. (2) Discussion on the Influence of Losses For the proposed method, depending on how the losses are modeled, the characteristic of the response from generators is different. If the losses are allocated to loads and are not explicitly modeled in the L1 DC OPF (as in MDC ), the response from generators for the marginal increase of loads at all buses is identical when there are no congestions. L2 However, if the losses are modeled using loss factors (as in MDC ), the generator response will be different for all buses because the marginal losses caused by a marginal increase in loads at different locations are not identical.

5. Case studies In this section, the performance of the proposed transmission cost allocation method is investigated. Following three methods are compared: (1) M_I: the MW-Mile method based on GSDF factors, which represents the prevalent marginal usage-based method using the artificially-selected slack bus (the distributed slack bus method shown in [9] is also discussed), (2) M_II: the power flow tracing method, which represents the prevalent average usage-based method, (3) M_III: the L1 proposed method using DC OPF model MDC , where losses are added to L2 , loads, and (4) M_IV: the proposed method using DC OPF model MDC where losses are linearized using loss factors. Data for all test cases are obtained from MATPOWER 4.1 [37]. The costs for transmission facilities are set to be proportional to their capacity [2].

(3) Discussion on the Costs of Congested Lines Transmission prices and energy prices together constitute the electricity price. The purpose of setting a suitable price is to induce the behavior of market participants. It should be noted that the electricity price provides the price signal, including the transmission prices and energy prices, as a whole. For congested transmission lines, the price signal of congested transmission lines given by LMP has been already embedded in the energy prices. Hence, it does not need to give repeated signals in transmission prices calculated by our method. In the proposed method, the costs of congested lines are allocated by the postage-stamp method because all the usage factors will turn out to be zeros for such lines. The price signals of congestion have already been reflected in LMP. (4) Economic Principle of the Proposed Method In the proposed method, the generators respond to the increase of loads with the objective of minimizing the operational costs. Hence, following equation should hold:

λNT = [∂h (PG)/ ∂PG ]TG×1 × (∂PG / ∂PD)G×N

(30)

Eq. (30) states that the minimum increase of system operational costs to serve the marginal increase of loads equals the LMP. It can be used to validate the obtained marginal network usage information. 4. Proposed transmission cost allocation method The proposed transmission cost allocation method based on the marginal usage of the power network can be obtained by using elements in matrix (∂f Kf/ ∂PD) K × N (or matrix (∂f tK/ ∂PD) K × N ) as the contribution factor of users at Bus i for Branch k mi − k in (1). The flowchart of the proposed approach as shown in Fig. 2. The relationship between the proposed method and the traditional MW-Mile method based on the GSDF is analyzed. According to the L1 analysis in Section 3.4, for MDC in which the losses are added to loads, if there are no congestions, the response from generators are identical for a marginal increase of loads at different locations (columns in matrix (∂PG / ∂PD)G × N are identical). Two scenarios are discussed: (1) The function of the generator production costs, i.e., hg (Pg ) is linear. In this

Fig. 2. Flowchart of the proposed approach. 6

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obtained from M_III and M_IV are unbiased. According to Fig. 4, the transmission prices are not much influenced by the modeling of losses for the PJM 5-bus system. The power flow tracing method is derived based on the proportional sharing principle. The fairness of this assumption is also hard to explain.

5.1. PJM 5-Bus system The PJM 5-bus system has 6 branches and 4 generators. The network configuration is illustrated in Fig. 3. Branch 6 is congested in the test system. (1) Investigation of the Network Usage

5.2. IEEE 118-Bus system

The response from generators for a unit increase of loads at Bus 2 is shown in Table 1. The marginal usage of the network, which is also the contribution factors of network users in (1), calculated by M_I to M_IV is illustrated in Table 2. The transmission prices of Branch 1 are provided in Table 3. For M_I, the results obtained by setting Bus 1 and Bus 5, respectively, as the slack bus are compared. The increase of loads is served by the generators at the slack bus. For setting Bus 1 as the slack bus, the generator response is the ratio of generation to the total generation at Bus 1, where two generators are located. As a result, the subjective choice of the slack bus causes a considerable difference in the evaluation of the network usage, which can be observed from Table 2. For example, for M_I, the total contribution factor of users at Bus 2 for Branch 3 is 0.1509 or −0.7371, respectively, using Bus 1 or Bus 5 as the slack bus. For M_II, the results obtained by the assumption of “the proportional sharing principle” are different from other methods. The reason for such distinctive differences can be easily understood from the loop network structure of the PJM 5-bus system illustrated by Fig. 3. For M_III and M_IV, the results are irrelevant with any subjective assumptions. The In the PJM 5-bus system, the objective function of generator outputs is quadratic and the increase of loads is served by two generators. The proportional share for different generators to serve the marginal increase of loads is determined by the coefficients of their cost functions. From Table 1, it can be observed that for M_IV, the summation of the generator responses for a unit load increase at Bus B is not equal to 1. The difference corresponds to the marginal increase in losses. The summations of the generator responses that serve a unit load increase at Bus 2, 3, and 4 are 1.0037, 1, and 0.9904, respectively. The increase of generator outputs for a marginal load at Bus 4 is smaller than 1, which indicates that the marginal increase in losses is negative. For M_III and M_IV, the Eq. (30) holds for the obtained solution, which validates that the proposed method is in accordance with the fairness preference in welfare economics.

The IEEE 118-bus system has 19 generators and 186 branches. There are five congested lines in the test system. The response from generators is different for the marginal increase of loads at different locations because of the congestion. Hence, there are no settings for the weights of the distributed slack buses that make M_I equivalent with M_III. The transmission prices obtained by M_I to M_IV are compared in Fig. 5. Distributed slack buses are used for M_I, where the weights are set according to the responses from the generators for the marginal increase of loads at Bus 2 and Bus 5, respectively. It can be observed from Fig. 5 that using the distributed slack buses, the difference between M_I and M_III becomes much smaller compared with using a single slack bus. However, subjective assumptions are still required for M_I, because the optimal responses from generators vary for the load increase at different locations of buses. In Fig. 5, two different settings for the weights of the distributed slack buses result in different transmission prices. For M_II, as an average usage-based method, the results are much more different from those from other methods. For the IEEE 118-bus system, the transmission prices obtained from M_III and M_IV are much more different because of the change in the system operating condition caused by different formulations of losses in the OPF model. It should be noted that whether M_III or M_IV should be used depends on the specific formulation of the OPF model. For example, in China, M_III should be used because the losses are modeled as L1 in MDC ; while in the US, M_IV should be used. 5.3. Other test systems Several other test systems are also investigated, including the IEEE and Polish benchmark systems. For the Polish systems, quadratic cost functions are added for all generators. Eq. (30) holds for M_III and M_IV on all test systems, which means the proposed method is in accordance with the fairness principle. The solution time on several standard test systems for obtaining the usage of the network is shown in Table 4. The solution time for M_III and M_IV are of the same magnitude as M_I and M_II. For the uncongested systems, such as the IEEE 300-bus system, M_III and M_IV are even more efficient than M_I because the structure of matrix R (N + G + 2K) × (N + G + 2K) in (23) is simpler. For congested systems, M_IV normally take a longer solution time. Even so, the computational efficiency of the proposed method is acceptable for real transmission systems with reasonable size and scope.

(2) Comparison of the Transmission Prices The transmission prices of Branch 1 calculated by M_I to M_IV are provided to compare the price signals. For M_I, it can be observed that different selection of the slack buses provide diverse price signals. Comparing the power flow tracing method (M_II) and our method (M_III and M_IV), totally different price signals are provided. For the transmission price, it should provide a price signal that prevents the overloading. The marginal usage at Bus 2, 3, and 4 for Branch 1 are 0.2188, 0, and −0.6018, respectively. The power usage at Bus 4 contributes most to the power flow on Branch 1, and hence, it should be charged with the highest transmission price. From this perspective, our method provides better price signal than M_II. The total transmission prices obtained by M_I to M_IV are shown in Fig. 4. Choosing different slack buses or methods results in totally different transmission prices. It can be observed from Fig. 4 that if Bus 1 is chosen as the slack bus, loads at Bus 2 are much closer to the power source than those at Bus 3. Hence, the “extent of use” of the power network by users at Bus 2 is much smaller than that by users at Bus 3, which explains the relationship of the transmission prices at buses 2 and 3 obtained from M_I by choosing Bus 1 as the slack bus. However, In M_III, the conclusion will be reversed. In spot markets, the principle of fairness is challenged in light of considering the different results caused by the subjective selection of the slack bus. By comparison, the results

Fig. 3. Network configuration of the PJM 5-bus system. 7

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network can be conveniently determined by the analysis of the marketclearing model, based on which an objective transmission cost allocation scheme can be designed. The influence of the loss modeling on the proposed method is discussed. Only matrix manipulations are involved for the proposed method. This paper provides a new transmission cost allocation method without involving any subjective assumptions in spot market. As the proposed method is computationally efficient, easy to apply, and in accordance with the current industry practice, the application of the proposed method in practical power systems warrants further explorations.

Table 1 Generator response for a unit load increase at Bus 2 in the PJM 5-Bus. Generator index

M_I

M_II

M_III

M_IV

1: slack bus

5: slack bus

1 2 3 4 5

0.1905 0.8095 0 0 0

0 0 0 0 1

0.0976 0.4149 0.1678 0 0.3197

0 0 0.8192 0 0.1808

0 0 0.8212 0 0.1825

Sum

1

1

1

1

1.0037

Table 2 Contribution factors by users at Bus 2 in the PJM 5-Bus. Branch index

1 2 3 4 5 6

M_I 1: slack bus

5: slack bus

0.6698 0.1792 0.1509 −0.3302 −0.3302 0.1509

0.6354 0.1017 −0.7371 −0.3646 −0.3646 −0.2629

CRediT authorship contribution statement

M_II

M_III

M_IV

0.2719 0.2043 0.1203 0.0092 0.0001 0.1279

0.2188 −0.0381 −0.1808 −0.7812 0.0381 0

0.2206 0.0386 −0.1820 −0.7824 0.0381 0

Zhifang Yang: Conceptualization, Methodology, Software, Validation, Formal analysis, Writing - original draft, Writing - review & editing. Xingyu Lei: Methodology, Software, Validation, Writing - review & editing. Juan Yu: Validation, Resources, Supervision. Jeremy Lin: Formal analysis, Writing - review & editing.

Declaration of Competing Interest The authors declared that there is no conflict of interest.

Table 3 Transmission prices of Branch 1 in the PJM 5-Bus. Bus index

2 3 4

Marginal usage of Branch 1

M_I

M_II

1: slack bus

5: slack bus

0.2188 0 −0.6018

0.0334 0.0271 0.0097

0.0343 0.0275 0.0086

M_III

Acknowledgment

M_IV

This work was supported by Natural Science Foundation of China (No. 51807014). 0.0420 0 0.0228

0.0157 0 0.0432

0.0157 0 0.0432

6. Conclusions In this paper, we found that the marginal usage of the power

Fig. 4. Comparison of the transmission prices for the PJM 5-bus system. 8

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Fig. 5. Comparison of the transmission prices in the IEEE 118-bus system. Table 4 Solution time for calculating the network usage. Test cases

Solution time (Seconds)

IEEE 300-bus Polish 2736-bus Polish 3120-bus a

M_Ia

M_II

M_III

M_IV

0.0389 1.5380 2.0263

0.0143 1.6479 2.6927

0.0172 1.0176 1.3289

0.0335 1.5103 2.3977

Slack bus provided by MATPOWER is used.

Appendix A. General formulation of DC OPF model

min

∑ hg (Pg ) (31)

g∈G

s.t.

∑ g ∈ Gi

Pg − Pi, D =

fijf −

∑ (i, j ) ∈ K

fijf = (θi − θj )/ x ij +

∑

fijt , [λi], i ∈ N

(32)

(j, i) ∈ K

1 loss f , [πijf ], (i, j ) ∈ K 2 ij

(33)

fijt = −(θi − θj )/ x ij +

1 loss f , [πijt], (i, j ) ∈ K 2 ij

(34)

− f ijmax ⩽ fijf ⩽ f ijmax ,

[ σ̲ ijf ,

(35)

σ¯ijf

] , (i, j ) ∈ K

− f ijmax ⩽ fijt ⩽ f ijmax , [ σ̲ ijt, σ¯ijt] , (i, j ) ∈ K

(36)

Pgmin ⩽ Pg ⩽ Pgmax , [ ρ̲ g , ρ¯g ], g ∈ G

(37)

The objective of the OPF model is to minimize the operational costs, which is normally described as the production costs of generators. Eq. (32) represents the power balancing equations. By definition, the Lagrangian multiplier λi is the LMP at bus i. Eqs. (33) and (34) are the expressions for branch flows, where losses are considered. Constraints (35) and (36) refer to the thermal limits of branch flows. Constraint (37) represents the generator output limitations. Appendix B. Deduction for branch flow expression The deduction for (19) and matrix DKf × N in (26) is taken as an example. By substituting (5) and (6) into (4), following equation can be obtained: f f f t t t AN×G PG − PD = AN ×K (N K×N θ N + f K,0) + AN×K (NK×N θ N + f K,0)

=MTN×N θ N + P0N

(38) 9

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where t f f t P0N = AN ×K f K,0 + AN×K f K,0

According to Section 3.2, the rank of matrix ¯ TN × N as follows: matrix M

(39)

MTN × N

is (N − 1). Without loss of generality, assuming the voltage angle of bus n is zero, define

MT 0 ¯ TN×N = ⎛ (N - 1) × (N - 1) N - 1⎞ M ⎜ T 0 1 ⎟⎠N×N N 1 ⎝

(40)

Hence, it can be obtained that

¯ TN×N)−1 (AN×G PG − PD) − (M ¯ TN×N)−1P0N θ N = (M

(41)

By substituting (41) into (5), it can be deduced that f ¯ TN×N)−1 (AN×G PG − PD) + f K,0 ¯ TN×N)−1P0N f Kf = N Kf ×N (M − N Kf ×N (M

(42)

f −1 ¯T The expression of (19) can be obtained by extracting the corresponding row from (42). Matrix DKf × N in (26) equals N N × N (MN × N)

Appendix C. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.ijepes.2019.105799.

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