Assessment of the collusion possibility and profitability in the electricity market: A new analytical approach

Electrical Power and Energy Systems 112 (2019) 381–392

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

Assessment of the collusion possibility and profitability in the electricity market: A new analytical approach

T

⁎

Mahdi Samadi , Mohammad Ebrahim Hajiabadi Department of Electrical Engineering, Hakim Sabzevari University, Sabzevar, Iran

A R T I C LE I N FO

A B S T R A C T

Keywords: Electricity market analysis Collusion LMP Jacobian Literature survey

This paper proposes an analytical approach for evaluating the potential of collusion in the electricity market, based on the Jacobian matrix of GenCos’ profit. To develop the proposed approach, two lemmas are presented with their proof. In the first lemma, a general quadratic programming problem is decomposed. This lemma is employed to decompose the market variables, which are provided by solving the optimal power flow (OPF) problem. The second one is used to calculate the Jacobian matrix, analytically. Finally, four indices are introduced to evaluate the collusion possibility and profitability. These indices quantify both financial and capacity withholding of the GenCos. The proposed approach allows market regulator to assess the degree of competition in the electricity market. The simulation results on the IEEE 24-bus and 300-bus test systems demonstrate the efficiency of the proposed approach.

1. Introduction

researches available on the market power.

1.1. Motivation and aim

1.2. Literature review

The principal goal of regulatory in the electricity market is to provide a perfect competitive environment [1]. Collusion is considered as a significant threat to a competitive market. Collusion is defined as any tacit or explicit agreement among Generation Companies (GenCos) to increase the profit by changing bids or reducing output [2]. The regulatory body should be able to assess the market and behavior of the players to identify the collusion possibility [3]. This helps the regulator to make preventive decisions to avoid market abuse. To this purpose, the magnitude and extent of collusion must be found. On the other hand, the ability of the GenCos to exert market power or collusive behavior depends on the generation and transmission constraints as well as the cost function of the units [4]. However, In general, recognizing the collusive behavior is not easy for the Independent System Operator (ISO) as there may be no explicit agreement between the market players [2]. Based on collusion definition, increasing the profit of each GenCo owing to a strategic action of another one demonstrates the potential of collusion. Now, a fundamental question arises: How does the profit change as market parameters change? The main motivation of the paper is to investigate this question. To the best knowledge of the authors, no major study has yet investigated the collusion possibility by an analytical method, compared to many analytical

The collusion identification and market power evaluation are closely related. Therefore, some works on these two topics are reviewed here. Market power is generally evaluated by some indices. These indices may be quantity-based (such as HHI, pivotal supplier index and residual supply index [5]) or price-based (such as output gap and Lerner index [6]). Generation companies with a great share of the market or located at a strategic bus of the network, are able to exercise market power through strategic behavior [7]. Strategic behavior is an action taken by a market player to increase their profit [3]. This action can be applied in two ways: capacity withholding (i.e., decreasing the power lower than the available capacity) and financial withholding (i.e., raising the bid price higher than the marginal cost) [8]. Usually, electricity producers can employ market power via capacity withholding in the transmission-constrained markets with inelastic demand and LMP-based pricing [9]. There are many studies about capacity withholding in the literature. The authors in [3], studied the strategic capacity withholding using an empirical analysis of German-Austrian electricity market. In [8], capacity withholding was analyzed in a day-ahead market using bi-level optimization. In [10,11], some indices were proposed to evaluate the capacity withholding of GenCos in an electricity market. However, the

⁎

Corresponding author. E-mail address: [email protected] (M. Samadi).

https://doi.org/10.1016/j.ijepes.2019.05.010 Received 12 February 2019; Received in revised form 17 April 2019; Accepted 5 May 2019 Available online 15 May 2019 0142-0615/ © 2019 Elsevier Ltd. All rights reserved.

Electrical Power and Energy Systems 112 (2019) 381–392

M. Samadi and M.E. Hajiabadi

A summary of the reviewed works is presented in Table 1. This table shows a category-based comparison between available methods and the proposed approach of this paper. The available approaches in the literatures have studied the collusion or market power, which are closely related. The collusion or market power can be evaluated using structural indices or behavioral indices. Structural indices are used to detect the potential of market power of GenCos. Behavioral indices estimate actual value of market power. The proposed approach in this paper is analytical and structural against other works, which are simulationbased and behavioral.

transmission constraints have not been considered in this model. The same authors proposed an enhancement of the aforementioned approach where the capacity withholding of GenCos is evaluated in the presence of network constraints [9]. There are some of works which have focused on the collusion of market players [12–15]. In [14] an agent-based model was introduced to study the market performance against capacity withholding as well as the collusion among the participants of market. In this study, a reinforcement-learning algorithm was utilized to model the interaction between agents. In [16], the effect of multi-market contact on the collusive behavior was investigated, but the impact of transmission constraints were neglected in GenCos’ strategies. In [17], an equilibrium-based model was proposed to analyze collusive behavior in the electricity market. Based on the findings of simulation-based models in [18], repetitive bargaining can result in tacit collusion. In [19], the tacit collusion in electricity market of Spain was analyzed. In [1], a model was proposed to assess the market power in electricity markets in presence of renewable energy resources. In this model, both explicit and tacit collusions have also been investigated. A particle swarm optimization algorithm was applied for updating the offering strategy of the market participants. Esmaeili et al., in [2], proposed a game-theoretic model to determine the conditions where generation companies may have a collusive behavior. They defined two collusive states (namely strong and weak) and used a bi-level optimization problem since the objective of the ISO is conflicted by the objective of GenCos. The authors in [20] proposed a virtual power market to investigate the explicit collusion from regulator's viewpoint. In such model, players participate in several coalitions to increase their benefits. In [21], the collusion in the electricity market was studied by considering the demand shock. In this regard, the effect of the demand level on the collusion profit was analyzed. In [22], the effect of regulatory intervention on capacity withholding and tacit collusion was examined from the non-pivotal firms’ point of view. The simulation results showed that the non-pivotal firms may have collusive behavior. In [23], the potential of strategic and collusive behavior of market players was investigated using an agent-based game-theoretic model. In [24], a multi-agent optimization model was proposed to study the possibility of tacit collusion. In this model, each market participant sets the production share to maximize its profit. The authors in [25] evaluated the impact of some characteristics of the market structure on the collusion by determining the joint profit. Although, some attributes that facilitate collusion are investigated in the literature, there is no work that has employed an analytical approach.

1.3. Approach and contributions The main goal of this paper is to evaluate collusion possibility and profitability based on a strong theoretical approach. Accordingly, we introduce the Jacobian matrix of GenCos’ profit to propose an analytical approach for evaluating the potential of collusion. It must be noted that, in vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. Hence, for each point, the Jacobian provides the best linear approximation of the function deviation around that point. The profit of each GenCo is a nonlinear function of the market variables. Therefore, in this paper, the GenCos’ profit is represented in vector form and the Jacobian concept is used to propose some useful indices for collusion assessment. To achieve this, the following steps are performed: (a) Mathematical development (in Section 2) (a.1) Formulation of full decomposition for a general Quadratic Programming (QP) problem is presented and proved in Lemma 1. (a.2) An analytical calculation of Jacobian matrix is proposed in Lemma 2. (b) Electricity market decomposition (b.1) Structural decomposition of optimal power flow output, particularly power generation and Locational Marginal Price (LMP), is performed based on Lemma 1. (b.2) Incremental profits of units are decomposed to effective factors using Lemmas 1 & 2. Therefore, the Jacobian matrix of profit is determined analytically.

Table 1 Literature Categorization. Ref no.

Subject

Detection criteria

Detection method

Structural or Behavioral

Considering Transmission constraints

Capacity withholding or Financial withholding

Proposing new index

[1]

profit-based

Simulation

Behavioral

✓

Financial withholding

✓

profit-based quantity-based quantity-based price-based quantity-based profit-based profit-based profit-based profit-based profit-based profit-based profit-based

Simulation Simulation Simulation Simulation Analytical Simulation Simulation Simulation Simulation Simulation Simulation Simulation

Behavioral Structural Structural Behavioral Behavioral Behavioral Behavioral Behavioral Behavioral Behavioral Behavioral Behavioral

✓ × ✓ × ✓ ✓ ✓ ✓ × ✓ × ×

Financial withholding Capacity withholding Both Capacity withholding Capacity withholding Both Financial withholding Financial withholding Both Financial withholding Financial withholding Financial withholding

× × ✓ ✓ ✓ × × × × × ✓ ✓

[27] [37]

Collusion & Market Power Collusion Market Power Market Power Market Power Market Power Collusion Collusion Collusion Collusion Collusion Collusion Collusion & Market Power Market Power Market Power

Analytical Simulation

Structural Both

✓ ×

Both Both

× ×

Our Method

Collusion

price-based price-based & quantity-based profit-based

Analytical

Structural

✓

Both

✓

[2] [3] [5] [8] [9] [14] [20] [21] [22] [23] [24] [25]

382

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⎡ Xmin ⎤ X = ⎢ Xmax ⎥, ⎢ Xmrg ⎥ ⎣ ⎦

⎡ LB min ⎤ LB=⎢ LB max ⎥, ⎢ LB mrg ⎥ ⎣ ⎦

UB

⎡ UB min ⎤ =⎢ UB max ⎥, ⎢ UB mrg ⎥ ⎣ ⎦

Γ−

+

−

⎡ Γ min ⎤ − = ⎢ Γ max ⎥, ⎢ Γ− ⎥ ⎣ mrg ⎦

Γ+

⎡ Γ min ⎤ = ⎢ Γ+max ⎥, ⎢ + ⎥ Γ ⎢ ⎣ mrg ⎥ ⎦

⎡ Fmin ⎤ F = ⎢ Fmax ⎥ ⎢ Fmrg ⎥ ⎣ ⎦

Fig. 1. Framework of the proposed approach.

(2)

An inequality constraint gi (X ) ≤ 0 is known as active at the optimal point X* if gi (X*) = 0 and as passive if gi (X*) < 0. In this research, active and passive inequalities are indicated by ‘act’ and ‘pas’ superscript, respectively, in the formulations. Considering this concept, the matrix A and the vectors B and Ω are also divided as (3). *

(c) Proposing four indices to evaluate collusion possibility using the Jacobian matrix elements in two categories: (c.1) Collusion indices based on the capacity withholding (c.2) Collusion indices based on the financial withholding Fig. 1 shows the framework of the proposed approach. In brief, changes in power generations and LMPs with respect to input parameters of OPF (i.e. unit parameters, network parameters and demands) are determined by Lemma 1. In addition, decomposition of Jacobian matrix of unit profits is clarified in Lemma 2. This approach is general and can be applied to any large power grid. It must be mentioned that some researches have been performed in the context of LMP decomposition ([26–30]). However, to the best of the authors' knowledge, the analytical decomposition of Jacobian matrix for collusion evaluation are specific to this paper and have not been used in any previous works.

act act Ωact ≠ 0 ⎤ A = ⎡ Apas ⎤, B=⎡ Bpas ⎤, Ω = ⎡ pas ⎣A ⎦ ⎣B ⎦ ⎣Ω = 0 ⎦

Aeq =

Λ

⎡ Fmrg ⎤ ⎡ Xmrg ⎤ ⎡ S11 S12 S13 S14 S15 ⎤ ⎢ LB min ⎥ ⎢ Λ ⎥ = ⎢ S 21 S 22 S 23 S 24 S 25 ⎥ ⎢ UB max ⎥ ⎢ Ωact ⎥ ⎢ S 31 S 32 S 33 S 34 S 35 ⎥ ⎢ Beq ⎥ ⎣ ⎦ ⎣ ⎥ ⎦⎢ act ⎥ ⎢ ⎦(n+ m+ k act) × 1 ⎣ B

LB ⩽ X ⩽ UB : (Γ−, Γ+)

(1d)

(6)

This decomposition form is used to determine the coefficients matrix of the electricity market variables. 2.2. Decomposition of Jacobian matrix The profit of each GenCo is a nonlinear function of the market variables. Therefore, the variation of GenCos’ profit, with respect to all variables may be shown by a Jacobian matrix form. In this paper, the Jacobian concept is used to develop new indices for collusion assessment. In this subsection, we attempt to decompose a special case of a nonlinear matrix function, using the result of Lemma 1. To this, we use “Hadamard product” definition between two matrices, denoted by A ⊙ B and corresponding to the element-wise multiplication between two matrices. i.e.: (A ⊙ B)ij = aij.bij [31,32]. Lemma 2 expresses the decomposition of the Jacobian matrix of a nonlinear matrix function if it is composed of two parts as (7).

(1b) (1c)

(5)

Proof:. The proof is provided in Appendix A.

(1a)

A. X ⩽ B : Ω

(4)

Lemma 1:. “For the QP problem (1) with Lagrangian function (5), the output of the optimization including marginal variables and Lagrangian multipliers can be decomposed into structural factors of QP problem as the matrix form (6):”

Subject to

X=

act A act max A mrg ]

For the QP problem (1) with the Lagrange equation (5), Lemma 1 expresses the decomposition of the optimization outputs into five type components, deduced from solving the KKT conditions at the optimal point.

The electricity market-clearing problem can be modeled by a QP structure (more details are provided in Section 3). Therefore, initially the decomposition of the QP is investigated. The general form of a QP problem including equality constraints, inequality constraints, lower/ upper bounds and the Lagrangian multipliers, is shown in (1a)–(1d).

Beq :

Aact =

+ (Ωact )T . (Aact . X − Bact ) + (Γ−min)T . (LB min − Xmin) + (Γ+max )T . (Xmax − UB max )

2.1. Decomposition of QP variables

Aeq .

, we have:

[ A act min

1

2. The mathematical development

1 T X . H. X + F T . X 2

A eq mrg ],

L(X,Λ, Ω, Γ) = 2 X T. H. X + F T . X + ΛT . (Aeq . X − Beq)

The rest of this paper is organized as follows: The mathematical developments including lemmas 1 and 2 are formulated in Section 2. The electricity market decomposition is explained in Section 3. In Section 4, the proposed indices are introduced. Simulation results are represented in Section 5. Finally, the paper is concluded in Section 6.

X

A eq max

act

Knowing that the passive constraints have null multipliers, the Lagrangian function of the optimization problem (1) is defined as (5):

1.4. Paper organization

Min

and A

Similarly, for the matrixes A eq [ A min

(3)

eq

After solving the QP problem and determining the values of variables and Lagrangian multipliers, decomposition of variables into the constructive components is performed. At the optimal point, Xmin and Xmax indicate the variables that are set to their lower and upper bounds, respectively. The others are specified as marginal variables and denoted by Xmrg. Therefore all vectors including X, LB, UB, Γ−, Γ+ and F can be divided into three sub-vectors as (2).

Lemma 2:. “Let vector U be a non-linear function of the variable vector Q 383

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M. Samadi and M.E. Hajiabadi

1 ̂ bi . Pgi2 2

as (7):

Costi = aî. Pgi +

[U] = [ K . Q ] ⊙ [ W . Q ] m×n n×1 m×n n×1 ⏟ m×1 ⏟ ⏟ m×1 m×1

where aî and bi ̂ are the coefficients of the function. So, the marginal cost

(7)

(12)

of unit i (MCi) is equal to:

Then, the Jacobian matrix of U, which is a function of the coefficients matrices K and W, as well as the Q, is calculated as (8a)–(8d):”

MCi = aî + bi .̂ Pgi

dU = (2(K ⊙ W). Diag(Q) + Y ) [J (K, W, Q)] = dQ m⏟ ×n m⏟ ×n m⏟ ×n n⏟ ×n

It is supposed that the GenCos submit their bids to the ISO in a linear function form as (14).

(8a)

bid i = ai + bi . Pgi

where:

Y = [ Y1 Y2 ⋯ Yi ⋯ Yn ] n

∑ j=1 j≠i

{( Ki ⊙ Wj + K j ⊙ Wi) ⊙ Qj } ⏟ ⏟ ⏟ ⏟ ⏟ m×1 m×1 m×1 m×1 m×1

bi = bi ̂ & ai = aî + ζi (8c)

(15)

The ξi is the level of deviation of bidi from MCi. The market problem under DC power flow constraints can be expressed as (16), In which: PG/PD/PL: The vector of power generation/demand/power line. MG/ML: Bus-unit / Bus-branch incidence matrix.

Ki = i th column of [K] , Wj = j th column of [W] , Qj = (qj). ones(m, 1) (8d) In summary:

Ng

(

1

Min ∑i = 1 ai . Pgi + 2 bi . Pgi2

[J (K, W, Q)] m⏟ ×n

PG, PL

(

n

)

+ [⋯ ∑ j ≠ i {(Ki ⊙ Wj + K j ⊙ Wi ) ⊙ Qj} ⋯]

PGMin ⩽ PG ⩽ PGMax (8e)

To clarify the application of the Lemma 2, suppose U is an m-dimensional vector as (9).

KaPg KmnPg KmxPg KdPg KlPg ⎤ ⎡ PG ⎤ = ⎡ . ⎥ Ka Kmn Kmx Kd Kl ⎣ LMP ⎦ ⎢ LMP LMP LMP LMP ⎦ ⎣ LMP

(10)

Then, using “Hadamard product”, U can be written as (7). Using Lemma 2 the sensitivity of each element of U with respect to each variable can be analytically calculated. For example, the element of row i and column j indicates:

dui , ∀ i = 1,2,. ..,m & ∀ j = 1,2,. ..,n dqj

− PLMax ⩽ PL ⩽ PLMax

(9)

If each element of U is a specific nonlinear multivariate function with respect to variables q1, q2, …, qn as follow:

ui = (ki1. q1 + ...+kin. qn ). (wi1. q1 + ...+win. qn )

&

(16)

Note that all demands are assumed inelastic. Using Lemma 1, the vectors of units’ generation, PG, and LMP can be decomposed to the market parameters as (17). These parameters include: bid of marginal units, the minimum/maximum generation of high/low cost units, loads and capacity of congested lines.

Proof:. Proof is provided in Appendix B.

U = [u1 u2 … um ]T

)

MG .PG − ML .PL =PD : LMP −1 =0 ∀l Pl − (θa − θb) xab

= 2(K ⊙ W). Diag(Q)

Ji, j =

(14)

Similar to [27,28,36], it is assumed that GenCos change their bidding strategies only by manipulating the intercept ai and bi remains constant. Therefore, it can be assumed:

(8b)

m×n

Yi = ⏟ m×1

(13)

a ⎡ mrg ⎤ ⎢ PGMin ⎥ ⎢ Max ⎥ ⎢ PG ⎥ ⎢ PD ⎥ ⎢ Max ⎥ ⎢ ⎣ PL ⎥ ⎦ (17)

It must be noted that LMPs and power generation of units are provided by the OPF problem (16) at the specified load level of the network. Therefore, the congested lines, fully dispatched units, marginal units and units with minimum generation are determined at the market equilibrium point. Then, the matrixes of coefficients (for LMP and PG) are calculated through Lemma 1. Afterwards, using Lemma 2, the incremental profit is decomposed to the main constructive factors. Thus, the proposed decomposition requires the results of auction problem (16) as input data.

(11)

In the next section, first, the electricity market problem is formulated and the market variables are decomposed using Lemma 1. Then, it is explained how to calculate the elements of the Jacobian matrix of profit using Lemmas 1 and 2. Lemma 2 is employed to determine the change in profit of each unit i due to the action of unit j, exactly.

3.2. Decomposition of incremental unit profit

3. Electricity market decomposition

To decompose the incremental profit, initially the profit function must be determined. For obtaining this, the revenue of each unit should be calculated. The revenue of unit i is:

3.1. Decomposition of LMP and power generation

Revenuei = (Pgi . LMPi )

There are two approaches that can be employed by an ISO to determine the locational marginal price, namely AC-OPF and DC-OPF [33]. Because of non-linearity and non-convexity of AC-OPF [3] and its computational burdens [34], the linear model, i.e., the DC-OPF, is currently preferred in the literature [2,33]. The results of [35] also showed that a DC-OPF is reasonable for market power evaluation in a transmission-constrained market. We also use the DC-OPF formulation as it common in the literature [2, 7, 9, 14, 27, 28 and 33]. DC-OPF problem is implemented by the ISO to minimize the generation cost (maximizing the social welfare) [26–28]. The cost function of the generating units is considered quadratic as (12):

(18)

LMPi in (18) is the electricity price at the bus where the unit i is located at. The profit of unit i (i.e. revenue minus production cost [9]) is calculated as (19).

1 Profiti = (Pgi . LMPi ) − ⎛aî. Pgi + bi .̂ Pgi2⎞ 2 ⎝ ⎠

(19)

Since the main goal of each GenCo is to increase its profit [1], the GenCo’s profit is used to evaluate the potential of collusion. Increasing the profit of each unit due to the strategy of other units, can indicate the collusion possibility and profitability. Now, an important question 384

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M. Samadi and M.E. Hajiabadi

∀ i = 1, 2, ...,Ng

arises: How will GenCo’s profit change as the input parameters change? To answer this question, the vector of all units’ profits are defined as (20).

⎡ Profit1 ⎤ [PR] = ⎢ ⋮ ⎥ ⎢ ProfitNg ⎥ ⏟× 1 Ng ⎦ ⎣

Ngmrg

ΔProfit i = ∑ j

Ngmax

+ ∑j Nb

(20)

(21)

Nq = (Ngmrg + Ngmin + Ngmax ) + Nb + Nlcong = Ng + Nb + Nlcong (22) To form the profit vector the LMP vector of the units, ULMP, must be calculated using bus-unit incidence matrix, MG, as (23): (23)

By substituting LMP from (17) into (23), we have:

[ULMP] = [W] . [Q] ⏟× 1 ⏟× 1 Ng Ng⏟ × Nq Nq

(24)

Therefore, the revenue vector can be written as (25) using the Hadamrd product definition:

[Revenue](Ng × 1) = [PG ] ⊙ [ULMP] = (K. Q) ⊙ (W. Q)

Nlcong

Cl (i, l). dPllmax

(30)

Therefore, the sensitivity of the incremental profit to each input parameter is evaluated. The interpretation of each coefficient in (30) is as follow: Ca(i,j): The change in incremental profit of unit i according to the variation in bid of marginal unit j. Cmn(i,j): The change in incremental profit of unit i according to the variation of the minimum generation capacity of high cost unit j. Cmx(i,j): The change in incremental profit of unit i according to the variation of the maximum generation capacity of low cost unit j. Cd(i,n): The change in incremental profit of unit i according to the variation of the load of bus n. Cl(i,l): The change in incremental profit of unit i according to the variation of the capacity of congested line l. These coefficients, determine the contribution of each factor to the profit in a transparent, fast, and precise manner. This is an important feature of the proposed approach as a powerful tool to specify the key factors affecting the profit of each unit. The Jacobin Matrix C is the basis for defining analytical indices for collusion evaluation.

The dimension of Q is:

[ULMP] = [MG]T . [LMP] ⏟× 1 ⏟× 1 Ng × Nb Ng⏟ Nb

Cmn (i , j ). dPg jmin

Cmx (i, j ). dPg jmax

+ ∑n Cd (i, n). dPdn + ∑l

The profit is a nonlinear function of LMP and power generation. By the way, according to (17), the vector of PG, can be rewritten as (21):

[PG ] = [K] [Q] , QT ≜ [ amrg PGMin PGMax PD PLMax ] ⏟× 1 ⏟× 1 Ng Ng⏟ × Nc Nq

Ngmin

Ca (i , j ). daj + ∑ j

4. The proposed collusion indices In this section, the Jacobian matrix is used and new indices are proposed to evaluate the collusion possibility based on the variation of incremental profit of GenCos. The aim of collusion is to increase profit by changing the offering [1,20]. Producers can collude together in two manners: capacity withholding and financial withholding. In capacity (or physical) withholding supplier reduces its production. The financial withholding is applied by raising the bid price [37], however, it will be shown that the profit can increase by decreasing the bid. The collusion between units i and j is possible if the total incremental profit of them due to the action of one of them be positive (i.e. ΔProfiti + ΔProfitj > 0). For clarity, it can be said that there are four cases for two units i and j because of the action of unit j as shown in Table 2. The action of unit j can be one of the following:

(25)

Moreover, the Hadamard product definition can be employed to express the units profit vector as (26).

1 [PR] = ([PG ] ⊙ [ULMP]) − ⎛ [â ] ⊙ [PG] + [b̂] ⊙ [PG ] ⊙ [PG] ⎞ 2 ⎠ ⎝ ⏟× 1 Ng (26) Considering the structural decomposition of ULMP and PG in (24) and (21), the profit vector in (26) can be rewritten as (27), using Lemma 1.

[PR] 1 = (K. Q) ⊙ (W. Q) − ⎛ [â ] ⊙ (K. Q) + [b̂] ⊙ (K. Q) ⊙ (K. Q) ⎞ 2 ⎠ ⎝ (27)

▪ Reduction in capacity if the unit j is a low-cost unit. ▪ Increasing or decreasing the bid if unit j is a marginal unit.

The coefficients which describe the linear relationship between the incremental profit and input parameters can be calculated using Lemma 2. Accordingly, the matrix C in (28) is defined as the Jacobin matrix of profit vector. Considering (27) and Lemma 2, the jacobian matrix C can be derived as (29).

These deductions are general and can be used for evaluating both capacity withholding and financial withholding. In this paper, the first row of Table 2 is named as “High Collusion” and the next case is named as “Low Collusion”. The total added profit obtained from collusion will be divided between i and j. As mentioned before, we use the incremental profit of GenCos to determine the potential of collusion. To this, the Jacobian matrix can be an effective instrument. Therefore, Matrix C in (29) is used to propose collusion indices. Corresponding to the five types of market parameters in Q, the jacobian matrix C can be divided into five parts as (31):

⎡ damrg ⎤ ⎢ dP Min ⎥ ⎢ G ⎥ [dPR] = [C] . [dQ] ⇒ [dPR] = [Ca Cmn Cmx Cd Cl ]. ⎢ dPGMax ⎥ ⎢ ⎥ ⏟× 1 ⏟× 1 Ng Ng⏟ × Nq Nq C] [⏟ ⎢ dPD ⎥ Max ⎢ dPL ⎥ ⎣ ⎦ (28)

Ca [C]Ng × Nq = ⎡ Ng ×⏟ Ngmrg ⎣

1 [C] = J (K, W, Q) − (â n × 1. I1×n) ⊙ K − ( b̂n × 1. I1×n) ⊙ J (K, K, Q) 2 (29)

Cmn

Ng ×⏟ Ngmin

Cmx

Ng ×⏟ Ngmax

Cd

Ng⏟ × Nb

Cl

Ng ×⏟ Nlcong

⎤ ⎦

(31)

The Ca and Cmx in (30) are the ones relating to amrg and PGMax, respectively. Therefore Matrices Ca and Cmx could be useful instruments to analytically evaluate the collusion, based on financial and capacity withholding, respectively. As an example, Cai,j shows the change in profit of unit i due to change of the bid of marginal unit j, i.e. Cai, j = ΔProfit i/Δaj . Similarly, Cmxi,j shows the change in profit of unit i

The computational complexity for calculating the elements of matrix C is moderate even for large scale system. For clarity, dPRi which is the i-th row of (28), indicates the incremental profit of unit i as is rewriten in a summation form as (30). 385

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Table 2 Different possible cases for action of unit j. Profiti due to action of j

Profitj due to action of j

(Profiti + Profitj) due to action of j

Corollary

1

increasing

increasing

increasing

High Collusion (This action is profitable for each unit i & j)

2

increasing

decreasing

increasing

Low Collusion (This action is not profitable for unit j but sum of their profits is increased) No Collusion (This action is not profitable for unit j and sum of their profits is decreased)

decreasing 3

decreasing

increasing

Not important

No Collusion (This action is not profitable for unit i.Unit j have market power)

4

decreasing

decreasing

decreasing

No Collusion (This action is never profitable for each i & j)

Table 3 The sign of profit changes of units i & j, due to the capacity withholding of unit j.

1 2 3 4

Cmxi,j

Cmxj,j

(Cmxi,j + Cmxj,j)

Collusion possibility

+ − − −

Not important + − +

Not important + − −

No Collusion No Collusion High Collusion Low Collusion

due to change of the capacity Cmx i, j = Δ(Profit i)/Δ(Pg Max ). j

of

low

cost

unit

j,

Table 4 The status of profit changes of units i & j, due to the financial withholding of unit j.

1 2 3

HCIibid ⎧ , j (Cai, j × Caj, j ) > 0 LCIibid = ,j ⎨|Cai, j + Caj, j| (Cai, j × Caj, j ) < 0 & |Cai, j | > |Caj, j | 0 else ⎩

(32)

else

(34)

i,j and Caj,j is different, i.e. with the change of the bid of marginal unit j, the profit of one unit will increase and the other will decrease. o Row 2: The profit variation of the unit i is higher than the profit variation of the unit j, by bid variation of unit j (|Cai,j| > |Caj,j|) Thus, there is a potential for a weak collusion. Therefore, the new collusion index LCIibid , j , is proposed as (35). o Row 3: in this, there is no potential for collusion.

• Row 4: The collusion with low confidence, ‘Low Collusion’, can be

(35)

It must be noted that the matrices Ca and Cmx are not symmetric. Therefore the collusion index of unit i due to action of unit j, is not necessarily the same as the collusion index of unit j due to action of unit i. The proposed indices can be used by regulators for collusion assessment in transmission-constrained electricity markets.

formed if the sum of the profits of units i and j is positive, while the profit of unit j is negative. In this state, unit j can gain the portion of benefit of unit i, based on collusion, despite own disadvantage. Therefore, the new collusion indexLCIicap , j , is proposed in (33).

5. Simulation results

(Cmx i, j + Cmx j, j ) < 0 & Cmx i, j < 0 else

5.1. Case study 1

(33) It is obvious that, the

High Collusion Low Collusion No Collusion

• Row 2 & 3: The sign of Ca

Cmx i, j < 0 & Cmx j, j ⩽ 0

HCIicap ,j

+ − −

⎧|Cai, j + Caj, j| (Cai, j × Caj, j ) > 0 HCIibid ,j = ⎨ 0 else ⎩

from this action. Therefore, there is no potential for collusion formation, if Cmxi,j is positive. Row 2: It is not possible to form collusion if the sum of the profits of units i and j is negative. Row 3: The collusion with high confidence, ‘High Collusion’, can be formed if both units i and j benefit from the capacity reduction of unit j. Therefore, the new collusion indexHCIicap , j , is proposed as (32) for∀ i ≠ j .

⎧|Cmx i, j + Cmx j, j| LCIicap ,j ≜ ⎨ 0 ⎩

Not important + −

of Cai,j and Caj,j is the same. In fact, in the positive case (Cai,j > 0 & Caj,j > 0) the bid rising, and in the negative case (Cai,j < 0 & Caj,j < 0) the bid reducing, will increase profit. Therefore, the new collusion index HCIibid , j , is proposed in (34) for ∀ i ≠ j .

• Row 1: The unit j reduces its capacity as a collusion if unit i benefits

|Cmx i, j + Cmx j, j| ≜ ⎧ ⎨ 0 ⎩

Collusion possibility

• Row 1: The collusion with high confidence can be formed if the sign

Table 3 shows the total possible states for the change in the profit of units i and j, resulting from the capacity reduction of low cost unit j. Since the capacity-based collusion can only be performed through capacity reduction, ΔPGjMax < 0, the negative sign in Table 3, indicates an increase in profit. The following can be explained about Table 3:

HCIicap ,j

(Cai, j × Caj, j )

the bid change in marginal unit j. The following can be explained about Table 4:

i.e.

4.1. Capacity-based collusion indices

• •

|Cai, j| − |Caj, j|

is a special case of the

LCIicap ,j .

In this research, the IEEE 24-bus Modified Reliability Test System (MRTS) with 32-generation units and 17-load points, is used to evaluate the proposed methodology [38]. The load and capacity of generators are doubled and the minimum generation of units are considered zero as in [27]. As illustrated in Table 5, at the market equilibrium point, 13 units are marginal, 5 expensive units are limited to their minimum generations and 14 low-cost units are bound by their generation caps.

4.2. Bid-based collusion indices The formation of collusion between two units i & j can be made by bid change (increase or decrease) of unit j. Table 4 shows the total possible states for the change in the profit of units i and j, resulting from 386

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Table 5 The status of units at the market equilibrium point. Mrg units

u1

u2

u3

u4

u5

u6

u7

u8

u9

u10

u11

u12

u13

real number Min units real number Max units real number

12 u14 7 u19 1

13 u15 8 u20 2

14 u16 9 u21 3

15 u17 10 u22 4

16 u18 11 u23 5

17

18

19

20

23

24

30

32

u24 6

u25 21

u26 22

u27 25

u28 26

u29 27

u30 28

u31 29

Table 6 The first section of Jacobian matrix (Ca ).

Table 7 The third section of Jacobian matrix (Cmx ).

387

u32 31

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Table 8 The capacity-based collusion indices between two low-cost units 25 and 27. Capacity change

ΔProfit25

Max ΔPg 25 = −1

−2.08

Max ΔPg 27 = −1

27.03

Max ΔPg 25 Max ΔPg 27

These cases are not allowed, because the maximum capacity of unit cannot be increased.

= +1

ΔProfit27

ΔProfit25 + ΔProfit27

High Collusion index

Low Collusion index

27.03

24.95

=0

cap LCI27,25 = 24.95

2.63

29.66

cap HCI27,25 cap HCI25,27

= 29.66

cap LCI25,27 = 29.66

= +1

Tables 6 and 7 show the first and third section of the Jacobian matrix in (31), corresponding to the action of marginal units and low cost units, respectively. The green-shaded cells in Tables 6 and 7, can be used to indicate the market power of units and the other values are useful to evaluate the collusion. Positive (negative) values in Table 6 imply that the profit is increased (decreased) by raising the bid of marginal units. For example, u32 (u13) benefits from increasing (decreasing) the bid of u12. The profit variations are often different for the mutual action of two marginal units. For instance, by a unit increment in bid of u1, 208.3$ is added to the profit of u13, but, by a unit increment in bid of u13, the profit of u1 is increased only 140.2$. However, the same type units (such as u2, u3, u4) have similar coefficients. As mentioned earlier, only negative values in Table 7 imply the possibility of employing capacity withholding. The most profitable gain can be earned by each of u27 and u27 if the capacity of another is reduced. Based on the sign and magnitude of the values in Table 7, the possibility and profitability of collusion can be determined. In order to better understand the effect of both sign and magnitude of coefficients on the collusion possibility, some numerical examples are provided in Tables 8 and 9. The values of these tables are taken from Tables 6 and 7. Table 8, shows the collusion profitability between two low-cost units 25 and 27. The necessary coefficients to this evaluation are: Cmx25,25 = 2.08, Cmx27,25 = −27.03, Cmx25,27 = −27.03 and Cmx27,27 = −2.63. From the Table 8, the following can be concluded:

economic withholding of unit 1 can be formed. 13 and (ΔProfit1 + ΔProfit13) are positive, both high and low collusion due to the economic withholding of unit 13, can be formed.

• Third row: Since both ΔProfit

Figs. 2 and 3 show the sorted values of the proposed bid based and capacity based collusion Indices, respectively. From these results, the profitability of all possible collusion can be identified. This collusion evaluation is performed in both low (weak) and high (strong) collusion. The most profitable collusion may be found when u12 changes its bid. According to Table 6, by a unit increment in bid of u12 the profits of u12 and u32 are increased by 593.2$ and 603.88$, respectively. Therefore, the total benefit of this collusion is equal to 1197.08 and this is a “high collusion”. However, if u1 added its bid the profits of u13 increases 208$ but its profits (i.e. u1) decreases by 72$. So, we conclude a “low collusion” can be created between these units. The total benefit of this collusion is equal to 136$. Given the Fig. 3, the highest collusion may be made between u27 and u28. 5.2. Case study 2 The IEEE 300-bus test system include has 69 units with 32678 MW installed capacity and 411 transmission lines. The peak load of this system is 29003 MW. The detail data of the system are available in [39]. The results show that, from total 69 × (69–1) = 4692 states to choose two units (to analysis the collusion), it can be said that:

• There are 285 states with LCI

• First row: Since ΔProfit

< 0, so the high collusion due to the capacity withholding of unit 25, cannot be occurred. However, because of (ΔProfit25 + ΔProfit27) > 0 the low collusion due to the capacity withholding of unit 25 can be formed. Second row: Since both ΔProfit27 and (ΔProfit25 + ΔProfit27) are positive, both high and low collusion due to the capacity withholding of unit 27, can be formed. The collusion between two units 25 and 27 is a directional relation. cap cap cap cap = 29.66 and LCI27,25 ≠ LCI25,27 = 0 while HCI25,27 Since HCI27,25 . 25

• •

• • •

Table 9, shows the collusion profitability between two marginal units 1 and 13. The necessary coefficients to this evaluation are: Ca1,1 = −72, Ca13,1 = 208.3, Ca1,13 = 140.2, Ca13,13 = 116.1. From the Table 9, the following can be concluded:

bid > 0 for low collusion by financial withholding. The range of indices value is between 0.08 and 1063.5 $. In the other states, LCIbid is equal to zero. This means that the collusion cannot be formed in other states. In 180 states (of pervious states), units can confidently collude by financial withholding, i.e. that states have HCIbid ≠ 0. (As it is mentioned in the paper, HCI is a special case of LCI). There are 153 states with LCIcap > 0 for low collusion by capacity withholding. The range of indices value is between 0.001 and 15.14 $. In the other states, LCIcap is equal to zero. In 92 states (of pervious states) units can confidently collude by capacity withholding, i.e. that states have HCIcap ≠ 0.

Figs. 4 and 5 show the sorted values of the proposed bid based and capacity based collusion Indices for this case study, respectively. We show 100 larger values of the indices for better visibility. In bid-based case, the maximum low collusion can be formed between units 9 and 39, by action of unit 9. In addition, the maximum high collusion can be formed between units 7 and 39, by action of unit

• First row: Since ΔProfit

1 < 0, so the high collusion due to the economic withholding of unit 1, cannot be occurred. However, because of (ΔProfit1 + ΔProfit13) > 0 the low collusion due to the

Table 9 The bid-based collusion indices for two marginal units 1 and 13. Bid change

ΔProfit1

ΔProfit13

ΔProfit1 + ΔProfit13

Δa1 = +1

−72

208.3

Δa1 = −1 Δa13 = +1

72 140.2

−208.3 116.1

Δa13 = −1

−140.2

−116.1

High Collusion index

Low Collusion index

136.3

bid HCI13,1 =0

bid LCI13,1 = 136.3

−136.3 256.3

Not profitable

−256.3

Not profitable

388

bid HCI1,13 = 256.3

bid LCI1,13 = 256.3

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Fig. 2. The bid based collusion Indices for IEEE 24-bus test system.

Fig. 3. The capacity based Collusion Indices for IEEE 24-bus test system.

equal to 1063.53 and 259.39 $, respectively. In cap-based case, the maximum low collusion can be formed between units 39 and 40, by action of unit 40. The total incremental profit of them is equal to 15.14 $. The maximum high collusion index is also similar to LCIcap.

6. Conclusion In this paper, a novel approach was proposed to evaluate the collusion possibility and profitability in the transmission-constrained electricity market. This goal was achieved by introducing two lemmas. Using the first lemma, output of OPF problem is decomposed with respect to input parameters. Using the second lemma, the Jacobian matrix of profit vector, which is a non-linear function of input parameters, is analytically calculated. In addition, four indices are proposed to assess the potential of both weak and strong collusions. The proposed indices can determine how much collusion is related to either capacity withholding or financial withholding of the Gencos. The proposed approach is general and can be applicable even for large-scale networks. Our findings is crucial from regulatory perspective since it can identify and evaluate the potential of GenCos' collusion.

Fig. 4. The bid based Collusion Indices for IEEE 300-bus test system.

Fig. 5. The capacity based Collusion Indices for IEEE 300-bus test system.

7. The total incremental profit of them for low and high collusion are Appendix A. The proof of Lemma 1 The problem (1a)–(1d) is convex. Therefore, the necessary Karush-Kuhn-Tucker (KKT) conditions are sufficient to find the global optimal solution of the problem [40–41]. Eqs. (A.1)–(A.3) represent the KKT conditions at the optimal point for the Lagrangian function:

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KKT1 : ∇x L (X∗,Λ∗,Ω∗) = 0

(A.1)

KKT2 : ∇Λ L

(X∗,Λ∗,Ω∗)

=0

(A.2)

KKT3 : ∇Ω L

(X∗,Λ∗,Ω∗)

=0

(A.3)

The first KKT condition is stated as (A.4): eq T − act T act ⎡ Hmin . Xmin +Fmin + (A min ) . Λ + (A min ) . Ω − Γmin ⎤ eq T act T ⎢H . X act +F + (A max ) . Λ + (A max ) . Ω + Γ+max ⎥ = 0 ⎢ max max max ⎥ act T act T ⎢ ⎥ Hmrg . Xmrg +Fmrg + (A eq mrg ) . Λ + (A mrg ) . Ω ⎣ ⎦

(A.4)

From the third row of (A.4) we have: act T T act = − F Hmrg . Xmrg + (A eq mrg mrg ) . Λ + (A mrg ) . Ω

(A.5)

In addition, the boundary variables are: (A.6)

Xmin =LB min and Xmax =UB max Eq. (A.7), is obtained by substituting the boundary variables into second KKT condition.

⎡ LB min ⎤ eq eq eq eq = 0 ⇒ A eq . X eq − A eq . LB KKT2 ⇒ [ A min A eq mrg =B min − A max . UB max max A mrg ]. ⎢ UB max ⎥ − B mrg min ⎢ Xmrg ⎥ ⎣ ⎦

(A.7)

Similarly, Eq. (A.8) are resulted from the KKT3. act act − A act . LB KKT3 ⇒ A act min − A max . UB max mrg . X mrg =B min

(A.8)

The KKT conditions can be expressed in matrix form as (A.9), based on (A.5, A.7 and A.8): eq T act T ⎡ Hmrg (A mrg ) (A mrg ) ⎤ ⎢ A eq 0 0 ⎥. ⎥ ⎢ mrg ⎢ A act 0 0 ⎥ mrg ⎦ ⎣ ⏟ E

⎡ Fmrg ⎤ − Fmrg −I 0 0 0 0 ⎤ ⎢ LBmin ⎥ X ⎤ ⎡ mrg ⎡ ⎡ ⎤ eq eq eq eq ⎢ Λ ⎥ = ⎢ Beq − A min. LB min − A max . UB max ⎥ = ⎢ 0 − Amin − Amax I 0 ⎥ ⎢UBmax ⎥ ⎥ ⎢ ⎥ ⎥⎢ act act ⎢ Ωact ⎥ ⎢ act act act 0 − Amin − Amax 0 I ⎦ ⎢ B eq ⎥ ⎣ ⎦ ⎢ ⎦ ⎣ ⎣ B − A min. LB min − A max . UB max ⎥ act ⎢ ⎥ ⎣ B ⎦

(A.9)

To prove Lemma 1, it is necessary to multiply the inverse of the matrix E, φ = E , to the right side of Eq. (A.9). Given that E is a block matrix, φ is also a block matrix which consisting of four parts. The relations in (A.10)–(A.14), explains how to calculate the matrix φ. -1

ϕ ≜ E−1

eq T act T ⎡ Hmrg (A mrg ) (A mrg ) ⎤ ϕ ϕ E E 1 2 1 2 ⎢ eq ⎤, E=⎡ ⎤ = A mrg 0 0 ⎥ =⎡ ⎢ ⎥ ⎢ ϕ3 ϕ4 ⎥ E3 E4 ⎦ ⎢ ⎥ ⎣ ⎣ ⎦ ⎢ A act 0 0 ⎥ mrg ⎣ ⎦

(A.10)

T T (A act )T E1 =Hmrg , E 2 = [(A eq mrg ) mrg ] , E 3 =E 2 , E 4=0

(A.11)

The relations of the inverse of block matrices are as [42]:

ϕ1 = E−1 1 − E−1 1 E 2 (E 3 E−1 1 E 2)−1E 3 E−1 1

(A.12)

ϕ2 = E−1 1 E 2 (E 3 E−1 1 E 2)−1

(A.13)

ϕ3 = (E 3 E−1 1 E 2)−1E 3 E−1 1 , ϕ4 = −(E 3 E−1 1 E 2)−1

(A.14)

Considering the dimension of matrix E, matrix φ can be extended to nine parts as (A.15):

(ϕ11 )Nmrg × Nmrg (ϕ12 )Nmrg ×m (ϕ13 )Nmrg × k act ⎤ (ϕ2 )Nmrg × (m + k act) ⎤ ⎡ ⎡ (ϕ1 )Nmrg × Nmrg ⎢ (ϕ ) (ϕ22 )m×m (ϕ23 )m×k act ⎥ ϕ=⎢ = 21 m×Nmrg ⎥ (ϕ3 )(m + k act )× Nmrg (ϕ4 )(m + k act )× (m + k act) ⎥ ⎢ ⎣ ⎦ ⎢ (ϕ31 )k act × Nmrg (ϕ32 )k act ×m (ϕ33 ) k×k act ⎥ ⎦ ⎣

(A.15)

Using φ, marginal variables and lagrangian multipliers are decomposed as (A.16).

⎡ Fmrg ⎤ ⎡ Fmrg ⎤ ϕ ϕ ϕ −I 0 0 0 0 ⎤ ⎢ LB ⎥ ⎡ S11 S12 S13 S14 S15 ⎤ ⎢ LB min ⎥ min ⎡ Xmrg ⎤ ⎡ 11 12 13 ⎤ ⎡ eq eq ⎢ Λ ⎥ = ⎢ ϕ21 ϕ22 ϕ23 ⎥ ⎢ 0 − A min − A max I 0 ⎥ ⎢ UB max ⎥ = ⎢ S 21 S 22 S 23 S 24 S 25 ⎥ ⎢ UB max ⎥ ⎢ Ωact ⎥ ⎢ ϕ ϕ ϕ ⎥ ⎢ 0 − A act − A act 0 I ⎥ ⎢ Beq ⎥ ⎢ S 31 S 32 S 33 S 34 S 35 ⎥ ⎢ Beq ⎥ ⎦ ⎣ 31 32 33 ⎦ ⎣ ⎣ ⎥ ⎣ ⎥ min max ⎦⎢ ⎦⎢ act ⎥ act ⎥ ⎢ ⎢ ⏟ ⎣ B ⎣ B ⎦ ⎦ S Thus, Lemma 1 is proved. □ B. The Proof of Lemma 2 According to the definition of U in (7) we have: 390

(A.16)

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… k1n ⎤ q1 w w12 ⎡ ⎤ ⎡ 11 w21 w22 … k2n ⎥ ⎢ q2 ⎥ ⎢ ⊙ ⋮ ⎢ ⋮ ⋱ ⋮ ⎥⎢ ⋮ ⎥ ⎥ ⎢q ⎥ ⎢ … kmn ⎦ ⎣ n ⎦ ⎣ wm1 wm2

⎡ k11 k12 ⎢k k U = ⎢ 21 22 ⋮ ⋮ ⎢ ⎣ k m1 k m 2

… w1n ⎡ q1 ⎤ ⎤ … w2n ⎥ ⎢ q2 ⎥ ⋱ ⋮ ⎥⎢ ⋮ ⎥ ⎢ ⎥ … wmn ⎥ ⎦ ⎣ qn ⎦

(B.1)

Multiplying the coefficient matrices by the vector Q results:

⎡ (k11 q1 + k12 q2 + ...+k1n qn ) ⎤ ⎡ (w11 q1 + w12 q2 + ...+w1n qn ) ⎤ ⎢ (k21 q1 + k22 q2 + ...+k2n qn ) ⎥ ⎢ (w21 q1 + w22 q2 + ...+w2n qn ) ⎥ U=⎢ ⎥ ⎥ ⊙ ⎢ ⋮ ⋮ ⎢ ⎥ ⎥ ⎢ ⎢ (km1 q1 + km2 q2 + ...+kmn qn ) ⎥ ⎢ (wm1 q1 + wm2 q2 + ...+wmn qn ) ⎥ ⎣ ⎦ ⎦ ⎣

(B.2)

Based on the “hadamard product” definition, the (B.3) is obtained. n

n

⎡ (k11 q1 + ...+k1n qn )(w11 q1 + ...+w1n qn ) ⎤ ⎡ (∑i = 1 k1i qi )(∑i = 1 w1i qi ) ⎤ ⎥ ⎥=⎢ ⋮ U=⎢ ⋮ ⎥ ⎢ (k q + ...+k q )(w q + ...+w q ) ⎥ ⎢ n n ∑ k q ∑ w q mn n m1 1 mn n ⎦ ⎢ ⎣ m1 1 ⎣ ( i = 1 mi i )( i = 1 mi i ) ⎥ ⎦

(B.3)

Eq. (B.3) can be rewritten as: n

n ⎡ ∑i = 1 k1i qi ( ∑ j = 1 w1j qj ) ⎤ n 2 ⎥ j≠i ⎡ ∑i = 1 (k1i w1i qi ) ⎤ ⎢ ⎥ ⎥+⎢ ⋮ ⋮ U=⎢ ⎥ ⎥ ⎢ n ⎢ n 2 n ⎢∑ k q ( ∑ j = 1 wmj qj ) ⎥ ∑ (k w q ) ⎢ ⎦ ⎢ i = 1 mi i ⎣ i = 1 mi mi i ⎥ ⎥ j≠i ⏟ Uα ⎣ ⎦ ⏟ Uβ

(B.4)

The elements of vector dUα can be calculated as below: n

⎡ ∑i = 1 2(k1i w1i qi ) dqi ⎤ ⎡ k11 w11 q1 dq1 + ...+k1n w1n qn dqn ⎤ ⎥=2×⎢ ⎥ dUα = ⎢ ⋮ ⋮ ⎢ n ⎥ ⎢ k w q dq + ...+k w q dq ⎥ 1 1 m m mn mn 2( k w q ) dq ∑ 1 1 n n⎦ ⎢ mi mi i i⎥ ⎣ ⎣ i=1 ⎦

(B.5)

The vector dQ and the diagonal matrix of Q are appeared in (B.6), by rewriting (B.5).

q … 0 ⎤ ⎡ dq1 ⎤ ⎡ k11 w11 q1 … k1n w1n qn ⎤ ⎡ dq1 ⎤ ⎡ k11 w11 … k1n w1n ⎤ ⎡ 1 ⎥×⎢ ⋮ ⎥=2×⎢ ⋮ ⋱ ⋮ ⎥ × ⎢⋮ ⋱ ⋮ ⎥ × ⎢ ⋮ ⎥ ⋮ ⋱ ⋮ dUα = 2 × ⎢ ⎢ k w q … k w q ⎥ ⎢ dq ⎥ ⎢ km1 wm1 … kmn wmn ⎥ ⎢ 0 … q ⎥ ⎢ dq ⎥ mn mn n ⎦ n⎦ ⎦ ⎣ ⎣ ⎣ n⎦ ⎣ n⎦ ⎣ m1 m1 1

(B.6)

So, in summary:

dUα = (2(K ⊙ W) × Diag(Q))×dQ

(B.7)

n×1

In addition, second part of (B.4), can be stated as follow: n

n

n

n

n

n

n

⎡ ∑i = 1 (k1i dqi ( ∑ j ≠ i w1j qj ) + k1i qi ( ∑ j ≠ i w1j dqj )) ⎤ ⎡ ∑i = 1 (k1i ∑ j ≠ i w1j qj ) dqi + ∑i = 1 ( ∑ j ≠ i k1i w1j qi dqj ) ⎤ ⎥=⎢ ⎥ ⋮ ⋮ dUβ = ⎢ ⎥ ⎢ n ⎢ n ⎥ n n n n n ⎢∑i = 1 (kmi dqi ( ∑ j ≠ i wmj qj ) + kmi qi ( ∑ j ≠ i wmj dqj )) ⎥ ⎢∑i = 1 (kmi ∑ j ≠ i wmj qj ) dqi + ∑i = 1 ( ∑ j ≠ i kmi wmj qi dqj ) ⎥ ⎦ ⎣ ⎣ ⎦

(B.8)

The second part of first row in (B.8), can be rewritten as: n

n

n

n

n

n

∑ ( ∑ k1i w1j qi dqj) = ∑ ( ∑ k1i w1j qi) dqj = ∑ ( ∑ k1j w1i qj) dqi i=1

j=1 j≠i

j=1

i=1 i≠j

i=1

j=1 j≠i

(B.9)

Eq. (B.8) can be reformulated as (B.10), according to (B.9).

(

)

(

)

(

)

(

)

⎡ ∑in= 1 k1i ∑nj ≠ i w1j qj dqi + ∑in= 1 ∑nj ≠ i k1j w1i qj dqi ⎤ ⎡ ∑in= 1 ∑nj ≠ i (k1i w1j + k1j w1i ) qj dqi ⎤ ⎥ ⎢ ⎢ ⎥ ⋮ ⋮ dUβ = ⎢ ⎥=⎢ ⎥ ⎥ ⎢ n ⎢ n ⎥ n n n n ⎢∑i = 1 ∑ j ≠ i (kmi wmj + kmj wmi ) qj dqi ⎥ ⎢∑i = 1 kmi ∑ j ≠ i wmj qj dqi + ∑i = 1 ∑ j ≠ i kmj wmi qj dqi ⎥ ⎦ ⎣ ⎣ ⎦

(

)

(

)

(B.10)

Eq. (B.11) is the expanded form of (B.10).

⎡ ∑j ≠ 1 (k11 w1j + k1j w11) qj dq1 ⋯ ∑j ≠ n (k1n w1j + k1j w1n ) qj dqn ⎤ ⎥ dUβ = ⎢ ⋮ ⋱ ⋮ ⎢ ⎥ ⎢∑j ≠ 1 (km1 wmj + kmj wm1) qj dq1 ⋯ ∑j ≠ n (k1n w1j + k1j w1n ) qj dqn ⎥ ⎣ ⎦

(B.11)

The vector dQ is appeared in (B.12), by rewriting (B.11).

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⎡ ∑j ≠ 1 (k11 w1j + k1j w11) qj ⋯ ∑j ≠ n (k1n w1j + k1j w1n ) qj ⎤ ⎡ dq1 ⎤ ⎥⎢ ⋮ ⎥ ⋮ ⋱ ⋮ dUβ = ⎢ ⎥⎢ ⎢ ⎥ ⎢∑j ≠ 1 (km1 wmj + kmj wm1) qj ⋯ ∑j ≠ n (k1n w1j + k1j w1n ) qj ⎥ ⎣ dqn ⎦ ⎦ ⎣

(B.12)

Eq. (B.13) represents the closed-form of (B.12):

dUβ = [∑j ≠ 1 (K1 ⊙ Wj + K j ⊙ W1) ⊙ Qj ⋯ ∑j ≠ n (Kn ⊙ Wj +K j ⊙ Wn ) ⊙ Qj] dQ

(B.13)

If the vector Yi is defined as (B.14), so dUβ in (B.13) can be rewritten as (B.15). Ki and Wi in (B.14), are the i-th columns of matrices K and W, respectively. n

Yi ≜ m×1

∑ j=1 j≠i

( Ki ⊙ Wj + K j ⊙ Wi ) ⊙ Qj m×1

m×1

m×1

m×1

(B.14)

dUβ = Y × dQ in which: Y = [ Y1 Y2 … Yn ]

(B.15)

m×n

Eq. (B.16) is obtained from (B.7) and (B.15).

dU = (2(K ⊙ W) × Diag(Q)) × dQ + Y × dQ ⏟β ⏟α dU dU

(B.16)

Consequently, the Jacobin matrix is equal to:

[J (K, W, Q)] =

dU = (2(K ⊙ W) × Diag(Q) + Y) dQ

(B.17)

So, the Lemma 2 is proved. □.

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