Research Article  Open Access
Lifeng Liu, Huisheng Gao, Yuwei Wang, Wei Sun, "Robust Optimization Model for Photovoltaic Power Producer’s Bidding DecisionMaking in Electricity Market", Mathematical Problems in Engineering, vol. 2020, Article ID 6109648, 8 pages, 2020. https://doi.org/10.1155/2020/6109648
Robust Optimization Model for Photovoltaic Power Producer’s Bidding DecisionMaking in Electricity Market
Abstract
With the deepening of electricity market (EM) reform and the high penetration of photovoltaic (PV) energy in power system, the uncertainties of a PV power output and fluctuation of EM prices would bring substantial financial risks for PV power producers (PPPs). This paper proposed a robust optimization model for PPP’s power bidding decisionmaking. Specifically, the random PV power outputs are modeled by the uncertainty set, which need no probabilistic information and can fully depict the continuous range of uncertainties. Subsequently, with respect to any scenario for dayahead EM prices, PPP’s optimal power bidding strategy is obtained under the worstcase realization within the uncertainty set, which guarantees the robustness in resisting the negative impact of random PV power outputs on PPP’s profit. Moreover, a reformulation approach was introduced for equivalently transforming our model into a tractable framework. Simulation was implemented to validate the feasibility and effectiveness of applying our proposed model.
1. Introduction
It is considered that the global climate changes and energy depletion, most serious threats to sustainable development, result from the overexploitation and utilization of fossil fuels [1–3]. Renewable energy, an alternative to fossil fuels, has been regarded as an effective way to fight against the energy shortage and environmental issues [4]. Photovoltaic (PV) power, which is an important part of renewable energy, has a promising future due to its emissionfree, mature technology and low cost [5, 6].
In China, according to the latest statistics of China’s electric power industry, the installed capacity of PV power is up to 174.63 million kilowatts, accounting for about 10% of national total capacity, and the PV power generation was 177.5 billion kilowatt/hour (kWh) at the end of 2018 [7]. A promising target has been set by the Chinese government for further expanding the installed capacity to 200 GW by 2020 [8]. The Chinese government has launched a series of policies and plans to support the development of the PV industry. However, the intermittency and instability (randomness) of solar energy resources, which are influenced by weather, season, and terrain, are gradually becoming a bottleneck that restricts the utilization of PV power. Especially, with the deepening of the electricity market (EM) reform and the high penetration of PV power in EM, the uncertainties of the PV power output and fluctuation of EM prices would bring substantial financial risks for PV power producers (PPPs). Therefore, it is a valuable and tough research issue for PPPs to optimize their power bidding strategies in EMs.
Apparently, quantitative analysis of the uncertainties is the important base for optimizing a bidding strategy for PPPs and other renewable energy sources (RESs). Gomes et al. [9] proposed a scenariobased stochastic optimization (SO) model for a PPP participating in the dayahead market, where the uncertainties of PV power and EM prices are simulated by multiple stochastic scenarios. Gomes et al. [10] established a twostage SO model for an integrated windphotovoltaic system to obtain its optimal dayahead offering strategy. In [11], the SObased PV power bidding model was modified by introducing a riskaversion term in a form of conditional value at risk (CVaR) to additionally provide flexibility in finding a tradeoff between profit maximization and risk management. The SO approaches should generate a large number of probabilistic scenarios according to the uncertainties, which would require considerable computational effort [12] or may result in nonconvex nonlinear programming [13].
Robust optimization (RO) has been verified as another brilliant method to account for uncertainties. Thatte et al. [14] proposed a RObased dayahead bidding strategy for a wind power plant in combination with a ESS device, given the inherent uncertainties in spot EM prices and the wind power output. In [15], an uncertainty set for wind and PV power outputs was constructed, and then a RO model was proposed for optimal dispatch of power system. Thatte and Xie [16] developed a RObased model predictive control (RMPC) scheme to determine the optimal offering strategies of an integrated wind farmenergy storage system, given the uncertainties in spot EM prices. In the RO model proposed by Wang et al. [17] for the dayahead scheduling of a microgrid, conservative parameters were introduced in the uncertainty set of distributed wind and PV power outputs. With the introduction of RO, multiple stochastic scenarios in SO are replaced with the continuous uncertainty set, which shows advantages of robustness improvement and relatively lower computational complexity [18].
Recently, some hybrid approaches with the RO method or idea have been introduced in the operation optimization issues of various energy systems. Wang et al. [12] proposed a stochasticrobustbased dayahead bidding strategy for a windESS system. By applying the similar modeling approach with [12], Wang et al. [19] established the optimal operation model for a combined cooling, heating, and power (CCHP) system based on the stochasticrobust hybrid optimization. In [20], with the help of the information gap decision theory (IGDT), both the robust and opportunity strategies are obtained for an energy hub to bid and operate under the EM circumstances. For summarizing the pros and cons of the abovementioned related works more intuitively, a comparative table is listed (Table 1).
According to the literature review and Table 1, with respect to modeling the uncertainties, RO and its related approaches are more desirable than SO due to the advantages mentioned above. However, as far as this paper can tell, the ROrelated studies for the optimal bidding strategy of a PPP in EM are still scarce at present. Apparently, due to the differences in system structures, models for other energy systems cannot be directly applied to the field of photovoltaic power plant. Therefore, the main contributions of this paper can be summarized as follows:(1)A RObased model is proposed and customized for solving the problem of PPP’s power bidding decisionmaking in dayahead EM. On one hand, the random PV power outputs are modeled by an (budget parameter embedded) uncertainty set, which need no probabilistic information and can fully depict the continuous range of uncertainties; on the other hand, with respect to any scenario for dayahead EM prices, PPP’s optimal power bidding strategy is obtained under the worstcase realization within the uncertainty set, which guarantees the robustness in resisting the negative impact of random PV power outputs on PPP’s profit.(2)A reformulation approach is proposed for equivalently transforming our proposed model into a tractable framework which can be easily solved by using the C&CG method.(3)Simulations are designed and implemented to validate the effectiveness of our proposed model in terms of profit optimization, computational time, and the ability of resisting uncertainties. Moreover, with the help of sensitivity analysis, the robustness of our proposed model is deeply studied, which is a necessary prerequisite for practical applications.

The rest of this paper is organized as follows. In Section 2, formulations for the PV power uncertainty set, a RO model for PPP’s dayahead bidding decisionmaking, and a reformulation approach for model solving are proposed, respectively. Simulations are implemented in Section 3 for verifying the feasibility and rationality of our method, and Section 4 concludes the paper.
2. Methodology
2.1. Relevant Explanation and Hypothesis
In this section, the robust optimization model is mathematically formulated for a PPP to make bidding decisions in spot EM. For the sake of simplicity and without loss of generality, we make some assumptions and explanations listed as follows before conducting any further research:(1)Similar to [21], this paper considers the PPP as a “price taker” participating in spot EMs. This is equivalent to that the behaviors of PPP will not affect EM clearing prices.(2)Operation cost of PV is neglected because obtaining radiation from the sun is for free.
2.2. Formulations for PV Power Producer’s Power Deviation
For time period t (), let be the dayahead scheduled power output of PPP and be its realtime power output. Then, the power output deviation can be formulated aswhere stands for the positive deviation and represents the negative deviation, which can be further formulated as
2.3. Uncertainties of PV Power Producer’s RealTime Power Outputs
This paper proposes an uncertainty set for modeling the stochastic feature of PPP’s realtime power output. In an uncertainty set, there is no need to estimate the probabilistic information, which makes the corresponding uncertainty model easy to be constructed and tractable. A typical uncertainty set for PPP’s realtime power outputs is formulated aswhere is the forecasted value of PPP’s realtime power output for time period t and denotes the radius of uncertainty interval for . Moreover, is a budget parameter for controlling the conservativeness of decisionmaking. Specifically, a larger corresponds to a larger uncertainty set , which means more uncertain scenarios should be considered and leads to more conservative in decisionmaking.
2.4. Robust Optimization Model for PV Power Producer’s Power Bidding DecisionMaking
In this paper, due to the lack of flexible sources (e.g., energy storage devices), PPP cannot control its realtime power output in the stage of balancing markets. That is to say, on the one hand, PPP can only strategically participate in the dayahead EM; on the other hand, realtime deviations from this participant’s dayahead scheduled power outputs are often inevitable and may cause extra cost. Hence, the key problem to be solved in this paper is how to optimally make the dayahead power bidding decisions for PPP under uncertainties.
For a delivery day, PPP pursues the maximization of its own profit:where stands for the daily profit of PPP participating in spot EMs; means revenue obtained from selling in dayahead EM; represents extra cost caused by deviation or in the realtime stage; and and are dayahead and balancing prices, respectively.
For time period t (), the dayahead scheduled power output should not exceed its maximum limitation:where can be deemed as the rated capacity of the PV unit.
It should be noted that PPP cannot maximize its profit by neglecting the impact of its stochastic realtime power outputs. Hence, this paper applies the robust optimization approach in modeling the optimal dayahead power bidding problem for PPP.
The goal of the robust optimization approach is to obtain the optimal decision under the worstcase realization of uncertainties. Therefore, when PPP makes its power bidding decisions using robust optimization, it is necessary to clarify what is the worst impact of uncertainties on profit. Although this worst impact may not be happened in reality, PPP still needs to focus on it to avoid the potential risk of profit loss caused by the worstcase realization of uncertainties.
Based on the robust optimization approach, our proposed model for PPP’s power bidding decisionmaking can be formulated as follows:s.t. equations (2) and (3) and (6)
Evidently, the inner min item in objective function equation (7) equals to find out the worstcase realization of which minimizes PPP’s profit; the outer max item in objective function equation (7) equals to obtain the optimal dayahead power biding strategy which maximizes PPP’s profit under the worstcase realization of .
2.5. Model Reformulation
To facilitate model solving, model equations (2),(3), and (6)–(8) should be further reformulated.Because the item is not directly related to , objective function equation (7) can be rewritten as
By introducing an auxiliary variable C, equation (9) can be further rewritten as
Evidently, equations (10) and (11) are equivalent to equation (9).
Based on equations (2), (3), and (9 ), the relationship among , , and can be reformulated as
Accordingly, the reformulation of model equations (2), (3), and (6)–(8) can be summarized as model equations (6), (8), (10), (11), and (13)–(16).
The robust optimization model equations (6), (8), (10), (11), and (13)–(16) is mathematically an NPhard problem. Therefore, this paper further decomposes model equations (6), (8), (10), (11), and (13)–(16) into a main problem (MP) and a subproblem (SP).
The MP, with P_{D} and C as the decision variables, determines the optimal power bidding strategy in all the worstcase realizations generated by the SP:where k and K represent the index and index set of the worstcase realizations generated by the SP.
The SP, with , , and as the decision variables, determines the worstcase realizations under an optimal power bidding strategy generated by the MP:
In summary, the final obtained MP and SP are equivalent to model equations (2), (3), and (6)–(8) and can be directly solved by using the following C&CG method [22]:(1) define feasibility tolerance ;(2)while do(3)solve (MP), obtain optimal solution of (MP) ;(4)solve (SP) with , get solution , , and ;(5);(6)end while.
3. Simulation and Comparisons
3.1. Case Design
In this subsection, for the purpose of demonstrating our simulation results more lucidly, we introduce an experimental case design concretely. In our case, a PPP, with a capacity of 100 MW, participates as a “price taker” in the dayahead electricity market. A delivery day is discretized into 24 time units with 1 hour for the duration of each time unit. The uncertainty set for realtime PV power outputs is constructed based on historical data. Specifically, the historical hourly PV power output data were generated by using the power curve function in [23] and the hourly mean radiation data in a Chinese city from July 1st to July 30th, 2018 [24]. Our model is solved based on the uncertainty set of PV power outputs and different scenarios for dayahead and realtime EM clearing prices. The scenarios, as listed in Table 2, were obtained from historical prices data from the DKWest area in the Nord Pool market during July 20th to July 30th, 2018 (because the Nord Pool market is of a twoprice balancing market, up/downregulation prices are different and one or the other of them is equal to the dayahead one at any specific time unit. Hence, we take the different one as the balancing price in the oneprice balancing market).
 
Note. “Da” means dayahead and “Re” means realtime. 
All simulations are implemented by utilizing the MATLAB R2014a software on a PC laptop with an Intel Core i7 at 2.1 GHz and 8 GB memory.
Moreover, the constructed uncertainty set (with ) is depicted in Figure 1. Besides, in Figure 1, PM is composed of 24 hourly average PV power outputs, with each one calculated by historical data for the same period; PMAX represents 24 hourly maximum PV power outputs, with each one obtained by comparing historical data for the same period.
3.2. Calculation Results Analysis
In this subsection, simulation of our proposed model is implemented to obtain the PPP’s optimal dayahead power bidding strategy. Relevant parameters are set as and . The obtained strategies under scenarios 1, 6, and 10 are demonstrated in Figures 2–4, respectively.
It can be found in Figures 2–4 that in the dayahead stages, PPP usually offers larger power outputs when the dayahead prices are relatively high and offers smaller power outputs when the dayahead prices are relatively low. Reasons for this is that the strategies of “selling more at relatively high prices” and “selling less at relatively low prices” in the dayahead market help to profit improvement. However, exceptions still exist. For example, in Figure 4, dayahead offerings at the 10^{th} to 15^{th} time periods are relatively large although the corresponding dayahead prices are relatively low. It is obvious that uncertainties of PV power outputs often cause power deviations which will bring excess costs in the presence of realtime EM prices. Therefore, the robust dayahead strategy based on our proposed model disobeys the abovementioned “selling more at relatively high prices” and “selling less at relatively low prices” rules sometimes for avoiding the potential risk of cost increase. Moreover, by implementing our simulation in this subsection, it takes an average of 11.7 seconds for obtaining our strategy. It is generally known that the dayahead market starts at least 12 hours before the delivery day and the balancing market begins few minutes to half an hour in advance; that is to say, low computational time makes our proposed model feasible in practice.
In order to facilitate the description, the strategy obtained by our proposed model is called strategy 1 in the rest of this paper. Moreover, a strategy obtained by a deterministic model is called strategy 2, and it is taken here for comparison. The difference between a deterministic model and our proposed model is that the former one is solved not based on the uncertainty set but only based on PM as presented in Figure 1. The obtained profits under 10 scenarios of these two strategies are demonstrated in Figures 5–7.
It can be found in Figures 5–7 that in Figure 5, the obtained profits by strategy 2 are generally higher than that by strategy 1. Conversely, in Figures 6 and 7, the obtained profits by strategy 1 are generally higher than that by strategy 2. Strategy 2 is obtained based on PM. Given that the realtime PV power outputs are presented by PM (as depicted in Figure 5), the obtained profits by strategy 2 are certainly the highest ones. In practice, the occurrence of PM in realtime is nearly impossible, which means the deviations of realtime PV power outputs from PM may cause substantial cost for strategy 2. However, strategy 1 is obtained based on the uncertainty set, which guarantees the robustness of PPP’s power bidding schedules in avoiding substantial cost brought by realtime PV power deviations. Hence, the obtained profits by strategy 1 are generally higher than that by strategy 2 in most realizations of realtime PV power outputs except for PM. In summary, by utilizing our proposed model for power bidding decisionmaking, PPP’s obtained profits can be further increased in the face of random fluctuations in PV power outputs.
3.3. Sensitivity Analysis
Parameter will significantly influence the result of our proposed model. In this subsection, the sensitivity analysis for is presented for further studying the performance of our proposed model, which is shown in Figure 8.
In Figure 8, the average profit equals to the mean value of the obtained profits under different prices scenarios. The range of the uncertainty set increases with the increase of the value and decreases with the decrease of the value. In extreme cases, the uncertainty set will shrink into PM when and will expand into a largest one as presented in Figure 1 when . A larger uncertainty set corresponds to a more conservative and robust strategy which obtains less profit when there is no realtime PV power deviation but guarantees less excess cost when there exist realtime PV power deviations. A smaller uncertainty set corresponds to a more aggressive strategy, which obtains more profits when there is no realtime PV power deviation, but brings more excess costs when there exist some realtime PV power deviations. Therefore, as shown in Figure 8, the average profit decreases with the increase of the value, given that the realtime PV power outputs are presented by PM. Conversely, the average profit increases with the increase of the value, given that the realtime PV power outputs are presented by the worstcase realization.
4. Conclusions
This paper proposed a robust optimization model for a PPP in determining the dayahead power bidding schedules under EM circumstances. In our proposed model, the stochastic feature of realtime PV power outputs is modeled as an uncertainty set. The optimal power bidding schedules are determined under the worstcase realizations within the uncertainty set, which guarantees the robustness in resisting the impact of realtime PV power deviations on profit. Moreover, a model reformulation approach based on dual theory was introduced for transforming our model into a MPSP problem which is tractable and easy to be solved by the C&CG method. Simulations have presented some interesting conclusions:(1)Our proposed model can make PPP strategically bids in dayahead EM according to prices signals (“selling more at relatively high prices” and “selling less at relatively low prices”).(2)By implementing our simulation, it takes an average of 11.7 seconds for obtaining our strategy. That is to say, low computational time makes our proposed model feasible in practice.(3)By utilizing our proposed model for power bidding decisionmaking, PPP’s obtained profits can be further increased in the face of random fluctuations in PV power outputs.(4)With increasing the values of the budget parameter in the uncertainty set, a more conservative and robust strategy can be obtained, which obtains less profit when there is no realtime PV power deviation, but guarantees less excess cost when there exist realtime PV power deviations.
Our future work will release the “price taker” assumption and focus on the “price maker” strategies of PPP. Moreover, extending the PPP to some hybrid energy systems such as microgrid, virtual power plant, and integrated energy system will also be the topic that we will focus on in the future.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This work was supported by the State Grid Corporation Science and Technology ProjectResearch on Time Sensitive Network Technology and Its Application in Distributed Energy Regulation (5700201919238A0000).
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 Hourly illumination data: https://wenku.baidu.com/view/dc8cc130c8d376eeafaa3183.html.
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Copyright © 2020 Lifeng Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.