Risk-based optimal bidding patterns in the deregulated power market using extended Markowitz model

Energy xxx (xxxx) xxx

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Risk-based optimal bidding patterns in the deregulated power market using extended Markowitz model Bakhtiar Ostadi*, Omid Motamedi Sedeh, Ali Husseinzadeh Kashan Faculty of Industrial and Systems Engineering, Tarbiat Modares University, Tehran, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 20 August 2019 Received in revised form 17 October 2019 Accepted 7 November 2019 Available online xxx

Deregulation of power industry has entailed important changes in the energy market. With the power industry being restructured, a generation company (GenCo) sells energy through auctions in a daily market, and submission of the appropriate amount of electricity with the right bidding price is important for a GenCo to maximize their profits and minimize the acceptance risk. The objective of this paper is to propose a novel approach for determination of the optimal biding patterns among GenCos in the deregulated power market using a hybrid of Markowitz Model and Genetic Algorithm (GA). While Markowitz Model as an optimization model considers the risk premium for biding patterns and GA as a search engine, considering the acceptance risk in deregulated market. A case study is used to examine the findings of the proposed approach. Also, to compare the proposed model, neural network by back propagation learning algorithm and real proposed pattern were considered. The numerical results indicate that the proposed model is statistically efficient and offers effective curves and biding patterns by lesser risk and equal profitability in day-ahead market as it is able to achieve better results compared to the neural network. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Markowitz model Deregulated market Risk estimation Value at risk (VaR) Energy generation cost

1. Introduction The advent of electricity markets in numerous countries around the world during the 1990s resulted in the introduction of the restructuring process of power industry with the goal of increasing economic efficiency and reducing the generation cost [1]. Electricity markets have been categorized as (a) singlesettlement markets (or real-time balancing market) and (b) two settlement markets (day ahead or pre-dispatch market). In a singlesettlement market, competitors purchase or sell wholesale electricity during the operating day, and electricity prices are set daily or hourly depending on the demand and available supply. In a twosettlement market, competitors purchase or sell wholesale electricity one day before the operating day on the day-ahead electricity market, and the difference between the proposed and actual demand on the operation day will be covered using a real-time market [2,3]. In the day-ahead market, every generation company (GenCo) (or participants) submits a bidding pattern and strategies to maximize

* Corresponding author. E-mail addresses: [email protected] (B. Ostadi), omid.motamedi@modares. ac.ir (O. Motamedi Sedeh), [email protected] (A. Husseinzadeh Kashan).

its profit by optimizing the bidding strategy [4e6] and to the Independent System Operator for each hour of the next day. The bid price should be greater than zero and lower than market price cap determined by the market ISO. Based on the predetermined market clearing mechanism, bid patterns are analyzed by the ISO and hourly MCP will be determined [7]. The day-ahead markets are under two market clearing payment mechanisms of uniform payment and a bid payment. In the uniform mechanism, payment will be made for all the accepted bid by market clearing price, but in pay as a bid mechanism, payment will be made for all the accepted levels of each participant based on its proposed price [8]. Therefore, participants in these competitive markets encounter daily challenges such as market rules, market clearing mechanism, bidding patterns of other competitors to optimize their biding patterns. Therefore, three fundamental issues must be considered [2,9]: 1) the risk of market participation or the proposed bidding pattern; 2) the profit or loss of the proposed bidding pattern; 3) uncertainty of prices both in day-ahead and real-time markets. In order to treat the first issue, mathematical models were

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utilized to forecast market clearing price for the next day and acceptance risk was evaluated based on the prediction result. In treating the second issue, biding pattern was compared to generation cost. Finally, a hybrid model of Markowitz Model and Genetic Algorithm (GA) has been applied to optimize the bidding pattern by considering both parameters of risk and profitability of biding patterns. To forecast MCP for the next day, several forecasting algorithms have been proposed in the literature, namely robust optimization [6], Times Series models [10], GARCH model [11], combination of Wavelet Transform and ARIMA [12], Fuzzy Auto Regression model [13], K-Nearest Neighborhood [14], combination of Multi Criteria Regression and Wavelet Transform [15], Game Theory [5,16e18], Bayesian Optimization [19], combination of SVM and Linear Regression [20,21], combination of Neural Network and Genetic Algorithm as well as Particle Swarm Algorithm [22], Fuzzy Particle Swarm Algorithm model [23], Markov Chain [24], Neural Network [25e27], Fuzzy Neural Network [28,29], combination of Neural Network model and Bat Optimization [30], combination of Neural Network model and Wavelet Transform [31]. In some research, bidding strategy modeling methods reviewed [5] and optimal bidding strategies determined using robust optimization, agent-based approach and numerical sensitivity analysis (NSA) [6,32]. All forecasting algorithms have been categorized in three groups of Game Theory Models, Times Series models and Simulations Models. Time series models have also been classified in three sets of Stochastic models, Artificial Intelligence models and Casual Models [4,5,32e35]. In the literature generation cost evaluation methods have been divided into two groups of accounting methods and mathematical methods. Jiang, for instance, calculated the Chinese natural gas power generation cost using accounting models [36]. Partridge estimated the energy generation cost, comparing wind and thermal power [37]. Motghare and Cham also estimated generation cost in 660 MW thermal power plants using the accounting model [1]. So far, many studies have been conducted separately to examine the generation cost and forecast the market clearing price of the next day, but there is no body to evaluate risk premium for each biding pattern in compare of other patterns, in this paper risk premium have been considered into a bidding pattern by a hybrid of Markowitz model and Genetic Algorithm. In the present study, Markowitz model was applied to determine optimum biding pattern, by consideration of acceptance risk and profitability result. The main contributions of this paper and the key advantages of the proposed model can be summarized as follows: a) application of Markowitz model to optimize biding strategy b) improvement of Markowitz model by Genetic Algorithm as a search engine c) estimation of generation cost function by heuristic algorithm in order to evaluate the profitability of each biding pattern The rest of this paper is organized as follows: The next section explains the Theoretical basis. Section 3 presents the Research methodology and proposed model. Numerical results are discussed in Section 4. Finally, Section 5 includes the concluding remarks. 2. Theoretical basis 2.1. Markowitz model Markowitz model or mean-variance analysis is a mathematical model for portfolio optimization by selecting assets as the expected return which is maximized for a given level of risk. This model was

developed by Harry Markowitz in 1952. He had believed that the portfolio selection process contains two stages that first stage starts with observation and ends with beliefs about the future and it cusses to choose of portfolio on second stage [38]. This model assuming that investors prefer more return and lesser risk. These preference are shown in the form of the following objective functions (Eq. (1) and Eq. (2)) and equations (Eq. (3) and Eq. (4)) [39]:

MAX

mp

(1)

Min

1 2 s p 2

(2)

where, mp and s2 p are expected return and risk of portfolio.

mp ¼

n X

xi mi

(3)

i¼1

s2 p ¼

n X i¼1

x2 i s 2 i þ

n X X xi xj Covði; jÞ

(4)

i¼1 j < i

where: mi is expected return of ith asset. P xi is weight of investment on ith asset as ni¼1 xi ¼ 1. 2 s i is the risk of ith asset. Cov(i,j) is covariance between the ith and ith assets.

2.2. Genetic Algorithm The Genetic Algorithm is a heuristic algorithm for random search, referring to the biology evolution. These algorithms have been successful in solving hard optimization problems by operations of initial population generation, fitness evaluation, selection, crossover and mutation [40]. In this paper GA will be applied to search, solution space to make smaller solution space as an input parameter for Markowitz model.

2.3. Particle swarm optimization algorithm PSO is a heuristic algorithm for finding a global optimum by iteratively Search to increase quality of candidate solutions. This algorithm foundation based on the group of birds to search for food in the solutions space randomly. Each bird is a single solution that moves in the solution space to look for better solutions [41]. In this paper PSO will be applied to search, solution space to make smaller solution space as an input parameter for Markowitz model.

2.4. Risk evaluation Value-at-Risk (VaR) is a statistical risk assessment tool developed by the financial institution to measure the riskiness of financial entities or portfolios of assets. It measures the expected maximum loss over a time interval within a specific confidence level [42]. VaR is applied to qualify risk of investment and measure how much a set of investments might lose with a given probability in a special time period in normal market conditions.

Please cite this article as: Ostadi B et al., Risk-based optimal bidding patterns in the deregulated power market using extended Markowitz model, Energy, https://doi.org/10.1016/j.energy.2019.116516

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3. Research methodology and proposed model

L X

In this study, each price value on bidding pattern has been considered as a kind of financial asset with deterministic risk and return, where each GENCO’s desire to choose the best combination of asset in their portfolio (bidding pattern) and the optimum combination of asset will be determined by Markowitz model. The proposed model consists of four parts: (1) generation cost estimation; (2) risk evaluation by VaR model; (3) combination of risk evaluation and cost evaluation by Markowitz model; and (4) application of GA to optimize the model as shown in Fig. 1.

3

pi ¼ Pmax

(7)

i¼1

where profitability of ith price is calculated as Eq. (8)

mi ¼ qi pi  f ðpi Þ ci ¼ 1; 2…L

(8)

and cost(x) is the cost of generation as much as x kWh. To evaluate the acceptance risk for ith price, we applied VAR models as s2i ¼ p (fMCP > Ci). By considering model parameter as following:

3.1. Extended Markowitz model To apply Markowitz model in a deregulated market, we consider each valid price (between zero and market cap) as an asset with deterministic risks (calculated by VaR model) and profitability (calculated by the difference between the proposed price and generation cost). By considering Ct ¼ {q1,q2, ……ql} as a set of valid prices in the tth hour of the next day then:

0  qi  qcap

ci ¼ 1; 2…L

T i L M qti

Hour of day Counter Number of biding level A very large number Proposed price for Hour Proposed Power for Hour

(5)

P ti pmax wi

(6)

X ti MCP t f(p) p(q < q2)

pi as a power with the biding rate of qi,

0  pi  pmax

ci ¼ 1; 2…L

s2 m

Maximum power Weight factor for ith level Variable to determine accept/reject for ith level Market clearing price on Hour t Cost function of power production Probability of market clearing price lower than q Value (power) at risk Expected return

Start

Determining Availability

Random generation of 1000 proposed prices between 0 and market cap (first generation)

Risk estimation for each Chromosome

Profitability estimation of Chromosomes

Markowitz Model Run Markowitz Model

Create next generation by Selection, Crossover and Mutation

Stop Condition Genetic Algorithm End

Fig. 1. Proposed model to optimize bidding pattern.

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A proposed optimization model for tth hour of next day can be formulated as bellow: n X ð1  Gi Þ

Max

i¼1 i1 i     i h X X  Pmax qi wi þ yi f Pmax wi  f Pmax wi j¼0 n X

Min

(9)

j¼0

Pmax Gi wi

(10)

i¼1

Step I: randomly generate a sample set of chromosomes as initial population, serving as a valid price (between 0 and market cap). Step II: Run Markowitz models. Step III: If the stop condition is fulfilled, stop, and resume in otherwise. Step IV: Create a new population through three operations of selection, crossover, and mutation, a) Selection: Select all the proposed prices in models. b) Crossover: randomly generate new chromosomes. c) Mutation: add or subtract the random value in the selected chromosome. Step V: go to Step II.

S.t.

X

wi ¼ 1

(11)

3.2. Extended VaR

Myi  wi

(12)

Mðyi  1Þ þ 1  wi

(13)

Generally, GENCOs do not have access to complete information of their opponent, so it is necessary for a GENCO to estimate opponents’ unknown information by risk. To evaluate the risk of each proposed price, Historical Simulation Method for VaR estimation has been applied as a comparison between the proposed price and historical changes in market clearing price to determine the ratio of times the proposed price is less than market clearing prices to times the proposed price is more than market clearing prices [42].

X

yi  10

(14)

yi ¼ bin

(15)

By considering the specific expected value (R), we can write the proposed model as Eq. (16) to Eq. (22). n i1 i      h X X X Max ð1  Gi Þ Pmax qi wi þ yi f Pmax wi  f Pmax wi i¼1



i

j¼0

j¼0

(16) S.t. n X

Pmax Gi wi ¼ R

(17)

wi ¼ 1

3.3. Profitability evaluation Electricity spot prices are often driven by natural gas prices during peak time periods, and generation of fixed cost is composed of three components: depreciation (f1), operation and maintenance (f2) and capital cost(f3)[43]. Let x (kWh) be the generation value, Cfuel be the gas price and f(x) be the fuel consumption function for generation of x kWh, and generation cost be evaluated by Eq (25). By considering Leontief, Quadratic, Cubic, and Trans-log Cost Functions for estimation of fuel consumption, and equations of 5e8, the results of parameters estimation are listed in Table 1

f ðxÞ ¼ pb0 X

(25)

(18)

  f ðxÞ ¼ P b0 þ b1 X þ b2 X 2

(26)

Myi  wi

(19)

  f ðxÞ ¼ p b0 þ b1 X þ b2 X 2 þ b3 X 3

(27)

Mðyi  1Þ þ 1  wi

(20)

  2 Lnf ðxÞ ¼ p b0 þ b1 Lnx þ b2 Lnx

(28)

(21)

As Table 1 shows fuel consumption for generation of x kWh is evaluated by Eq (29).

(22)

f ðxÞ ¼ 0:24 þ 2:3x þ 0:011x2  0:00000008x3

i¼1

X

X

yi  10

yi ¼ bin We can write Eq. (17) as below: n X

Pmax Gi wi  R  εR

By considering Eq. (25), generation cost of x kWh is calculated to

(23)

i¼1 n X

Pmax Gi wi  R þ εR

(29)

(24)

i¼1

In order to solve Markowitz model, Genetic Algorithm was utilized in pursuing the solution space through the following steps:

Table 1 Result of the parameter estimation for evaluation of generation cost. Parameter

Leontief

Quadratic

Cubic

Trans-log

b0 b1 b2 b3

3.60E-01

1.40E-03 4.00E-05 3.80E-03

7.20E-01 1.60Eþ00 2.00Eþ00

Error (MSE)

20%

19%

2.40E-01 2.30Eþ00 1.10E-02 8.00E-08 15%

17%

Please cite this article as: Ostadi B et al., Risk-based optimal bidding patterns in the deregulated power market using extended Markowitz model, Energy, https://doi.org/10.1016/j.energy.2019.116516

10 8 6 4 2 0

$

$

B. Ostadi et al. / Energy xxx (xxxx) xxx

0

25

50

75 MWh

100

125

10 8 6 4 2 0

150

0

10 8 6 4 2 0 25

50

75 MWh

100

125

0

150

$

$ 50

75 MWh

100

125

0

150

$

$ 50

75 MWh

100

125

125

150

25

50

75 MWh

100

125

150

25

50

75 MWh

100

125

150

Fig. 9. Result of proposed model for profitability of 560$.

10 8 6 4 2 0 25

100

10 8 6 4 2 0

Fig. 4. Result of proposed model for profitability of 260$.

0

75 MWh

Fig. 8. Result of proposed model for profitability of 500$.

10 8 6 4 2 0 25

50

10 8 6 4 2 0

Fig. 3. Result of proposed model for profitability of 175$.

0

25

Fig. 7. Result of proposed model for profitability of 425$.

$

$

Fig. 2. Result of proposed model for profitability of 85$.

0

5

10 8 6 4 2 0 0

150

25

50

75 MWh

100

125

150

Fig. 10. Result of proposed model for profitability of 580.

10 8 6 4 2 0

$

$

Fig. 5. Result of proposed model for profitability of 320$.

0

25

50

75 MWh

100

125

Fig. 6. Result of proposed model for profitability of 375$.

150

10 8 6 4 2 0 0

25

50

75 MWh

100

125

150

Fig. 11. Result of proposed model for profitability of 620$.

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Table 2 Result of proposed model in comparison to NN based on risk. Set No.

# of records in test set

1 2 3 4 5 6 7 8 9 10

Average income($/mwh)

166 104 94 132 100 91 96 127 87 98

5.03 4.65 3.11 3.33 3.94 1.63 4.13 4.7 5.78 2.8

Average of acceptance risk Proposed model

Neural network

39% 33% 14% 18% 26% 5% 25% 56% 21% 18%

51% 55% 18% 51% 45% 35% 42% 57% 34% 22%

Table 3 Result of proposed model in comparison to NN based on hourly income. Set No.

# of records in test set

Average of income by running each model($/Hour) NN model

Markowitz model by VaR < 30 MWh

Markowitz model by VaR <60 MWh

Markowitz model by VaR < 90 MWh

1 2 3 4 5 6 7 8 9 10

166 104 94 132 100 91 96 127 87 98

544.32 457.38 142.38 165.06 569.52 618.2 475.02 564.48 645.12 667.8

473.76 395.64 131.04 214.2 269.64 315 585.9 715.68 253.26 509.04

573.76 674.1 417.3 337.68 635.04 672.84 277.2 471.24 323.82 783.72

211.24 211.29 577.08 754.74 681.26 549.36 476.28 554.4 185.15 682.92

Table 4 Comparison of proposed model with real biding patterns in Iran deregulated market. Set No.

# of records in test set

1 2 3 4 5 6 7 8 9 10

166 104 94 132 100 91 96 127 87 98

Average of income by running each model($/Hour) Case Study result

Markowitz model by VaR < 60 MWh

256.73 229.86 141.89 152.44 524.82 404.28 156.97 396.57 181 517.37

573.76 674.1 417.3 337.68 635.04 672.84 277.2 471.24 323.82 783.72

be Eq. (30).

GA

PSO

  CostðxÞ ¼ 0:295 þ Cfuel 024 þ 2:3x þ 0:011x2  0:00000008x3

10

(30)

8 $

6 4. Numerical results

4 2 0 0

25

50

75 MW

100

125

Fig. 12. Comparison of GA and PSO with VaR <75 MWh

150

The present case study involves a gas power plant by capacity of 146 MW in Iran’s deregulated market. Iran’s energy market is a pool-based day-ahead market with a pay as bid settlement mechanism. The competition in Iran day-ahead market is in generation side, and GENCOs should offer biding curves in rising steps with a maximum of ten steps and a cap price limit of 8.9 $/MWh in 2018. The structured data set used for parameter estimation has been

Please cite this article as: Ostadi B et al., Risk-based optimal bidding patterns in the deregulated power market using extended Markowitz model, Energy, https://doi.org/10.1016/j.energy.2019.116516

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GA

PSO

10 8 $

6 4 2 0 0

25

50

75 MW

100

125

150

Fig. 13. Comparison of GA and PSO with VaR <45 MWh

GA

PSO

10 8 $

6 4 2 0 0

25

50

75 MW

100

125

150

Fig. 14. Comparison of GA and PSO with VaR <30 MWh

gathered from summer 2012 until summer 2016. The model is programmed using Matlab for a set of random days of proposed power plant in Iran’s power market with different profitability values. To apply proposed model, different value will be apply for s2 and mp will be determined by running model. Figs. 2e11 present the results of the model for difference mp at a specific hour of a specific day. By consideration of Eq. (9) to Eq. (15) as proposed model with single objective function as Eq. (10), and consideration of Eq. (9) as a constrain with specific value of profitability (which has been determined based on result of NN model), and solving proposed model, the best biding pattern by lowest value of expected risk for chosen profitability value, will be obtain. Figs. 2e11 present the

7

optimum biding patterns for difference values of profitability (Eq. (9)) in specific day. For example, Fig. 2 is a result of proposed model by consideration value of 85 for Eq. (9) and solving model by Eq. (9) to Eq. (15) It can be observed that optimum bidding pattern relies on the proposed profitability values. To compare the proposed model, neural network by back propagation learning algorithm was applied on the same data. The data set was divided into two parts, the training data and testing data. In the first stage, performance of NNs was evaluated in all test sets, preliminary profitability and risk of each solution of NN has been evaluated, and the proposed model was run for the test set by minimizing the acceptance risk with the deterministic profitability as equal as NN results. Table 2 shows the result of the proposed model in comparison to simple NN for different sub sets of data. Table 2 shows the proposed model and Neural Network model performance for optimizing biding problem in Iran’s deregulated market. It can be observed that proposed model offers a biding pattern with lower risk than NN with the same profitability for each MWh since the acceptance risk shows a corresponding improvement of 18% by utilizing Markowitz model. Finally, in order to evaluate the proposed model performance, this model with three difference variance (VAR) range (low, medium and high) has been compared with NN. Table 3 shows the result of the proposed model in comparison with simple NN. According to Table 3 for each subset of test set (except subset 9), the proposed model offers a better solution and higher profitability than NN model. The weighted average of income for each hour in NN model is 479 $, while in Markowitz model with varying VaR this value is different: VaR<30 MWh (397$), VaR<60 MWh (516$), and VaR<90 MWh (482$). In the other hand, for the evaluation of proposed model performance in real world, we compare result of proposed model by VaR <60 MWh with biding patterns of power plant in the real case study and Table 4 shows the result of comparison in 10 sets. Based on Table 4, if the case study power plant used the proposed model instead of its own model, its profitability has been increased more than 200 $/Hour. Finally, in order to evaluation of impact of the heuristic algorithm in proposed model performance, the proposed model has been executed by two different heuristic algorithms of GA and PSO with the same variance for all 10 sets. Figs. 12e14 are show samples of result of proposed model by application of GA and PSO algorithm as a search engine. Table 5 shows the avarage result of proposed model by deployment of GA and PSO as a search engine. Based on Table 5, It can be observed that there is no high difference in performance of proposed model by replacing PSO as a

Table 5 Comparison of GA and PSO Algorithm as a search engine of proposed model. Set No.

1 2 3 4 5 6 7 8 9 10

# of records in test set

166 104 94 132 100 91 96 127 87 98

Average of income by running proposed model($/Hour) GA

GA

GA

PSO

PSO

PSO

VaR < 30 MWh

VaR < 60 MWh

VaR < 90 MWh

VaR < 30 MWh

VaR < 60 MWh

VaR < 90 MWh

473.76 395.64 131.04 214.2 269.64 315 585.9 715.68 253.26 509.04

573.76 674.1 417.3 337.68 635.04 672.84 277.2 471.24 323.82 783.72

211.24 211.29 577.08 754.74 681.26 549.36 476.28 554.4 185.15 682.92

467.51 384.43 150.03 219.13 278.43 273.67 664.93 811.12 278.84 512.36

524.51 533.40 324.87 304.12 558.69 525.89 342.80 635.49 325.55 648.15

234.16 260.86 431.72 600.16 553.84 429.57 452.85 608.93 232.43 602.61

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search engine and none of the method completely overcomes the other. Based on results value, average of income by running proposed model with GA as a search engine for values of VaR<30 MWh, VaR<60 MWh, and VaR<90 MWh are 386 $/Hour, 516 $/Hour and 488 $/Hour and average of income by running proposed model with PSO as a search engine for values of VaR<30 MWh, VaR<60 MWh, and VaR<90 MWh are 404 $/Hour, 472 $/Hour and 440 $/Hour. 5. Conclusion This paper presented a new hybrid model based on Markowitz model and Genetic Algorithm to minimize the acceptance risk of bidding pattern and maximize profitability. Two parameters of acceptance risk and profitability of bidding patterns were estimated by mathematical methods. Markowitz model was applied to optimize biding patterns and maximize profitability based on a specific risk for GenCos in deregulated power markets. Numerical testing results show that the proposed model is statistically efficient, and offers effective curves in Iran’s day-ahead market as it is able to achieve better results compared to the neural network. The results obtained from the proposed method indicate maximization of profits and benefits over other methods. It can be easily used to determine the optimal bidding strategy in different market rules, risk evaluation, and profitability evaluation. The results show that proposed approach is an appropriate model for solving optimization problem of bidding strategy in deregulated power market by Markowitz model and there is no big change in performance of proposed model by changing search engine model. References [1] Motghare VS, Cham R. Generation cost calculation for 660 MW thermal power plants. Int. J. Innov. Science, Eng. Technol. 2014;1. [2] Fazlalipour P, Ehsan M, Mohammadi-Ivatloo B. Risk-aware stochastic bidding strategy of renewable micro-grids in day-ahead and real-time markets. Energy 2019;171:689e700. [3] Jeon HW, Taisch M, Prabhu V. Measuring variability on electrical power demands in manufacturing operations. J Clean Prod 2016;137:1628e46. [4] Davatgaran V, Saniei M, Mortazavi SS. Optimal bidding strategy for an energy hub in energy marke. Energy 2018;148:482e93. [5] Li G, Shi J, Qu X. Modeling methods for GenCo bidding strategy optimization in the liberalized electricity spot markete-A state-of-the-art review. Energy 2011;36:4686e700. [6] Nojavan S, Najafi-Ghalelou A, Majidi M, Zare K. Optimal bidding and offering strategies of merchant compressed air energy storage in deregulated electricity market using robust optimization approach. Energy 2018;142:250e7. [7] Marquardt J. Conceptualizing power in multi-level climate governance. J Clean Prod 2017;154:167e75. [8] Grilli L. Deregulated electricity market and auctions: the Italian case, vol. 2. Sientific research an academic publisher; 2010. p. 238e42. [9] Deng J, Wang H-B, Wang C-M, Zhang G-W. A novel power market clearing model based on the equilibrium principle in microeconomics. J Clean Prod 2017;142:1021e7. [10] Nogales FJ, Contreras J, Conejo AJ, Espinola R. Forecasting next-day electricity prices by time series models. IEEE Trans Power Syst 2002;17:342e8. [11] Zhang J, Tan Z, Yang S. Day-ahead electricity price forecasting by a new hybrid method. Comput Ind Eng 2012;63:695e701. [12] Yang Z, Ce L, Lian L. Electricity price forecasting by a hybrid model, combining wavelet transform, ARMA and kernel-based extreme learning machine methods. Appl Energy 2017;190:291e305. [13] Khashei M, Mokhatab rafiei F, Bijari M. Hybrid Fuzzy auto-regressive integrated moving average (FARIMAH) model for forecasting the foreign exchange markets. 2013. [14] Lora AT, Santos JMR, Exposito AG, Ramos JLM, Santos JCR. Electricity market price forecasting based on weighted nearest neighbors techniques. IEEE Trans Power Syst 2007;22:1294e301. [15] Nogales FJ, Conejo AJ. Electricity price forecasting through transfer function models. J Oper Res Soc 2006;57:350e6.

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Please cite this article as: Ostadi B et al., Risk-based optimal bidding patterns in the deregulated power market using extended Markowitz model, Energy, https://doi.org/10.1016/j.energy.2019.116516