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Wind power bidding strategy in the short-term electricity market
Accepted Manuscript Wind Power Bidding Strategy in the Short-term Electricity Market
Shaomao Li, Chan S. Park PII: DOI: Reference:
S0140-9883(18)30359-1 doi:10.1016/j.eneco.2018.08.029 ENEECO 4142
To appear in:
Energy Economics
Received date: Revised date: Accepted date:
11 September 2016 24 June 2018 26 August 2018
Please cite this article as: Shaomao Li, Chan S. Park , Wind Power Bidding Strategy in the Short-term Electricity Market. Eneeco (2018), doi:10.1016/j.eneco.2018.08.029
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Wind Power Bidding Strategy in the Short-term Electricity Market Shaomao Lia,∗, Chan.S Parka Department of Industrial and Systems Engineering, Auburn University, Auburn, AL 36830, USA
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Abstract
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This paper presents an analytical trading electricity model for wind power producers (WPPs) in the short-term electricity market in the U.S. This model addresses four specific uncertainties: real-time (RT) wind power generation, day-ahead (DA) locational marginal prices (LMPs), RT LMPs, and deviation penalty rates. The model is designed to find the optimal bidding strategy to maximize the expected revenue under these uncertainties. In addition, this paper shows advanced forecasting techniques could be used with the proposed bidding strategy to help WPPs trade energy in short-term markets. A case study is presented to illustrate the effectiveness of this proposed bidding strategy and advanced forecasting techniques using a set of real data taken from a wind farm in the PJM electricity market.
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Keywords: wind power, bidding strategy, forecasting model, short-term market, analytical method
1. Introduction
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Use of wind energy has rapidly expanded around the world. Wind power is expected to continue its fast growth in many countries, including the U.S. and China. However, the uncertainty and variability of wind power give rise to several challenges of power systems and the electricity market. From WPPs’ point of view, it is essential to participate in the electricity market and gain as much revenue as possible. In general, WPPs in the U.S. prefer to commit to power purchasing agreements (PPAs) to sell the generated wind power, as these arrangements bring in stable cash flow for a relatively long term, typically over 10 years. Besides PPAs, WPPs can also participate in two other short-term markets, the day-ahead (DA) market and the real-time (RT) market. The DA market operates like a forward market, and the RT market is often referred to as a balancing market. When generation companies ∗ Corresponding
author Email address: [email protected] (Shaomao Li )
Preprint submitted to Journal of energy economics
September 3, 2018
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commit to sell a certain amount of power in the DA market, imbalances between the DA committed quantity and the actual wind power generation must be rectified in the RT market. Due to the uncertainties of wind power generation and electricity market prices, it is challenging for WPPs to develop a bidding strategy to sell power in those two short-term electricity markets. In this paper, we intend to find how much power they should commit to the DA market. In general, WPPs will generate at its maximum available capacity in RT market due to the low marginal production cost. Since electricity markets around the world have been reformed and wind generation has penetrated the market only recently, an optimal bidding strategy for WPPs to maximize profits in the short-term electricity market is still a relatively new problem. There are some approaches to deal with the bidding strategy problem for WPPs. Bathurst et al. [1] developed a bidding strategy using Markov probabilities for a wind farm in the United Kingdom to determine the best energy contract level; this study also demonstrated how a change in market closure delay and in the number of trades per day impacts the overall expected imbalance costs. Matevosyan and Soder [2] developed a bidding strategy as a stochastic mixed integer programming problem to maximize expected profit and minimize the imbalance costs for WPPs in the Nordic electricity market. ARMA model was used to simulate scenarios for wind speed forecasting errors. Pinson et al. [3] proposed different strategies by the analytical method for point prediction and probabilistic wind forecasting cases. They followed the Dutch electricity pool but used a wind farm in Ireland in their case study. Pousinho, et al. [4] proposed a two-stage stochastic programming model to deal with two kinds of uncertainty: wind power and electricity market prices. A wind farm in Portugal was used in the case study. The aforementioned literatures focus on maximizing the profit for WPPs in short-term electricity markets. Optimal bidding strategies for WPPs taking into risks have been extensively researched. Morales et al. [5] and Hosseini-Firouz [6] built a multi-stage stochastic programming model which considered risk management. This model applied ARIMA techniques to predict electricity prices and wind speeds. Their case study used wind speed of the state of Kansas and historical prices of the Iberian Peninsula electricity market for the fitting process in their case study. Catalao, et al. [7] proposed a two-stage stochastic programming model which used a hybrid intelligent approach to generate the scenario tree. A wind farm in Portugal was used in the case study. Botterud, et al. [8] derived optimal DA bids under different assumptions for risk preferences and deviation penalty schemes in the U.S. market. Their model used a generalized reduced gradient algorithm to solve the nonlinear programming problem. They emphasized benefits of using advanced wind power forecasting methods. Dent et al. [9] built a bidding strategy for WPPs in the Great Britain market using the analytic method, and presented analytical expressions for determining the optimal forward market strategy under the risk-neutral and risk-averse cases. Zhang et al. [10] proposes three different bidding strategies, i.e., the expected profit-maximization strategy, the chance-constrained programming-based strat2
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egy and the multi-objective bidding strategy. Analytical solutions under the three strategies were obtained and they are based on assuming normal distribution of wind power generation. They tested bidding strategies on a hypothetical wind farm following Spanish market rules. Vilim and Botterud [11] proposed a stochastic model with considering the price effects of wind power production and consumption. They developed and compared two bidding algorithm in the Nord Pool electricity market. Dai and Qiao [12] proposed a bilevel stochastic optimization model to obtain the optimal bidding strategy for a strategic WPP with considering the uncertainties in the demand, the wind power production, and the bidding strategies of the strategic conventional power producers. The bilevel model is transferred into a mixed-integer linear problem by using the duality theory and Karush-Kuhn-Tucker condition. Guerrero-Mestre, et al. [13] proposed an optimal joint offer to the DA market through an external agent for a group of wind farms and compared it with other offering strategies using a stochastic programming approach. A case study in the electricity market in Spain was carried out with extended results to compare different strategies for offering energy to the DA market considering different risk aversions. Baringo and Conejo [14] proposed a multi-stage risk-constrained stochastic complementarity model for the optimal offering strategy of a WPP with a large amount of installed capacity, which behaves as a price-maker in both the DA and the balancing markets. There is also literature discussing other ways of maximizing profit for WPPs, such as combining wind power with energy storage [15, 16, 17] or hydro power [18, 19, 20]. This paper not only presents an analytical model of the optimal DA market bidding strategy for WPPs in U.S. electricity markets; it also takes advantage of forecasting techniques, historical data and valuable market information to help bid like WPPs do in real world. This paper suggests the following improvements:
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1 No need to generate scenarios because it does not require generating a great number of scenario trees or solving complex LP problems.
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2 Test the effectiveness of the strategy with real world data provided by a wind power generation company in the U.S. electricity market.
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3 Utilize advanced forecasting techniques to improve the accuracy of prediction of revenue streams. 4 Incorporate valuable market information, such as forecasted ISO demands, forecasted and generated hourly wind power generations, to help WPPs bid to the short-term market.
The rest of the paper is organized as follows: Section II briefly outlines the market structure in the U.S. short-term electricity markets and the bidding strategy problem faced by WPPs. Section III presents an analytic method for developing a bidding strategy model for WPPs. This section also introduces beta distribution to represent the probability density function (pdf) of wind power generation prediction errors, and various forecasting techniques to predict
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electricity prices and generation outputs. Section V provides case studies with result analysis and comparison. Finally, Section VI presents several relevant conclusions and possibilities for future work.
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2. Market structure and problem description
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The main symbols used in this paper:
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πh Revenue from energy market, hour h (h = 1 . . . 24), [$]. qDA,h Quantity bid into DA market, hour h, [MWh]. qRT,h Actual delivery, hour h, [MWh]. qmax Wind farm generation capacity, [MW]. pDA,h , pRT,h DA and RT LMPs, hour h, [$/MWh]. pen deviation penalty rates for deviation between DA schedule and RT delivery, [$/MWh]. f (qRT,h ) Probability density function of RT wind power generation. F (qRT,h ) Cumulative distribution function of RT wind power generation. qpred Normalized predicted power
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Electricity markets in the U.S. are operated by independent system operators (ISOs) or regional transmission organizations (RTOs), which typically provide a two-settlement system for market participants to trade electricity. The short-term electricity Markets consists of two markets, a DA market and a RT balancing market. The DA market is a forward market in which hourly clearing prices are calculated for each hour of the next operating day based on generation offers, demand bids, increment offers, decrement bids and bilateral transaction schedules submitted into the DA market. The balancing market is the RT energy market in which the clearing prices are calculated every five minutes based on the actual system operations security-constrained economic dispatch. The DA market settlement is based on scheduled hourly quantities and on DA hourly prices, the Real-time settlement is based on actual hourly (integrated) quantity deviations from DA scheduled quantities and on RT prices integrated over the hour. [21] To recover uplift costs due to deviations, some ISOs impose deviation penalty charges to generation companies which cause such problems. Unlike the conventional power generator, WPPs must pay these penalty fees frequently due to the unpredictability of wind power generation. The revenue gained from the two-settlement electricity revenue for an hour, h, includes three parts: DA market revenue, RT market revenue, and deviation penalty charge, as shown in Eq.(1): πh = pDA,h · qDA,h + pRT,h (qRT,h − qDA,h ) − pen|qRT,h − qDA,h |
(1)
ISOs in the U.S. share similar time frameworks for the short-term electricity market. This paper uses PJM as an example of how the short-term electricity market operates in the U.S. The Day-ahead Energy Market bid/offer period closes at 10:30am on the day before the Operating Day. At 10:30am PJM 4
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begins to run the DA market clearing software to determine the hourly commitment schedules and the LMPs for the DA market. PJM posts the DA hourly schedules and LMPs By 13:30pm. During 13:30pm and 14:15pm, the balancing market offer period are opened for market participants to submit revised offers for resources not selected in the DA market. At 14:15pm, PJM performs a second resource commitment, which includes the updated offers, updated resource availability information, and updated PJM load forecast information and load forecast deviation. PJM may perform additional resource commitment runs, as necessary, based on updated PJM load forecasts and updated resource availability information. During the operating day, the RT market is organized to guarantee real time balance between power supplies and demands. If generation companies produce less power than their DA scheduled quantities, they must buy power from the RT market at RT LMPs; if they produce more power than the DA scheduled quantities, they are paid by the market at RT LMPs. This arrangement is represented by the second part of Eq.(1).[21] In PJM, deviation charges are collected to pay for Balancing Operating Reserve in the balancing market. Total daily costs of balancing operating reserve for the Operating Day is allocated and charged to PJM Members in proportion to their locational realtime deviations from day-ahead schedules. As accounting for Operating Reserve is performed on a daily basis, pen is identical for every hour in the same day. WPPs must decide every day how much power they should commit to the DA market for each hour of the next operating day. They face four main uncertainties: pDA,h , pRT,h , qRT,h and pen. When WPPs submit offers to DA markets for the next operation day, they need to forecast wind power generation 13.5 to 37.5 hours ahead, and in the meantime, they are also unaware of hourly DA LMPs, RT LMPs and deviation penalty rates, all uncertainties make the bidding problem complex. 3. Mathematical formulation
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In this paper, we assume that WPPs are price takers, i.e., their bidding strategies do not affect market LMPs, as generation outputs of most wind farms are relatively small compared to generation outputs of other resources. Also, wind power is usually not a marginal cost in the electricity market. In reality, the above assumption may not hold all the time, especially for those regions with high share of wind power. This paper targets at those WPPs who do not have market power. We also assume WPPs are risk-neutral, meaning they make hourly decisions based on the expected value of revenue without considering risks. The reason for this risk-neutral assumption is that hourly profits are less important for risk preference because of the large number of hourly trades in a year. 3.1. Analytical model development This section proposes an analytical method to solve the bidding strategy problem. Eq.(1) is the revenue function used to calculate revenue from shortterm markets for WPPs, whom are assumed to be risk-neutral. The goal of the 5
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proposed model is to maximize the expected value of the revenue: M ax
E[πh (qDA,h )]
(2)
s.t. 0 ≤ qDA,h ≤ qmax
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ifE[pRT,h ] − E[pDA,h ] > E[pen] ifE[pDA,h ] − E[pRT,h ] > E[pen] if 0 ≤ a ≤ 1
E[pen] + E[pDA,h ] − E[pRT,h ] 2E[pen]
(4)
(5)
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problem can be solved as follows:
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As described in Appendix, this 0, * (qDA,h ) = qmax , −1 F (a),
(3)
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Where F −1 (a) is the inverse distribution function (also called quantile function) of qRT,h . From the above results, we can see that once a is obtained in Eq.(5), the optimal (qDA,h )* can be easily obtained according to Eq.(4). For instance, if a is 0.35, then the optimal DA bid for that specific hour should be the 35th percentile of real-time wind power generation distribution. When the expected real-time price is higher than the sum of expected day-ahead price and deviation penalty rate, WPPs should allocate 0 in the DA market as they get paid higher in RT market even they need to pay the penalty cost for the deviation. When the expected day-ahead Price is higher than the sum of expected real-time price and deviation Penalty rate, WPPs should bid their full capacity in the DA market as they get paid higher in DA market even they need to pay the penalty cost for the deviation.
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3.2. Bidding strategy process
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The process of the proposed optimal bidding strategy can be summarized in four steps:
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Step 1 Develop f (qRT,h ) by the historical data. In this paper, the pdf of RT wind power generations and the pdf of wind power generations prediction errors are used as f (qRT,h ). Step 2 Estimate the expected values of pDA,h , pRT,h and pen for each hour of the next operating day, either by prediction techniques or by calculating means of the historical data. Step 3 Plug those expected values into Eq.(5) to obtain a. Step 4 Refer to Eq.(4) to find (qDA,h )* .
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3.3. f (qRT,h ) and beta distribution
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From the above analysis, we know that it is critical to develop an appropriate f (qRT,h ) to find the optimal (qDA,h )* . In this paper, two pdfs are used: the pdf of historical actual wind power generation and the pdf of wind power generation prediction errors. The reason these two pdfs are used is that the historical data of point forecasted hourly wind power generation and actual generated wind power generation is available. Both pdfs are applied in the case study. Although the Weibull distribution is commonly employed to describe wind power generation, it might be inappropriate to describe wind power generation prediction errors. This paper uses a beta distribution function to describe wind power generation prediction errors by analyzing the historical hourly data of forecasted wind power generation and generated wind power generation. Beta distribution is used for two reasons. First, the beta distribution can accurately represent the pdf of the wind power prediction errors. Second, this mathematical method can easily find the percentile value (qDA,h )* of other general wind farms if two parameters of the beta distribution are roughly known, rather than exact historical data. Beta function that models the occurrence of RT wind power generation x if a certain prediction power is given is as follows: (6)
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fq (x) = xα−1 (1 − x)β−1 ∗ n
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where α and β are two parameters used in beta function, and n is a constant [22]. In this paper, α and β are calculated based on the available data. Once α and β are fixed, the inverse distribution function of beta distribution can be easily used to find (qDA,h )* in Matlab.
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3.4. Forecasting techniques
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LMPs and deviation penalty rates forecasting are essential in the proposed bidding strategy, as a derived in Eq.(5) is determined by their expected values. We introduce four prediction techniques in this work: the cubic spline, ARMA, ARMAX and state space models (SSM). The latter three models are time series models. Except for ARMA, the other three prediction models include forecasted ISO demands as an explanatory variable.
3.4.1. ARMA model ARMA relates current prices to past prices, and current errors to previous errors as well. The general ARMA (p,q) model has the following form yt = φ1 yt−1 + · · · + φp yt−p − (θ1 εz−1 + · · · + θq εt−q ) + εt
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where εt is a white noise process, and φp and θq are coefficients of backshift operators [23]. ARMA(3,1) is used in the case study.
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3.4.2. ARMAX model An ARMAX model simply adds one explanatory variable to the right side of an ARMA model; an ARMAX model relates current price to past prices and current forecasted demands. The general ARMAX model has the following form (8)
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yt = βxt + φ1 yt−1 + · · · + φp yt−p − (θ1 εz−1 + · · · + θq εt−q ) + εt
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where xt is the explanatory variable, which represents the forecasted demand in our case, whose coefficient is β. [24].
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3.4.3. State space model The state space model (also called the dynamic linear model) provides a mathematically rigorous treatment of time series modeling based on a Bayesian approach. By allowing for variability in the regression coefficients, this model can let the system properties change in time. The general state space model has the following form ( Yt = Ft θt + t , t ∼ N (0, Vt ) (9) θt = Gt θt−1 + ωt ωt ∼ N (0, Wt )
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where Yt is the observation at time t, θt is the vector of parameters, Ft is the row vector of covariates at time t, Gt is the evolution matrix, and t and ωt are the observation error and the evolution error at time t, respectively. The upper part is called the observation equation and the lower part is called the evolution equation[25]. In the case study, ARMAX in the state space model is used to forecast LMPs and deviation penalty rates.
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3.4.4. Cubic spline A cubic spline model is not a time series model, as it only relates current prices to forecasted demands. A cubic spline is a special function defined piecewise by a cubic polynomial. General cubic spline model has the following form
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yh = β0 +β1 xh +β2 x2h +β3 x3h +β4 (xh −a1 )3+ +β5 (xh −a2 )3+ +· · ·+βk+3 (xh −ak )3+ (10) where k is the number of knots, ak is the value of kth knot, and β is the coefficient [26]. The above four forecasting models are applied to predict LMPs and deviation penalty rates in the case study in R software, and the relevant results are presented in the following section. 4. Case study In this section, we apply the proposed bidding strategy model to a largescale wind farm, which is within PJM in the U.S. and has a total installed 8
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generation capacity of 102 MW. This wind farm trades the generated electricity in the short-term electricity market operated by PJM. We first describe and analyze the historical data of this wind farm. Then we present the results of the proposed bidding strategy model test. Finally, we compare the results of bidding strategies with different forecasting techniques.
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4.1. Description of data
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Hourly data from Jan. 1st to Dec. 31th, 2013 is used, including DA LMPs, RT LMPs, deviation penalty rates, forecasted ISO demands, forecasted wind power generation and generated RT wind power generation. To make analyses and predictions more accurately, we divide those hourly data into 24 groups, one for each hour, and we treat the first nine months’ data as the training data and the rest as the testing data.
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Figure 1: Histogram and statics of DA and RT LMPs
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Figure 2: Histogram and statics of deviation penalty rates
Figure 3: Histogram of generated RT wind power generation
The histograms and statics of hourly DA and RT LMPs, deviation penalty rates, and generated wind power generation in 2013 are shown in Figs.1-3, respectively. From Fig.1 we can see the mean of DA LMPs is about $1 higher than that of RT LMPs; meanwhile, the volatility of DA LMPs is smaller than that of RT LMPs. Number of negative prices of RT LMPs is much larger than 10
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Table 1: Correlation coefficients among data
RT LMP
pen
Pred. Load
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0.61
0.33
0.64
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0.28
0.52
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0.49
DA LMP RT LMP
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DA LMP
Pred. Load
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Pred. MW
Gen. MW
-0.02
-0.04
-0.05
-0.08
0.01
-0.02
0.01
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DA LMPs. Negative prices mean that producers were actually paying for their energy to be taken to ensure electricity dispatch. This makes an economic sense because of the federal production tax credit. Even if WPPs give the power away or offer the system money to take it, they still receive a tax credit equal to $23/MWh. In Fig.2, most deviation penalty rates are not more than $5/MWh. Fig.3 shows that about half of the time, this wind farm generated less than 20 MWh in 2013.
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Table 1 indicates correlation coefficients between each data item. Pred. denotes Predicted, Gen. denotes generated, MW denotes wind power output, and Load means ISO demand. The correlation coefficient between generated RT wind power generation and LMPs is relatively small and negative. The predicted ISO demand is highly correlated with DA LMPs, RT LMPs and deviation penalty rates. Thus, including this factor in price forecasting models is both reasonable and necessary, and WPPs in the real world can also access the information about predicted ISO demands before they make bidding decisions in the short-term electricity markets. 4.2. The pdf of wind power generation prediction errors
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When we compare the hourly forecasted wind power generation and generated wind power generation data, it is found that the point forecast accuracy is not good enough to be used for the bidding purpose. To make the bidding strategy perform better, we develop the pdf of wind power generation Prediction Errors, which is expressed in the Beta distribution function to represent RT wind power generation distribution when forecasted wind power generation are given. We divide the qpred into 10 ranges with an interval of 0.1. α and β values of beta pdfs for these 10 qpred ranges were calculated in Matlab and are presented in Table 2. We choose 10 ranges because if we chose too many ranges, the available data would be not enough to develop an accurate distribution function for individual ranges, but if we chose too few ranges, the distribution function for individual ranges would become too rough to represent real distributions.
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Table 2: Parameters for 10 individual ranges of qpred
0∼0.1
0.1∼0.2
0.2∼0.3
0.3∼0.4
0.4∼0.5
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0.35
1.17
1.45
1.71
1.36
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0.80
6.57
4.52
3.96
2.19
qpred
0.5∼0.6
0.6∼0.7
0.7∼0.8
0.8∼0.9
0.9∼1.0
α
1.17
1.09
1.39
1.52
1.00
β
1.30
0.97
0.80
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qpred
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0.31
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The real distributions of wind power generation prediction errors for qpred = [0, 0.1], [0.1, 0.2], [0.6, 0.7] and [0.9, 1.0] are presented as examples in Fig.4. Beta distributions of these four ranges are shown in Fig.5, accordingly.
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Figure 4: Real pdfs of wind power generation Prediction Error for qpred = [0, 0.1], [0.1, 0.2], [0.6, 0.7] and [0.9, 1.0]
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Figure 5: Beta pdfs of wind power generation Prediction Error for qpred = [0, 0.1], [0.1, 0.2], [0.6, 0.7] and [0.9, 1.0]
4.3. Results of bidding strategy model tests
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This section tests the effectiveness of the bidding strategy proposed in this paper. We conduct 24 hourly model tests separately by following the steps mentioned in the bidding strategy process section to see whether the optimal (qDA,h ) by the proposed model can achieve the maximum revenue. Because the purpose of this part of the case study is to test the proposed model, we do not take advantage of information of forecasted wind power generation and advanced prediction techniques in this part. Therefore electricity LMPs, deviation penalty rates and wind power generations are treated as stochastic variables, whose probability distributions are based on historical data of those variables in 2013. In the test, the study time is one hour and the goal is to find the optimal (qDA,h ) to maximize the expected one-hour revenue. We first use the whole year real data to do the test, so there are 365 scenarios for individual hours test. The result is showed in Table 3, where (qDA,h )* and (qDA,h )** are the optimal solution obtained by the proposed bidding strategy model and the actual optimal solution by enumeration method, respectively; π(qDA,h )* and π(qDA,h )** denote the revenue by bidding (qDA,h )* and (qDA,h )** to the market, respectively; Diff.q and Diff.π denotes the difference between (qDA,h )* and (qDA,h )** , and the difference between π(qDA,h )* and π(qDA,h )** , respectively. Means of historical data are used as the expected value of DA LMPs, RT LMPs, and deviation penalty rates to calculate a in step 3 of the bidding strategy process. The pdf of RT wind power generation showed in Fig.3 is used as f (qRT,h ) in step 4 to obtain the (qDA,h )* .
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Table 3: Comparison for individual hours, real data
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π ** 859.0 713.7 651.8 638.7 593.7 637.4 787.4 904.4 826.0 844.9 881.8 883.5 919.7 945.8 1169.3 1197.6 1193.8 1358.5 1299.4 1143.7 1122.4 1087.2 1056.7 907.0 $
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π* 859.0 713.7 651.8 638.7 593.7 636.7 787.4 904.4 825.0 844.4 881.8 881.9 919.7 945.3 1169.3 1197.6 1193.7 1358.5 1299.4 1143.7 1122.3 1087.1 1056.7 907.0 $
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Diff.q 0 1 0 0 1 10 2 0 8 3 0 7 0 5 0 0 5 0 0 0 4 2 0 0 MWh
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(qDA,h )** 102 28 67 102 63 55 8 101 39 9 0 10 0 4 102 102 94 102 102 84 44 17 102 91 MWh
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(qDA,h )* 102 27 67 102 64 65 10 101 47 12 0 17 0 9 102 102 99 102 102 84 48 15 102 91 MWh
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hours 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 unit
Diff.π 0.0 0.0 0.0 0.0 0.0 0.7 0.0 0.0 1.0 0.5 0.0 1.6 0.0 0.5 0.0 0.0 0.1 0.0 0.0 0.0 0.1 0.1 0.0 0.0 $
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Table 4: Comparison for individual hours, simulated data
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π ** 860.4 715.8 660.0 653.9 581.6 624.9 768.6 862.7 815.3 824.1 913.5 909.0 989.4 1019.4 1195.7 1216.9 1278.7 1372.5 1296.0 1136.4 1107.1 1080.2 1035.7 924.4 $
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π* 860.4 715.8 660.0 653.9 581.6 624.8 768.6 862.7 815.2 824.1 912.9 909.0 989.4 1019.4 1195.7 1216.9 1278.7 1372.5 1296.0 1136.3 1107.1 1080.2 1035.7 924.4 $
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Diff.q 0 0 2 0 1 1 1 2 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 MWh
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(qDA,h )** 102 29 65 102 60 40 6 96 35 7 0 13 1 1 102 102 101 102 102 81 33 15 102 99 MWh
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(qDA,h )* 102 29 67 102 59 39 5 98 34 7 1 13 1 1 102 102 101 102 102 81 33 15 102 99 MWh
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hours 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 unit
Diff.π 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.1 0.0 0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 $
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From Table 3, we can find that for most hours the difference between (qDA,h )* and (qDA,h )** are fairly small. Even for those hours whose optimal qDA,h differences are relatively large, such as 6am and 12pm, their one-hour revenue differences turn out to be small. The reason these differences exist is partly because the sample size for each hour is not large enough; some extreme values of variables will affect the final result to some extent. When we enlarge the scenario number from 365 to 1,000,000 by Monte Carlo simulation method, both qDA,h differences and the revenue differences are largely reduced, which can be found in Table 4. The distributions of those random variables in the simulation case are almost the same as the ones in the previous case. Both the method and procedure to find (qDA,h )* and (qDA,h )** in this simulation are the same as the ones used in the previous case. The test of significance indicates that (qDA,h )* and (qDA,h )** are equivalent to each other in a statistic sense. The above results indicate the proposed optimal bidding strategy works well in the real world as intended. 4.4. Results of wind power generation prediction errors
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In this part of case study, we compare the revenues by applying the same proposed bidding strategy but using different f (qRT,h ) in step 4 of the bidding strategy process: the pdf of RT wind power generation and the pdf of wind power generation forecasting errors. The main difference between these two pdfs is that the latter one includes information of forecasted wind power generation and relevant historical distribution, whereas the former only represents the distribution of historical wind power generation outputs. Note that in this paper, we apply no forecasting technique to predict wind power generation; all the forecasted wind power generation data is given. It needs to be pointed out that both the pdf of RT wind power generation and the pdf of wind power generation forecasting errors are expressed in the Beta distribution function, and both models use historical means as the expected values of pDA,h , pRT,h and pen in Eq.(5) to obtain the optimal (qDA,h )* .
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Figure 6: Revenues comparison between bidding strategies with the pdf of RT wind power generation and with the pdf of wind power generation prediction errors
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Fig.6 shows bidding results, where the vertical axis is the whole year revenue for each hour. In Fig.6, most groups with the pdf of wind power generation prediction errors outperform the ones with the pdf of wind power generation, and the total revenue difference in 2013 is $434,578, which accounts for about 6.7% of the total revenue. From the above results, we can see that information of forecasted wind power generation does help WPPs gain the extra revenue in the short-term electricity market if applying the proposed bidding strategy.
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Figure 7: Revenues comparison between bidding the predicted wind power generation and bidding strategy with the pdf of wind power generation prediction errors
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Fig.7 shows results of bidding the predicted wind power generation model and the proposed bidding strategy with the pdf of wind power generation prediction errors as f (qRT,h ). The former just bids the forecasted wind power generation quantity to the DA market without using the proposed strategy model. The latter is the same as in the previous case. In Fig.7, nearly all groups of bidding strategies, with the pdf of wind power generation prediction errors, outperform the ones of bidding predicted wind power generation, and the total revenue difference in 2013 is $249,797. In other words, the proposed bidding strategy model with the pdf of wind power generation prediction errors as f (qRT,h ) helps WPPs improve total revenue of bidding predicted wind power quantity to the DA market in 2013 by approximately 3.7%. The above results indicate again that the proposed bidding strategy model works well in the real world.
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4.5. Results comparison of different bidding strategies and forecasting techniques In previous cases, we assumed the expected values of LMPs for the same hour remain constant in 2013, which is oversimplified in the real world, where WPPs could take advantage of valuable information, like forecasted ISO demands, and advanced forecasting techniques to predict next day’s LMPs and deviation penalty rates accurately. In this part of the case study, we will show how to incorporate forecasting techniques into the proposed optimal bidding strategy. The major difference between these bidding strategies with forecasting techniques and previous ones is that the expected values of DA LMPs, RT LMPs, and deviation penalty rates will be predicted rather than being means of historical data. Therefore, for individual hours, each day’s qDA,h could be different
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Table 5: Results of different bidding strategies
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Difference /$ 18,096 8,728 6,803 0 -19,024 -55,854 -63,351 -103,029 -121,893
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Profit /$ 1,915,076 1,905,708 1,903,783 1,896,980 1,877,956 1,841,126 1,833,629 1,793,951 1,775,087
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Bidding Strategies perfect forecasting model cubic spline model state space model bid historical mean model ARMAX Model bid predicted wind power generation model ARMA Model Bid full capacity model bid 0 MW model
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if the calculated a in Eq.(5) changes. Four advanced forecasting models are adopted in this work: cubic spline, ARMA, ARMAX and state space models. Besides these four models, we also present five other bidding strategy models for comparison. The result of 3-month revenues by these nine bidding strategy models are presented in Table 5.
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In Table 5, bid 0 MWh model and bid full capacity model mean that WPPs bid 0 MWh and 102MWh to the DA market, respectively. Historical mean model applies the proposed bidding strategy and uses the means of historical data as the expected values of those random variables in Eq.(5), which is the same as the bidding strategy with the pdf of wind power generation prediction errors in the previous section. Perfect forecasting model assumes that wind power generation forecasting is 100% accurate, and WPPs bid these forecasted quantities to the DA market. There will be no imbalance cost in this case, and it serves as a benchmark of perfect information, since it is unrealistic in the real world. Note further that, within these nine bidding strategies, five follow the same proposed optimal bidding strategy process but with different approaches to obtain the expected values of LMPs and deviation penalty rates. These five strategies include four forecasting models and historical mean model. Beta distribution function of wind power generation prediction errors is used in these models. To make accurate predictions, within ARMA, ARMAX and state space models, we develop 24 hourly forecasting models to predict the DA LMPs, RT LMPs and deviation penalty rates of next operation day’s individual hours. Table 5 shows that either bidding nothing or bidding full capacity to DA market are bad strategies, because, as predicted, their deviation costs become relatively high in these cases. Except for the ARMA model, all the models that apply the proposed bidding strategy gain revenues higher than the bid predicted wind power generation model. Bid historical mean model serves as a benchmark to calculate revenue difference between those bidding models in Table 5. Among those models with forecasting techniques, the state space model and the cubic spine model achieve the highest two revenues, gaining about $6,803 and $8,728 over the bid historical 19
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mean model, respectively. This promising result indicates some advanced forecasting techniques would be beneficial to the proposed optimal bidding strategy. The state space model and the cubic spine model are recommended to predict market prices in this paper.
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5. Conclusion
6. Appendix
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This paper presents an optimal bidding strategy model for WPPs to trade power in the short-term electricity market. We test it in a case study with real-world data and compare it with the model by the Monte Carlo simulation method. The results of both cases indicate the proposed model looks promising. The analytic model is much simpler and more computationally efficient than the simulation method which needs to develop a large scenarios set. The proposed model can also utilize the information of forecasted ISO demands, forecasted and generated wind power generations to help bid. To make the model more accurate and efficient, fitted beta distribution functions are used to describe the error between forecasted value and realized value of wind power generation. This work also shows that advanced forecasting techniques would help the proposed optimal bidding strategy. The state space model and cubic spline model gain higher revenues than other forecasting models in the case study. Further research will be conducted to improve the accuracy of prediction techniques, and to extend the proposed model to other electricity markets with different market rules.
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The goal of the proposed model is to maximize the expected value of the revenue: M ax
E[πh (qDA,h )]
(11)
s.t. 0 ≤ qDA,h
(12)
qDA,h ≤ qmax
(13)
Due to the assumption that WPPs are price takers, Eq.(11) can be converted
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as follows: E[πh (qDA,h )]
qDA,h
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Z qDA,h − E[pen] · (− (qRT,h − qDA,h )f (qRT,h )d(qRT,h ) 0 Z qmax (qRT,h − qDA,h )f (qRT,h )d(qRT,h )) +
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= E[pDA,h · qDA,h ] + E[pRT,h (qRT,h − qDA,h )] − E[pen · |qRT,h − qDA,h |] (14) Z qDA,h = E[pDA,h · qDA,h ] + E[pRT,h ] · ( (qRT,h − qDA,h )f (qRT,h )d(qRT,h ) 0 Z qmax + (qRT,h − qDA,h )f (qRT,h )d(qRT,h ))
qDA,h
Z
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= E[pDA,h · qDA,h ] qDA,h
+ (E[pRT,h ] + E[pen]) ·
(15)
+ (E[pRT,h ] − E[pen]) ·
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(qRT,h − qDA,h )f (qRT,h )d(qRT,h ) Z0 qmax
(qRT,h − qDA,h )f (qRT,h )d(qRT,h ) qDA,h
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We know that :
(16)
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Rq d( 0 DA,h (qRT,h − qDA,h )f (qRT,h )d(qRT,h )) d(qDA,h )
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= (qDA,h − qDA,h )f (qDA,h ) · 1 − (0 − qDA,h )f (0) · 0 Z qDA,h + (−1)f (qRT,h )d(qRT,h ) Z0 qDA,h =− f (qRT,h )d(qRT,h )
(17)
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R qmax d( qDA,h (qRT,h − qDA,h )f (qRT,h )d(qRT,h )) d(qDA,h )
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= (qmax − qDA,h ) · f (qmax ) · 0 − (qDA,h − qDA,h ) · f (qDA,h ) · 1 Z qmax + (−1)f (qRT,h )d(qRT,h ) qDA,h qmax
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f (qRT,h )d(qRT,h )
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qDA,h
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With Eq.(17) and Eq.(18), the first derivative of Eq.(11) with respect to qDA,h : Z qDA,h d(E[πh (qDA,h )]) =E[pDA,h ] − (E[pRT,h ] + E[pen]) · ( f (qRT,h )d(qRT,h )) d(qDA,h ) 0 Z qmax + (E[pRT,h ] − E[pen]) · (− f (qRT,h )d(qRT,h )) =E[pDA,h ] − (E[pRT,h ] + E[pen]) · (F (qDA,h ))
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+ (E[pRT,h ] − E[pen]) · (−(1 − F (qDA,h )))
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qDA,h
(19)
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=E[pen] − E[pRT,h ] + E[pDA,h ] − 2E[pen] · F (qDA,h )
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Note that F (qDA,h ) is the cumulative distribution function of qRT,h when qRT,h is equal to qDA,h . The second derivative of Eq.(11) with respect to qDA,h : d2 (πh (qDA,h )) = −2E[pen] · f (qDA,h ) ≤ 0 d(qDA,h )2
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L(qDA,h , λ, ϕ) =
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The above result indicates that the objective function is concave in qDA,h . Define the Lagrangian as E[πh (qDA,h )] − λ(qDA,h − qmax ) − ϕ(0 − qDA,h )
(21)
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The Karush−Kuhn−Tucker conditions:
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d(L) d(E[πh (qDA,h )]) = −λ+ϕ=0 d(q d(qDA,h ) DA,h ) λ > 0, qDA,h = qmax ; or λ = 0 ϕ > 0, qDA,h = 0; or ϕ = 0
(22)
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By plugging Eq.(19) into Eq.(22), we can obtain the optimal solution: ifE[pRT,h ] − E[pDA,h ] > E[pen] 0, * (23) (qDA,h ) = qmax , ifE[pDA,h ] − E[pRT,h ] > E[pen] −1 F (a), if 0 ≤ a ≤ 1 a=
E[pen] + E[pDA,h ] − E[pRT,h ] 2E[pen]
(24)
Where F −1 (a) is the inverse distribution function of qRT,h .
References [1] G. N. Bathurst, J. Weatherill, G. Strbac, Trading wind generation in short term energy markets, IEEE Trans. Power Syst. 17 (3) (2002) 782–789. [2] J. Matevosyan, L. Sder, Minimization of imbalance costs trading wind power on the short-term power market, IEEE Trans. Power Syst. 21 (3) (2006) 1396–1404. 22
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[3] P. Pinson, C. Chevallier, G. N. Kariniotakis, Trading wind generation from short-term probabilistic forecasts of wind power, IEEE Trans. Power Syst. 22 (3) (2007) 1148–1156.
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[4] H. M. I. Pousinho, V. M. F. Mendes, J. P. S. Catalao, A stochastic programming approach for the development of offering strategies for a wind power producer, Electric Power Systems Research 89 (2012) 45–53.
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[5] J. M. Morales, A. J. Conejo, J. Prez-Ruiz, Short-term trading for a wind power producer, IEEE Trans. Power Syst. 25 (1) (2010) 554–564.
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[6] M. Hosseini-Firouz, Optimal offering strategy considering the risk management for wind power producers in electricity market, Int J Electr Power Energy Syst. 49 (2013) 359–368.
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[7] J. P. S. Catalao, H. M. I. Pousinho, V. M. F. Mendes, Optimal offering strategies for wind power producers considering uncertainty and risk, IEEE Trans. Power Syst. 6 (2) (2012) 270–277.
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[8] A. Botterud, Z. Zhou, J. Wang, R. J. Bessa, H. Keko, J. Sumaili, V. Miranda, Wind power trading under uncertainty in LMP markets, IEEE Trans. Power Syst. 27 (2) (2012) 894–903.
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[9] C. J. Dent, J. W. Bialek, B. F. Hobbs, Opportunity cost bidding by wind generators in forward markets: Analytical results, IEEE Trans. Power Syst. 26 (3) (2011) 1600–1608.
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[10] H. Zhang, F. Gao, J. Wu, K. Liu, X. Liu, Optimal bidding strategies for wind power producers in the day-ahead electricity market, Energies 5 (2012) 4804–4823. [11] M. Vilim, A. Botterud, Wind power bidding in electricity markets with high wind penetration., Applied Energy 118 (2014) 141–155.
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[12] T. Dai, W. Qiao, Optimal bidding strategy of a strategic wind power producer in the short-term market, IEEE Transactions on Sustainable Energy 6 (2015) 707–719.
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[13] V. Guerrero-Mestre, A. A. S. de la Nieta, J. Contreras, J. P. S. Catalo, Optimal bidding of a group of wind farms in day-ahead markets through an external agent, IEEE Transactions on Power Systems 31 (2016) 2688– 2700. [14] L. Baringo, A. J. Conejo, Offering strategy of wind-power producer: A multi-stage risk-constrained approach, IEEE Transactions on Power Systems 31 (2016) 1420–1429. [15] G. N. Bathurst, G. Strbac, Value of combining energy storage and wind in short-term energy and balancing markets, Electric Power Syst. Res. 67 (2003) 1–8. 23
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[16] I. D. A. Dukpa, B. Venkatesh, L. Chang, Optimal participation and risk mitigation of wind generators in an electricity market, IET Renewable Power Gener. 4 (2) (2010) 165–175.
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[17] S. Y. Wang, J. L. Yu, Optimal sizing of the caes system in a power system with high wind power penetration, Int. J. Electrical Power & Energy Syst. 37 (1) (2012) 117–125.
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[18] A. Jaramillo, E. D. Castronuovo, I. Snchez, J. Usaola, Optimal operation of a pumped-storage hydro plant that compensates the imbalances of a power producer, Electr. Power Syst. Res. 81 (9) (2011) 176–177.
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[19] E. D. Castronuovo, J. A. P. Lopes, On the optimization of the daily operation of a wind-hydro power plant, IEEE Transactions on Power Systems 19 (3) (2004) 1599–1606.
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[20] A. A. S. de la Nieta, J. Contreras, J. I. Muoz, Optimal coordinated wind-hydro bidding strategies in day-ahead markets, IEEE Transactions on Power Systems 28 (2) (2013) 798–809. [21] Pjm manual 11: Energy & ancillary services market operations (2017). URL http://www.pjm.com/library/manuals.aspx
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[22] A. Fabbri, T. G. S. Romn, J. R. Abbad, V. H. M. Quezada, Assessment of the cost associated with wind generation prediction errors in a liberalized electricity market, IEEE Trans. Power Syst. 20 (3) (2005) 1440–1446.
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[23] R. J. Hyndman, G. Athanasopoulos, Forecasting: principles and practice (2012). URL https://www.otexts.org/fpp [24] R. J. Hyndman, The arimax model muddle (2010). URL http://www.r-bloggers.com/the-arimax-model-muddle/
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[25] G. Petris, S. Petrone, P. Campagnoli, Dynamic Linear Models with R, Springer, 2009.
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[26] M. Kuhn, K. Johnson, Applied Predictive Modeling, Springer, 2013.
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Highlights • No need to generate scenarios because it does not require generating a great number of scenario trees or solving complex LP problems.
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• Test the effectiveness of the strategy with real world data provided by a wind power generation company in the U.S. electricity market.
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• Utilize advanced forecasting techniques to improve the accuracy of prediction of revenue streams.
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• Incorporate valuable market information, such as forecasted ISO demands, forecasted and generated hourly wind power generations, to help WPPs bid to the short-term market.
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Shaomao Li, Chan S. Park PII: DOI: Reference:
S0140-9883(18)30359-1 doi:10.1016/j.eneco.2018.08.029 ENEECO 4142
To appear in:
Energy Economics
Received date: Revised date: Accepted date:
11 September 2016 24 June 2018 26 August 2018
Please cite this article as: Shaomao Li, Chan S. Park , Wind Power Bidding Strategy in the Short-term Electricity Market. Eneeco (2018), doi:10.1016/j.eneco.2018.08.029
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Wind Power Bidding Strategy in the Short-term Electricity Market Shaomao Lia,∗, Chan.S Parka Department of Industrial and Systems Engineering, Auburn University, Auburn, AL 36830, USA
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Abstract
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This paper presents an analytical trading electricity model for wind power producers (WPPs) in the short-term electricity market in the U.S. This model addresses four specific uncertainties: real-time (RT) wind power generation, day-ahead (DA) locational marginal prices (LMPs), RT LMPs, and deviation penalty rates. The model is designed to find the optimal bidding strategy to maximize the expected revenue under these uncertainties. In addition, this paper shows advanced forecasting techniques could be used with the proposed bidding strategy to help WPPs trade energy in short-term markets. A case study is presented to illustrate the effectiveness of this proposed bidding strategy and advanced forecasting techniques using a set of real data taken from a wind farm in the PJM electricity market.
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Keywords: wind power, bidding strategy, forecasting model, short-term market, analytical method
1. Introduction
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Use of wind energy has rapidly expanded around the world. Wind power is expected to continue its fast growth in many countries, including the U.S. and China. However, the uncertainty and variability of wind power give rise to several challenges of power systems and the electricity market. From WPPs’ point of view, it is essential to participate in the electricity market and gain as much revenue as possible. In general, WPPs in the U.S. prefer to commit to power purchasing agreements (PPAs) to sell the generated wind power, as these arrangements bring in stable cash flow for a relatively long term, typically over 10 years. Besides PPAs, WPPs can also participate in two other short-term markets, the day-ahead (DA) market and the real-time (RT) market. The DA market operates like a forward market, and the RT market is often referred to as a balancing market. When generation companies ∗ Corresponding
author Email address: [email protected] (Shaomao Li )
Preprint submitted to Journal of energy economics
September 3, 2018
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commit to sell a certain amount of power in the DA market, imbalances between the DA committed quantity and the actual wind power generation must be rectified in the RT market. Due to the uncertainties of wind power generation and electricity market prices, it is challenging for WPPs to develop a bidding strategy to sell power in those two short-term electricity markets. In this paper, we intend to find how much power they should commit to the DA market. In general, WPPs will generate at its maximum available capacity in RT market due to the low marginal production cost. Since electricity markets around the world have been reformed and wind generation has penetrated the market only recently, an optimal bidding strategy for WPPs to maximize profits in the short-term electricity market is still a relatively new problem. There are some approaches to deal with the bidding strategy problem for WPPs. Bathurst et al. [1] developed a bidding strategy using Markov probabilities for a wind farm in the United Kingdom to determine the best energy contract level; this study also demonstrated how a change in market closure delay and in the number of trades per day impacts the overall expected imbalance costs. Matevosyan and Soder [2] developed a bidding strategy as a stochastic mixed integer programming problem to maximize expected profit and minimize the imbalance costs for WPPs in the Nordic electricity market. ARMA model was used to simulate scenarios for wind speed forecasting errors. Pinson et al. [3] proposed different strategies by the analytical method for point prediction and probabilistic wind forecasting cases. They followed the Dutch electricity pool but used a wind farm in Ireland in their case study. Pousinho, et al. [4] proposed a two-stage stochastic programming model to deal with two kinds of uncertainty: wind power and electricity market prices. A wind farm in Portugal was used in the case study. The aforementioned literatures focus on maximizing the profit for WPPs in short-term electricity markets. Optimal bidding strategies for WPPs taking into risks have been extensively researched. Morales et al. [5] and Hosseini-Firouz [6] built a multi-stage stochastic programming model which considered risk management. This model applied ARIMA techniques to predict electricity prices and wind speeds. Their case study used wind speed of the state of Kansas and historical prices of the Iberian Peninsula electricity market for the fitting process in their case study. Catalao, et al. [7] proposed a two-stage stochastic programming model which used a hybrid intelligent approach to generate the scenario tree. A wind farm in Portugal was used in the case study. Botterud, et al. [8] derived optimal DA bids under different assumptions for risk preferences and deviation penalty schemes in the U.S. market. Their model used a generalized reduced gradient algorithm to solve the nonlinear programming problem. They emphasized benefits of using advanced wind power forecasting methods. Dent et al. [9] built a bidding strategy for WPPs in the Great Britain market using the analytic method, and presented analytical expressions for determining the optimal forward market strategy under the risk-neutral and risk-averse cases. Zhang et al. [10] proposes three different bidding strategies, i.e., the expected profit-maximization strategy, the chance-constrained programming-based strat2
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egy and the multi-objective bidding strategy. Analytical solutions under the three strategies were obtained and they are based on assuming normal distribution of wind power generation. They tested bidding strategies on a hypothetical wind farm following Spanish market rules. Vilim and Botterud [11] proposed a stochastic model with considering the price effects of wind power production and consumption. They developed and compared two bidding algorithm in the Nord Pool electricity market. Dai and Qiao [12] proposed a bilevel stochastic optimization model to obtain the optimal bidding strategy for a strategic WPP with considering the uncertainties in the demand, the wind power production, and the bidding strategies of the strategic conventional power producers. The bilevel model is transferred into a mixed-integer linear problem by using the duality theory and Karush-Kuhn-Tucker condition. Guerrero-Mestre, et al. [13] proposed an optimal joint offer to the DA market through an external agent for a group of wind farms and compared it with other offering strategies using a stochastic programming approach. A case study in the electricity market in Spain was carried out with extended results to compare different strategies for offering energy to the DA market considering different risk aversions. Baringo and Conejo [14] proposed a multi-stage risk-constrained stochastic complementarity model for the optimal offering strategy of a WPP with a large amount of installed capacity, which behaves as a price-maker in both the DA and the balancing markets. There is also literature discussing other ways of maximizing profit for WPPs, such as combining wind power with energy storage [15, 16, 17] or hydro power [18, 19, 20]. This paper not only presents an analytical model of the optimal DA market bidding strategy for WPPs in U.S. electricity markets; it also takes advantage of forecasting techniques, historical data and valuable market information to help bid like WPPs do in real world. This paper suggests the following improvements:
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1 No need to generate scenarios because it does not require generating a great number of scenario trees or solving complex LP problems.
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2 Test the effectiveness of the strategy with real world data provided by a wind power generation company in the U.S. electricity market.
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3 Utilize advanced forecasting techniques to improve the accuracy of prediction of revenue streams. 4 Incorporate valuable market information, such as forecasted ISO demands, forecasted and generated hourly wind power generations, to help WPPs bid to the short-term market.
The rest of the paper is organized as follows: Section II briefly outlines the market structure in the U.S. short-term electricity markets and the bidding strategy problem faced by WPPs. Section III presents an analytic method for developing a bidding strategy model for WPPs. This section also introduces beta distribution to represent the probability density function (pdf) of wind power generation prediction errors, and various forecasting techniques to predict
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electricity prices and generation outputs. Section V provides case studies with result analysis and comparison. Finally, Section VI presents several relevant conclusions and possibilities for future work.
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2. Market structure and problem description
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The main symbols used in this paper:
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πh Revenue from energy market, hour h (h = 1 . . . 24), [$]. qDA,h Quantity bid into DA market, hour h, [MWh]. qRT,h Actual delivery, hour h, [MWh]. qmax Wind farm generation capacity, [MW]. pDA,h , pRT,h DA and RT LMPs, hour h, [$/MWh]. pen deviation penalty rates for deviation between DA schedule and RT delivery, [$/MWh]. f (qRT,h ) Probability density function of RT wind power generation. F (qRT,h ) Cumulative distribution function of RT wind power generation. qpred Normalized predicted power
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Electricity markets in the U.S. are operated by independent system operators (ISOs) or regional transmission organizations (RTOs), which typically provide a two-settlement system for market participants to trade electricity. The short-term electricity Markets consists of two markets, a DA market and a RT balancing market. The DA market is a forward market in which hourly clearing prices are calculated for each hour of the next operating day based on generation offers, demand bids, increment offers, decrement bids and bilateral transaction schedules submitted into the DA market. The balancing market is the RT energy market in which the clearing prices are calculated every five minutes based on the actual system operations security-constrained economic dispatch. The DA market settlement is based on scheduled hourly quantities and on DA hourly prices, the Real-time settlement is based on actual hourly (integrated) quantity deviations from DA scheduled quantities and on RT prices integrated over the hour. [21] To recover uplift costs due to deviations, some ISOs impose deviation penalty charges to generation companies which cause such problems. Unlike the conventional power generator, WPPs must pay these penalty fees frequently due to the unpredictability of wind power generation. The revenue gained from the two-settlement electricity revenue for an hour, h, includes three parts: DA market revenue, RT market revenue, and deviation penalty charge, as shown in Eq.(1): πh = pDA,h · qDA,h + pRT,h (qRT,h − qDA,h ) − pen|qRT,h − qDA,h |
(1)
ISOs in the U.S. share similar time frameworks for the short-term electricity market. This paper uses PJM as an example of how the short-term electricity market operates in the U.S. The Day-ahead Energy Market bid/offer period closes at 10:30am on the day before the Operating Day. At 10:30am PJM 4
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begins to run the DA market clearing software to determine the hourly commitment schedules and the LMPs for the DA market. PJM posts the DA hourly schedules and LMPs By 13:30pm. During 13:30pm and 14:15pm, the balancing market offer period are opened for market participants to submit revised offers for resources not selected in the DA market. At 14:15pm, PJM performs a second resource commitment, which includes the updated offers, updated resource availability information, and updated PJM load forecast information and load forecast deviation. PJM may perform additional resource commitment runs, as necessary, based on updated PJM load forecasts and updated resource availability information. During the operating day, the RT market is organized to guarantee real time balance between power supplies and demands. If generation companies produce less power than their DA scheduled quantities, they must buy power from the RT market at RT LMPs; if they produce more power than the DA scheduled quantities, they are paid by the market at RT LMPs. This arrangement is represented by the second part of Eq.(1).[21] In PJM, deviation charges are collected to pay for Balancing Operating Reserve in the balancing market. Total daily costs of balancing operating reserve for the Operating Day is allocated and charged to PJM Members in proportion to their locational realtime deviations from day-ahead schedules. As accounting for Operating Reserve is performed on a daily basis, pen is identical for every hour in the same day. WPPs must decide every day how much power they should commit to the DA market for each hour of the next operating day. They face four main uncertainties: pDA,h , pRT,h , qRT,h and pen. When WPPs submit offers to DA markets for the next operation day, they need to forecast wind power generation 13.5 to 37.5 hours ahead, and in the meantime, they are also unaware of hourly DA LMPs, RT LMPs and deviation penalty rates, all uncertainties make the bidding problem complex. 3. Mathematical formulation
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In this paper, we assume that WPPs are price takers, i.e., their bidding strategies do not affect market LMPs, as generation outputs of most wind farms are relatively small compared to generation outputs of other resources. Also, wind power is usually not a marginal cost in the electricity market. In reality, the above assumption may not hold all the time, especially for those regions with high share of wind power. This paper targets at those WPPs who do not have market power. We also assume WPPs are risk-neutral, meaning they make hourly decisions based on the expected value of revenue without considering risks. The reason for this risk-neutral assumption is that hourly profits are less important for risk preference because of the large number of hourly trades in a year. 3.1. Analytical model development This section proposes an analytical method to solve the bidding strategy problem. Eq.(1) is the revenue function used to calculate revenue from shortterm markets for WPPs, whom are assumed to be risk-neutral. The goal of the 5
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proposed model is to maximize the expected value of the revenue: M ax
E[πh (qDA,h )]
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s.t. 0 ≤ qDA,h ≤ qmax
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ifE[pRT,h ] − E[pDA,h ] > E[pen] ifE[pDA,h ] − E[pRT,h ] > E[pen] if 0 ≤ a ≤ 1
E[pen] + E[pDA,h ] − E[pRT,h ] 2E[pen]
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(5)
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problem can be solved as follows:
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As described in Appendix, this 0, * (qDA,h ) = qmax , −1 F (a),
(3)
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Where F −1 (a) is the inverse distribution function (also called quantile function) of qRT,h . From the above results, we can see that once a is obtained in Eq.(5), the optimal (qDA,h )* can be easily obtained according to Eq.(4). For instance, if a is 0.35, then the optimal DA bid for that specific hour should be the 35th percentile of real-time wind power generation distribution. When the expected real-time price is higher than the sum of expected day-ahead price and deviation penalty rate, WPPs should allocate 0 in the DA market as they get paid higher in RT market even they need to pay the penalty cost for the deviation. When the expected day-ahead Price is higher than the sum of expected real-time price and deviation Penalty rate, WPPs should bid their full capacity in the DA market as they get paid higher in DA market even they need to pay the penalty cost for the deviation.
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3.2. Bidding strategy process
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The process of the proposed optimal bidding strategy can be summarized in four steps:
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Step 1 Develop f (qRT,h ) by the historical data. In this paper, the pdf of RT wind power generations and the pdf of wind power generations prediction errors are used as f (qRT,h ). Step 2 Estimate the expected values of pDA,h , pRT,h and pen for each hour of the next operating day, either by prediction techniques or by calculating means of the historical data. Step 3 Plug those expected values into Eq.(5) to obtain a. Step 4 Refer to Eq.(4) to find (qDA,h )* .
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3.3. f (qRT,h ) and beta distribution
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From the above analysis, we know that it is critical to develop an appropriate f (qRT,h ) to find the optimal (qDA,h )* . In this paper, two pdfs are used: the pdf of historical actual wind power generation and the pdf of wind power generation prediction errors. The reason these two pdfs are used is that the historical data of point forecasted hourly wind power generation and actual generated wind power generation is available. Both pdfs are applied in the case study. Although the Weibull distribution is commonly employed to describe wind power generation, it might be inappropriate to describe wind power generation prediction errors. This paper uses a beta distribution function to describe wind power generation prediction errors by analyzing the historical hourly data of forecasted wind power generation and generated wind power generation. Beta distribution is used for two reasons. First, the beta distribution can accurately represent the pdf of the wind power prediction errors. Second, this mathematical method can easily find the percentile value (qDA,h )* of other general wind farms if two parameters of the beta distribution are roughly known, rather than exact historical data. Beta function that models the occurrence of RT wind power generation x if a certain prediction power is given is as follows: (6)
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3.4. Forecasting techniques
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LMPs and deviation penalty rates forecasting are essential in the proposed bidding strategy, as a derived in Eq.(5) is determined by their expected values. We introduce four prediction techniques in this work: the cubic spline, ARMA, ARMAX and state space models (SSM). The latter three models are time series models. Except for ARMA, the other three prediction models include forecasted ISO demands as an explanatory variable.
3.4.1. ARMA model ARMA relates current prices to past prices, and current errors to previous errors as well. The general ARMA (p,q) model has the following form yt = φ1 yt−1 + · · · + φp yt−p − (θ1 εz−1 + · · · + θq εt−q ) + εt
(7)
where εt is a white noise process, and φp and θq are coefficients of backshift operators [23]. ARMA(3,1) is used in the case study.
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3.4.2. ARMAX model An ARMAX model simply adds one explanatory variable to the right side of an ARMA model; an ARMAX model relates current price to past prices and current forecasted demands. The general ARMAX model has the following form (8)
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yt = βxt + φ1 yt−1 + · · · + φp yt−p − (θ1 εz−1 + · · · + θq εt−q ) + εt
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3.4.3. State space model The state space model (also called the dynamic linear model) provides a mathematically rigorous treatment of time series modeling based on a Bayesian approach. By allowing for variability in the regression coefficients, this model can let the system properties change in time. The general state space model has the following form ( Yt = Ft θt + t , t ∼ N (0, Vt ) (9) θt = Gt θt−1 + ωt ωt ∼ N (0, Wt )
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where Yt is the observation at time t, θt is the vector of parameters, Ft is the row vector of covariates at time t, Gt is the evolution matrix, and t and ωt are the observation error and the evolution error at time t, respectively. The upper part is called the observation equation and the lower part is called the evolution equation[25]. In the case study, ARMAX in the state space model is used to forecast LMPs and deviation penalty rates.
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3.4.4. Cubic spline A cubic spline model is not a time series model, as it only relates current prices to forecasted demands. A cubic spline is a special function defined piecewise by a cubic polynomial. General cubic spline model has the following form
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yh = β0 +β1 xh +β2 x2h +β3 x3h +β4 (xh −a1 )3+ +β5 (xh −a2 )3+ +· · ·+βk+3 (xh −ak )3+ (10) where k is the number of knots, ak is the value of kth knot, and β is the coefficient [26]. The above four forecasting models are applied to predict LMPs and deviation penalty rates in the case study in R software, and the relevant results are presented in the following section. 4. Case study In this section, we apply the proposed bidding strategy model to a largescale wind farm, which is within PJM in the U.S. and has a total installed 8
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generation capacity of 102 MW. This wind farm trades the generated electricity in the short-term electricity market operated by PJM. We first describe and analyze the historical data of this wind farm. Then we present the results of the proposed bidding strategy model test. Finally, we compare the results of bidding strategies with different forecasting techniques.
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4.1. Description of data
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Hourly data from Jan. 1st to Dec. 31th, 2013 is used, including DA LMPs, RT LMPs, deviation penalty rates, forecasted ISO demands, forecasted wind power generation and generated RT wind power generation. To make analyses and predictions more accurately, we divide those hourly data into 24 groups, one for each hour, and we treat the first nine months’ data as the training data and the rest as the testing data.
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Figure 1: Histogram and statics of DA and RT LMPs
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Figure 2: Histogram and statics of deviation penalty rates
Figure 3: Histogram of generated RT wind power generation
The histograms and statics of hourly DA and RT LMPs, deviation penalty rates, and generated wind power generation in 2013 are shown in Figs.1-3, respectively. From Fig.1 we can see the mean of DA LMPs is about $1 higher than that of RT LMPs; meanwhile, the volatility of DA LMPs is smaller than that of RT LMPs. Number of negative prices of RT LMPs is much larger than 10
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Table 1: Correlation coefficients among data
RT LMP
pen
Pred. Load
1
0.61
0.33
0.64
1
0.28
0.52
1
0.49
DA LMP RT LMP
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DA LMP
Pred. Load
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Pred. MW
Gen. MW
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-0.04
-0.05
-0.08
0.01
-0.02
0.01
-0.04
1
0.53 1
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DA LMPs. Negative prices mean that producers were actually paying for their energy to be taken to ensure electricity dispatch. This makes an economic sense because of the federal production tax credit. Even if WPPs give the power away or offer the system money to take it, they still receive a tax credit equal to $23/MWh. In Fig.2, most deviation penalty rates are not more than $5/MWh. Fig.3 shows that about half of the time, this wind farm generated less than 20 MWh in 2013.
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Table 1 indicates correlation coefficients between each data item. Pred. denotes Predicted, Gen. denotes generated, MW denotes wind power output, and Load means ISO demand. The correlation coefficient between generated RT wind power generation and LMPs is relatively small and negative. The predicted ISO demand is highly correlated with DA LMPs, RT LMPs and deviation penalty rates. Thus, including this factor in price forecasting models is both reasonable and necessary, and WPPs in the real world can also access the information about predicted ISO demands before they make bidding decisions in the short-term electricity markets. 4.2. The pdf of wind power generation prediction errors
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When we compare the hourly forecasted wind power generation and generated wind power generation data, it is found that the point forecast accuracy is not good enough to be used for the bidding purpose. To make the bidding strategy perform better, we develop the pdf of wind power generation Prediction Errors, which is expressed in the Beta distribution function to represent RT wind power generation distribution when forecasted wind power generation are given. We divide the qpred into 10 ranges with an interval of 0.1. α and β values of beta pdfs for these 10 qpred ranges were calculated in Matlab and are presented in Table 2. We choose 10 ranges because if we chose too many ranges, the available data would be not enough to develop an accurate distribution function for individual ranges, but if we chose too few ranges, the distribution function for individual ranges would become too rough to represent real distributions.
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Table 2: Parameters for 10 individual ranges of qpred
0∼0.1
0.1∼0.2
0.2∼0.3
0.3∼0.4
0.4∼0.5
α
0.35
1.17
1.45
1.71
1.36
β
0.80
6.57
4.52
3.96
2.19
qpred
0.5∼0.6
0.6∼0.7
0.7∼0.8
0.8∼0.9
0.9∼1.0
α
1.17
1.09
1.39
1.52
1.00
β
1.30
0.97
0.80
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qpred
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The real distributions of wind power generation prediction errors for qpred = [0, 0.1], [0.1, 0.2], [0.6, 0.7] and [0.9, 1.0] are presented as examples in Fig.4. Beta distributions of these four ranges are shown in Fig.5, accordingly.
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Figure 4: Real pdfs of wind power generation Prediction Error for qpred = [0, 0.1], [0.1, 0.2], [0.6, 0.7] and [0.9, 1.0]
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Figure 5: Beta pdfs of wind power generation Prediction Error for qpred = [0, 0.1], [0.1, 0.2], [0.6, 0.7] and [0.9, 1.0]
4.3. Results of bidding strategy model tests
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This section tests the effectiveness of the bidding strategy proposed in this paper. We conduct 24 hourly model tests separately by following the steps mentioned in the bidding strategy process section to see whether the optimal (qDA,h ) by the proposed model can achieve the maximum revenue. Because the purpose of this part of the case study is to test the proposed model, we do not take advantage of information of forecasted wind power generation and advanced prediction techniques in this part. Therefore electricity LMPs, deviation penalty rates and wind power generations are treated as stochastic variables, whose probability distributions are based on historical data of those variables in 2013. In the test, the study time is one hour and the goal is to find the optimal (qDA,h ) to maximize the expected one-hour revenue. We first use the whole year real data to do the test, so there are 365 scenarios for individual hours test. The result is showed in Table 3, where (qDA,h )* and (qDA,h )** are the optimal solution obtained by the proposed bidding strategy model and the actual optimal solution by enumeration method, respectively; π(qDA,h )* and π(qDA,h )** denote the revenue by bidding (qDA,h )* and (qDA,h )** to the market, respectively; Diff.q and Diff.π denotes the difference between (qDA,h )* and (qDA,h )** , and the difference between π(qDA,h )* and π(qDA,h )** , respectively. Means of historical data are used as the expected value of DA LMPs, RT LMPs, and deviation penalty rates to calculate a in step 3 of the bidding strategy process. The pdf of RT wind power generation showed in Fig.3 is used as f (qRT,h ) in step 4 to obtain the (qDA,h )* .
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Table 3: Comparison for individual hours, real data
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π ** 859.0 713.7 651.8 638.7 593.7 637.4 787.4 904.4 826.0 844.9 881.8 883.5 919.7 945.8 1169.3 1197.6 1193.8 1358.5 1299.4 1143.7 1122.4 1087.2 1056.7 907.0 $
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π* 859.0 713.7 651.8 638.7 593.7 636.7 787.4 904.4 825.0 844.4 881.8 881.9 919.7 945.3 1169.3 1197.6 1193.7 1358.5 1299.4 1143.7 1122.3 1087.1 1056.7 907.0 $
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Diff.q 0 1 0 0 1 10 2 0 8 3 0 7 0 5 0 0 5 0 0 0 4 2 0 0 MWh
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(qDA,h )** 102 28 67 102 63 55 8 101 39 9 0 10 0 4 102 102 94 102 102 84 44 17 102 91 MWh
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(qDA,h )* 102 27 67 102 64 65 10 101 47 12 0 17 0 9 102 102 99 102 102 84 48 15 102 91 MWh
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hours 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 unit
Diff.π 0.0 0.0 0.0 0.0 0.0 0.7 0.0 0.0 1.0 0.5 0.0 1.6 0.0 0.5 0.0 0.0 0.1 0.0 0.0 0.0 0.1 0.1 0.0 0.0 $
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Table 4: Comparison for individual hours, simulated data
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π ** 860.4 715.8 660.0 653.9 581.6 624.9 768.6 862.7 815.3 824.1 913.5 909.0 989.4 1019.4 1195.7 1216.9 1278.7 1372.5 1296.0 1136.4 1107.1 1080.2 1035.7 924.4 $
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π* 860.4 715.8 660.0 653.9 581.6 624.8 768.6 862.7 815.2 824.1 912.9 909.0 989.4 1019.4 1195.7 1216.9 1278.7 1372.5 1296.0 1136.3 1107.1 1080.2 1035.7 924.4 $
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Diff.q 0 0 2 0 1 1 1 2 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 MWh
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(qDA,h )** 102 29 65 102 60 40 6 96 35 7 0 13 1 1 102 102 101 102 102 81 33 15 102 99 MWh
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(qDA,h )* 102 29 67 102 59 39 5 98 34 7 1 13 1 1 102 102 101 102 102 81 33 15 102 99 MWh
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hours 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 unit
Diff.π 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.1 0.0 0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 $
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From Table 3, we can find that for most hours the difference between (qDA,h )* and (qDA,h )** are fairly small. Even for those hours whose optimal qDA,h differences are relatively large, such as 6am and 12pm, their one-hour revenue differences turn out to be small. The reason these differences exist is partly because the sample size for each hour is not large enough; some extreme values of variables will affect the final result to some extent. When we enlarge the scenario number from 365 to 1,000,000 by Monte Carlo simulation method, both qDA,h differences and the revenue differences are largely reduced, which can be found in Table 4. The distributions of those random variables in the simulation case are almost the same as the ones in the previous case. Both the method and procedure to find (qDA,h )* and (qDA,h )** in this simulation are the same as the ones used in the previous case. The test of significance indicates that (qDA,h )* and (qDA,h )** are equivalent to each other in a statistic sense. The above results indicate the proposed optimal bidding strategy works well in the real world as intended. 4.4. Results of wind power generation prediction errors
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In this part of case study, we compare the revenues by applying the same proposed bidding strategy but using different f (qRT,h ) in step 4 of the bidding strategy process: the pdf of RT wind power generation and the pdf of wind power generation forecasting errors. The main difference between these two pdfs is that the latter one includes information of forecasted wind power generation and relevant historical distribution, whereas the former only represents the distribution of historical wind power generation outputs. Note that in this paper, we apply no forecasting technique to predict wind power generation; all the forecasted wind power generation data is given. It needs to be pointed out that both the pdf of RT wind power generation and the pdf of wind power generation forecasting errors are expressed in the Beta distribution function, and both models use historical means as the expected values of pDA,h , pRT,h and pen in Eq.(5) to obtain the optimal (qDA,h )* .
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Figure 6: Revenues comparison between bidding strategies with the pdf of RT wind power generation and with the pdf of wind power generation prediction errors
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Fig.6 shows bidding results, where the vertical axis is the whole year revenue for each hour. In Fig.6, most groups with the pdf of wind power generation prediction errors outperform the ones with the pdf of wind power generation, and the total revenue difference in 2013 is $434,578, which accounts for about 6.7% of the total revenue. From the above results, we can see that information of forecasted wind power generation does help WPPs gain the extra revenue in the short-term electricity market if applying the proposed bidding strategy.
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Figure 7: Revenues comparison between bidding the predicted wind power generation and bidding strategy with the pdf of wind power generation prediction errors
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Fig.7 shows results of bidding the predicted wind power generation model and the proposed bidding strategy with the pdf of wind power generation prediction errors as f (qRT,h ). The former just bids the forecasted wind power generation quantity to the DA market without using the proposed strategy model. The latter is the same as in the previous case. In Fig.7, nearly all groups of bidding strategies, with the pdf of wind power generation prediction errors, outperform the ones of bidding predicted wind power generation, and the total revenue difference in 2013 is $249,797. In other words, the proposed bidding strategy model with the pdf of wind power generation prediction errors as f (qRT,h ) helps WPPs improve total revenue of bidding predicted wind power quantity to the DA market in 2013 by approximately 3.7%. The above results indicate again that the proposed bidding strategy model works well in the real world.
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4.5. Results comparison of different bidding strategies and forecasting techniques In previous cases, we assumed the expected values of LMPs for the same hour remain constant in 2013, which is oversimplified in the real world, where WPPs could take advantage of valuable information, like forecasted ISO demands, and advanced forecasting techniques to predict next day’s LMPs and deviation penalty rates accurately. In this part of the case study, we will show how to incorporate forecasting techniques into the proposed optimal bidding strategy. The major difference between these bidding strategies with forecasting techniques and previous ones is that the expected values of DA LMPs, RT LMPs, and deviation penalty rates will be predicted rather than being means of historical data. Therefore, for individual hours, each day’s qDA,h could be different
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Table 5: Results of different bidding strategies
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Difference /$ 18,096 8,728 6,803 0 -19,024 -55,854 -63,351 -103,029 -121,893
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Profit /$ 1,915,076 1,905,708 1,903,783 1,896,980 1,877,956 1,841,126 1,833,629 1,793,951 1,775,087
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Bidding Strategies perfect forecasting model cubic spline model state space model bid historical mean model ARMAX Model bid predicted wind power generation model ARMA Model Bid full capacity model bid 0 MW model
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if the calculated a in Eq.(5) changes. Four advanced forecasting models are adopted in this work: cubic spline, ARMA, ARMAX and state space models. Besides these four models, we also present five other bidding strategy models for comparison. The result of 3-month revenues by these nine bidding strategy models are presented in Table 5.
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In Table 5, bid 0 MWh model and bid full capacity model mean that WPPs bid 0 MWh and 102MWh to the DA market, respectively. Historical mean model applies the proposed bidding strategy and uses the means of historical data as the expected values of those random variables in Eq.(5), which is the same as the bidding strategy with the pdf of wind power generation prediction errors in the previous section. Perfect forecasting model assumes that wind power generation forecasting is 100% accurate, and WPPs bid these forecasted quantities to the DA market. There will be no imbalance cost in this case, and it serves as a benchmark of perfect information, since it is unrealistic in the real world. Note further that, within these nine bidding strategies, five follow the same proposed optimal bidding strategy process but with different approaches to obtain the expected values of LMPs and deviation penalty rates. These five strategies include four forecasting models and historical mean model. Beta distribution function of wind power generation prediction errors is used in these models. To make accurate predictions, within ARMA, ARMAX and state space models, we develop 24 hourly forecasting models to predict the DA LMPs, RT LMPs and deviation penalty rates of next operation day’s individual hours. Table 5 shows that either bidding nothing or bidding full capacity to DA market are bad strategies, because, as predicted, their deviation costs become relatively high in these cases. Except for the ARMA model, all the models that apply the proposed bidding strategy gain revenues higher than the bid predicted wind power generation model. Bid historical mean model serves as a benchmark to calculate revenue difference between those bidding models in Table 5. Among those models with forecasting techniques, the state space model and the cubic spine model achieve the highest two revenues, gaining about $6,803 and $8,728 over the bid historical 19
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mean model, respectively. This promising result indicates some advanced forecasting techniques would be beneficial to the proposed optimal bidding strategy. The state space model and the cubic spine model are recommended to predict market prices in this paper.
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5. Conclusion
6. Appendix
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This paper presents an optimal bidding strategy model for WPPs to trade power in the short-term electricity market. We test it in a case study with real-world data and compare it with the model by the Monte Carlo simulation method. The results of both cases indicate the proposed model looks promising. The analytic model is much simpler and more computationally efficient than the simulation method which needs to develop a large scenarios set. The proposed model can also utilize the information of forecasted ISO demands, forecasted and generated wind power generations to help bid. To make the model more accurate and efficient, fitted beta distribution functions are used to describe the error between forecasted value and realized value of wind power generation. This work also shows that advanced forecasting techniques would help the proposed optimal bidding strategy. The state space model and cubic spline model gain higher revenues than other forecasting models in the case study. Further research will be conducted to improve the accuracy of prediction techniques, and to extend the proposed model to other electricity markets with different market rules.
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The goal of the proposed model is to maximize the expected value of the revenue: M ax
E[πh (qDA,h )]
(11)
s.t. 0 ≤ qDA,h
(12)
qDA,h ≤ qmax
(13)
Due to the assumption that WPPs are price takers, Eq.(11) can be converted
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as follows: E[πh (qDA,h )]
qDA,h
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Z qDA,h − E[pen] · (− (qRT,h − qDA,h )f (qRT,h )d(qRT,h ) 0 Z qmax (qRT,h − qDA,h )f (qRT,h )d(qRT,h )) +
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= E[pDA,h · qDA,h ] + E[pRT,h (qRT,h − qDA,h )] − E[pen · |qRT,h − qDA,h |] (14) Z qDA,h = E[pDA,h · qDA,h ] + E[pRT,h ] · ( (qRT,h − qDA,h )f (qRT,h )d(qRT,h ) 0 Z qmax + (qRT,h − qDA,h )f (qRT,h )d(qRT,h ))
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We know that :
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Rq d( 0 DA,h (qRT,h − qDA,h )f (qRT,h )d(qRT,h )) d(qDA,h )
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With Eq.(17) and Eq.(18), the first derivative of Eq.(11) with respect to qDA,h : Z qDA,h d(E[πh (qDA,h )]) =E[pDA,h ] − (E[pRT,h ] + E[pen]) · ( f (qRT,h )d(qRT,h )) d(qDA,h ) 0 Z qmax + (E[pRT,h ] − E[pen]) · (− f (qRT,h )d(qRT,h )) =E[pDA,h ] − (E[pRT,h ] + E[pen]) · (F (qDA,h ))
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+ (E[pRT,h ] − E[pen]) · (−(1 − F (qDA,h )))
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qDA,h
(19)
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=E[pen] − E[pRT,h ] + E[pDA,h ] − 2E[pen] · F (qDA,h )
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Note that F (qDA,h ) is the cumulative distribution function of qRT,h when qRT,h is equal to qDA,h . The second derivative of Eq.(11) with respect to qDA,h : d2 (πh (qDA,h )) = −2E[pen] · f (qDA,h ) ≤ 0 d(qDA,h )2
(20)
L(qDA,h , λ, ϕ) =
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The above result indicates that the objective function is concave in qDA,h . Define the Lagrangian as E[πh (qDA,h )] − λ(qDA,h − qmax ) − ϕ(0 − qDA,h )
(21)
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The Karush−Kuhn−Tucker conditions:
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d(L) d(E[πh (qDA,h )]) = −λ+ϕ=0 d(q d(qDA,h ) DA,h ) λ > 0, qDA,h = qmax ; or λ = 0 ϕ > 0, qDA,h = 0; or ϕ = 0
(22)
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By plugging Eq.(19) into Eq.(22), we can obtain the optimal solution: ifE[pRT,h ] − E[pDA,h ] > E[pen] 0, * (23) (qDA,h ) = qmax , ifE[pDA,h ] − E[pRT,h ] > E[pen] −1 F (a), if 0 ≤ a ≤ 1 a=
E[pen] + E[pDA,h ] − E[pRT,h ] 2E[pen]
(24)
Where F −1 (a) is the inverse distribution function of qRT,h .
References [1] G. N. Bathurst, J. Weatherill, G. Strbac, Trading wind generation in short term energy markets, IEEE Trans. Power Syst. 17 (3) (2002) 782–789. [2] J. Matevosyan, L. Sder, Minimization of imbalance costs trading wind power on the short-term power market, IEEE Trans. Power Syst. 21 (3) (2006) 1396–1404. 22
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[3] P. Pinson, C. Chevallier, G. N. Kariniotakis, Trading wind generation from short-term probabilistic forecasts of wind power, IEEE Trans. Power Syst. 22 (3) (2007) 1148–1156.
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[4] H. M. I. Pousinho, V. M. F. Mendes, J. P. S. Catalao, A stochastic programming approach for the development of offering strategies for a wind power producer, Electric Power Systems Research 89 (2012) 45–53.
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[5] J. M. Morales, A. J. Conejo, J. Prez-Ruiz, Short-term trading for a wind power producer, IEEE Trans. Power Syst. 25 (1) (2010) 554–564.
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[6] M. Hosseini-Firouz, Optimal offering strategy considering the risk management for wind power producers in electricity market, Int J Electr Power Energy Syst. 49 (2013) 359–368.
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[7] J. P. S. Catalao, H. M. I. Pousinho, V. M. F. Mendes, Optimal offering strategies for wind power producers considering uncertainty and risk, IEEE Trans. Power Syst. 6 (2) (2012) 270–277.
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[8] A. Botterud, Z. Zhou, J. Wang, R. J. Bessa, H. Keko, J. Sumaili, V. Miranda, Wind power trading under uncertainty in LMP markets, IEEE Trans. Power Syst. 27 (2) (2012) 894–903.
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[9] C. J. Dent, J. W. Bialek, B. F. Hobbs, Opportunity cost bidding by wind generators in forward markets: Analytical results, IEEE Trans. Power Syst. 26 (3) (2011) 1600–1608.
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[10] H. Zhang, F. Gao, J. Wu, K. Liu, X. Liu, Optimal bidding strategies for wind power producers in the day-ahead electricity market, Energies 5 (2012) 4804–4823. [11] M. Vilim, A. Botterud, Wind power bidding in electricity markets with high wind penetration., Applied Energy 118 (2014) 141–155.
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[12] T. Dai, W. Qiao, Optimal bidding strategy of a strategic wind power producer in the short-term market, IEEE Transactions on Sustainable Energy 6 (2015) 707–719.
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[13] V. Guerrero-Mestre, A. A. S. de la Nieta, J. Contreras, J. P. S. Catalo, Optimal bidding of a group of wind farms in day-ahead markets through an external agent, IEEE Transactions on Power Systems 31 (2016) 2688– 2700. [14] L. Baringo, A. J. Conejo, Offering strategy of wind-power producer: A multi-stage risk-constrained approach, IEEE Transactions on Power Systems 31 (2016) 1420–1429. [15] G. N. Bathurst, G. Strbac, Value of combining energy storage and wind in short-term energy and balancing markets, Electric Power Syst. Res. 67 (2003) 1–8. 23
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[16] I. D. A. Dukpa, B. Venkatesh, L. Chang, Optimal participation and risk mitigation of wind generators in an electricity market, IET Renewable Power Gener. 4 (2) (2010) 165–175.
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[17] S. Y. Wang, J. L. Yu, Optimal sizing of the caes system in a power system with high wind power penetration, Int. J. Electrical Power & Energy Syst. 37 (1) (2012) 117–125.
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[18] A. Jaramillo, E. D. Castronuovo, I. Snchez, J. Usaola, Optimal operation of a pumped-storage hydro plant that compensates the imbalances of a power producer, Electr. Power Syst. Res. 81 (9) (2011) 176–177.
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[19] E. D. Castronuovo, J. A. P. Lopes, On the optimization of the daily operation of a wind-hydro power plant, IEEE Transactions on Power Systems 19 (3) (2004) 1599–1606.
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[20] A. A. S. de la Nieta, J. Contreras, J. I. Muoz, Optimal coordinated wind-hydro bidding strategies in day-ahead markets, IEEE Transactions on Power Systems 28 (2) (2013) 798–809. [21] Pjm manual 11: Energy & ancillary services market operations (2017). URL http://www.pjm.com/library/manuals.aspx
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[22] A. Fabbri, T. G. S. Romn, J. R. Abbad, V. H. M. Quezada, Assessment of the cost associated with wind generation prediction errors in a liberalized electricity market, IEEE Trans. Power Syst. 20 (3) (2005) 1440–1446.
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[23] R. J. Hyndman, G. Athanasopoulos, Forecasting: principles and practice (2012). URL https://www.otexts.org/fpp [24] R. J. Hyndman, The arimax model muddle (2010). URL http://www.r-bloggers.com/the-arimax-model-muddle/
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[25] G. Petris, S. Petrone, P. Campagnoli, Dynamic Linear Models with R, Springer, 2009.
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[26] M. Kuhn, K. Johnson, Applied Predictive Modeling, Springer, 2013.
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Highlights • No need to generate scenarios because it does not require generating a great number of scenario trees or solving complex LP problems.
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• Test the effectiveness of the strategy with real world data provided by a wind power generation company in the U.S. electricity market.
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• Utilize advanced forecasting techniques to improve the accuracy of prediction of revenue streams.
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• Incorporate valuable market information, such as forecasted ISO demands, forecasted and generated hourly wind power generations, to help WPPs bid to the short-term market.
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