Stochastic multi-criteria decision-making for scheduling of wind–photovoltaic–hydropower systems

In response to global climate change, more and more countries are developing new energy sources such as wind power and photovoltaic power generation (Rigatos et al. 2019; Russo et al. 2023; Xia et al. 2023). The scientific development of wind–photovoltaic–hydropower system scheduling scheme is an important basis for guaranteeing system efficiency, system safety, downstream river health, reservoir safety, and other goals (Zhang et al. 2021). However, the wind–photovoltaic–hydropower system scheduling scheme is affected by many uncertainties in the scheduling process, such as the subjective preference of decision-makers and the importance of the indicators themselves (Liu et al. 2020).

A number of multi-criteria decision-making methods have been applied to scheduling of wind–photovoltaic–hydropower systems. In the deterministic field, Kang et al. (2011) used the Fuzzy Analytical Hierarchy Process (FAHP) to analyze the indicators of benefits, opportunities, costs and risks of wind farms to assess the expected comprehensive benefits of wind farm projects in order to select the most appropriate wind farm construction option in the comprehensive wind farm siting decision. Based on the subjective preference of the decision-maker and the improved entropy weight to determine the target weights, combined with the fuzzy set theory, Lu et al. (2011) proposed a decision-making method for multi-objective joint scheduling of a group of reservoirs. Perera et al. (2013) obtained Pareto frontiers for the objectives of energy cost, load deficit, energy wastage and fuel consumption through multi-objective optimization during the design of a hybrid energy systems (HESs), and then considered the existence of ambiguity in the level of importance between the indicators, the weights of the indicators were processed using the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS). Sanchez-Lozano et al. (2016) analyzed 10 indicators related to wind farm siting by coupling FAHP and fuzzy TOPSIS (FTOPSIS) methods in wind farm siting assessment. The study shows that the method is suitable not only for evaluating quantitative indicators, but also for qualitative indicators.

In the field of uncertainty, Liu et al. (2019) considered different decision-makers' knowledge and preference of indicators in the group decision-making process in the formulation of scheduling scheme for wind and hydropower system, and considered that the weights of indicators obeyed the uniform distribution. The second generation of SMAA (SMAA-2) was used for decision-making, and a scheme that meets the preferences of decision-makers was obtained. Qin et al. (2010) described the indicators such as maximum water level, maximum discharge flow, annual power generation and minimum output as random variables obeying interval normal distribution, and proposed a risk-based multi-criteria decision-making method based on the combination of dominant likelihood degree and comprehensive assignment. Zhu et al. (2017) proposed a stochastic multi-criteria decision-making model for reservoir flood control scheduling based on Qin's work by assuming the indicator weights to be uniformly and normally distributed, in addition to considering that the indicator values obey the normal distribution, and that there may be conflicts in the indicator weights. The results show that the model has sorted acceptability indicators, center vectors and global acceptability indicators, which can provide rich decision-making information for decision-makers when making decisions.

The development of scheduling schemes for wind–photovoltaic–hydropower systems involves knowledge from a number of fields, including wind power, photovoltaic power, hydropower, ecology and group decision-making, and requires a high level of knowledge on the part of the decision-maker in order to develop a scientifically sound scheduling scheme (Tan et al. 2021). At the same time, it is difficult to obtain a scheduling scheme that satisfies the preferences of the decision-making group due to conflicting preferences and competing objectives in the group decision-making process (Wang et al. 2022). The subjective preferences of different interest decision-makers are not consistent, and the weights of indicators derived from different objective weighting methods are not the same, leading to a certain degree of uncertainty in the weights of indicators (Chen et al. 2023). Therefore, this paper proposes a stochastic multi-criteria decision-making framework for wind–photovoltaic–hydropower systems. In response to the decision-making process, decision-makers are afraid to express their own preference information due to their own knowledge limitations, SMAA-2 and VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) models are reviewed, and the coupled SMAA–VIKOR model is used to analyze the inverse weight space of each completely unknown indicator weight, and to clarify the indicator weight space. In the later stages of decision-making, as information availability continues to improve, the decision-maker's understanding of the options and indicator weights becomes clearer, but there is still a corresponding ambiguity in the indicator weights, which relies on the partly clear and partly ambiguous subjective opinions of the decision-making group. For this reason, the Intuitionistic Fuzzy Analytic Hierarchy Process (IFAHP) is introduced to allow decision-making groups to express their fuzzy preference information. Finally, the IFAHP- SMAA–VIKOR model is developed to scientifically develop long-term scheduling schemes for wind–photovoltaic–hydropower systems. The proposed stochastic multi-criteria decision-making framework for wind–photovoltaic–hydropower systems overcomes the difficulty of uncertainty in indicator weights or even completely unknown information of indicator weights when making decisions. This study enriches and refines the stochastic multi-criteria decision-making methods under uncertainties.

The rest of this paper is organized as follows. Section 2 provides the details of the SMAA-2 Model, VIKOR Model, and group decision-making model for wind–photovoltaic–hydropower systems. Section 3 provides the details of the case study, which includes the analysis of sampling method, utility functions, and deterministic and stochastic multi-criteria decision-making. Section 4 presents the discussion and conclusions of this study.

The development of scheduling scheme for wind–photovoltaic–hydropower systems involves knowledge from multiple disciplines such as wind power, photovoltaic power, hydropower, ecology, and group decision-making, which requires the decision-making group to be familiar with these disciplines, and it is difficult to develop a scientifically sound scheduling scheme. Decision-attribute information accompanied by the subjective will of the decision-maker and its own attributes exists a certain uncertainty, even the indicator weight information is completely unknown. In view of this, we reviewed the SMAA-2 model to carry out inverse weight space analysis of indicator weight space to overcome the obstacle of no weight information, reviewed the VIKOR model to improve the linear summation type utility function in SMAA-2, and established a new SMAA–VIKOR model to derive a scientific and reasonable scheduling scheme for wind–photovoltaic–hydropower systems.

Lahdelma & Salminen (2001) proposed a SMAA-2 that can take into account the weight information and the uncertainty of the alternatives. SMAA-2 is based on a decision model that performs inverse weight space analysis on indicator weights and calculates the probability of each alternative becoming the optimized one, which overcomes the problem that traditional deterministic decision models are not applicable to multi-criteria decision problems with unknown weight information.

In the stochastic multi-attribute decision-making process, the indicator weight space W can be represented by a probability density function ⁠, and there are usually four possibilities, which are (1) the indicator weight values are uniquely determined, (2) the indicator weight values obey a uniform distribution in the specified interval, (3) the indicator weight values obey an arbitrary type of distribution in the specified interval, and (4) the information about the indicator weights is completely unknown (Barron & Barrett 1996).

• (1)

  The indicator weight values are uniquely determined.

The VIKOR method is a multi-criteria decision-making method for trade-off ranking. It starts by identifying a set of positive and negative ideal solutions, and subsequently calculates the distances between the alternatives and the positive and negative ideal solutions, and performs a trade-off ranking of the alternatives based on maximizing the group benefits and minimizing the individual regrets (Opricovic & Tzeng 2004). Both the VIKOR method and the TOPSIS method are trade-off ranking methods for near-ideal solutions. The TOPSIS method bases the trade-off ranking on the fact that the alternatives are the closest to the positive ideal solution and the farthest from the negative ideal solution. This may lead to a reverse order result, while the VIKOR method does not need to consider the problem that the closest alternative needs to be closest to the ideal point and farthest from the negative ideal point, which is a good way to avoid the reverse order problem.

A linear utility function is used in the original SMAA-2 model to evaluate the utility values of the alternatives. Due to the characteristic of non-metricity of multi-objective problems in wind–photovoltaic–hydropower systems, the decision-making directly through the linear sum-type utility function in the SMAA-2 model may not be able to obtain a fair solution. However, the SMAA-2 model has good expandability and is easy to be coupled with other multi-attribute decision-making methods (Corrente et al. 2014). And the VIKOR model is widely used in the field of multi-attribute decision-making, which has been proved to have better maneuverability, convenience and robustness by calculating the proximity of the evaluated value of each alternative to the ideal solution and avoiding the inverse order problem. Therefore, we try to review the VIKOR model into the SMAA-2 model of inverse weight space analysis, and propose a stochastic multi-criteria decision-making method, i.e., the SMAA–VIKOR model.

The SMAA–VIKOR model inherits the SMAA-2 model with the characteristic of inverse weight space analysis, and the specific operation procedure is as follows:

• (1)

  Normalize the decision matrix of the alternatives to obtain a standardized decision matrix ⁠;

• (2)

  Determine the distribution of the feasible weight space and perform Latin Hypercube Sampling (LHS) based on its probability density function to randomly generate feasible weights w;

• (3)

  Call the VIKOR model and calculate to get the corresponding ranking of each scheme;

• (4)

  Determine whether the number of iterations is satisfied; if so, next step; if not, return to step (2);

• (5)

  Calculate the ranked acceptability indicators and global acceptability indicators and center weight vectors based on the ranking of the alternatives obtained during the cycle;

• (6)

  End.

The proposed methodology is applied to the Yalong River Basin. The experimental runoff data were measured at the hydrological station in 2016. The wind speed, solar radiation, and temperature were obtained from National Meteorological Administration (http://data.cma.cn/data/detail/dataCode/A.0012.0001.html). A more detailed description of the study area and data can be found in our previous study (Liu et al. 2019). In this study, on the basis of the non-inferior solution set of the wind–photovoltaic–hydropower system obtained after the multi-objective optimization by Liu et al. (2019), 20 representative schemes were selected uniformly on the Pareto frontier surface obtained in that study, as shown in Figure 6. In order to efficiently solve the multi-criteria decision model, the Monte Carlo (MC) and LHS were compared for the SMAA-2 and SMAA–VIKOR models in Section 3.1. In Section 3.2, the SMAA-2 and SMAA–VIKOR models were compared to verify the validity of the VIKOR method as a utility function for the decision model over the simple linear summation type utility function. Finally, the SMAA–VIKOR was compared with the VIKOR in Section 3.3 to confirm the superiority of SMAA–VIKOR over the deterministic VIKOR method.

The SMAA–VIKOR was compared with VIKOR in this section to validate SMAA–VIKOR and to further confirm the superiority of SMAA–VIKOR over the deterministic VIKOR. Since the deterministic VIKOR model has to be input with indicator weight information for multi-criteria decision-making, in this section, experts are invited to give the corresponding preference information for the three indicators, as shown in Table 1. The three indicators correspond to the three objectives of multi-objective optimization, with weights of 0.26, 0.36, and 0.38, respectively.

By inputting the indicator weight information in Table 1 into the deterministic VIKOR model for decision-making, the values of the integrated indicators can be obtained as shown in Table 2. As can be seen from Table 2, the comprehensive ranking structure of each scheme is A2A6A3A5A7A1A9A10A8A4A11A12.

With the above results, it can be seen that the SMAA–VIKOR model can obtain the probability of each scheme to achieve different ranking results, while the deterministic VIKOR model can only obtain the unique ranking results. In addition, the deterministic VIKOR model requires the decision-makers to give decision information before the multi-criteria decision-making, which creates a big difficulty for the decision-makers. which does not require information about the weights of the indicators given by the decision-making group, has a certain superiority over the deterministic VIKOR model and can provide more useful information for the decision-makers when making decisions.

In group decision-making for scheduling schemes of wind–photovoltaic–hydropower systems, there exists a certain uncertainty in the indicator weights or even no information about the indicator weights due to the knowledge level of the decision-makers and the attribute information of the scheme set itself. This study proposes a stochastic multi-criteria decision-making framework for wind–photovoltaic–hydropower systems to overcome the difficulty of uncertainty in indicator weights or even completely unknown information of indicator weights when making decisions. Under this framework, SMAA theory and VIKOR model are reviewed, and the SMAA–VIKOR model is proposed to clarify the indicator weight space. The study shows that the proposed SMAA–VIKOR model can overcome the obstacle of decision-makers' lack of information on indicator weights. The ranking acceptability indicators calculated by the model show a more obvious trend of advantages and disadvantages, which gives full confidence to the decision-making group to formulate a plan to be implemented. It breaks through the bottleneck of group decision-making, which is difficult to make effective decisions due to the condition of incomplete information, and enriches the library of stochastic multi-criteria decision-making methods for the scientific formulation of scheduling schemes of wind–photovoltaic–hydropower systems under uncertainty conditions. Several findings can be revealed as follows:

In solving the SMAA-2 and SMAA–VIKOR models, the global acceptability indicators obtained by the LHS method are generally better than those obtained by the MC method, and the stability of the LHS method is also better than that of the MC method.

The SMAA–VIKOR model has a probability of 0.69 to rank first globally for Scheme A2, which is higher than the probability of 0.46 for the SMAA-2 model to rank first for Scheme A2. Additionally, the schemes obtained from the SMAA–VIKOR model are more concentrated in their rankings compared to the SMAA-2 model. This concentration can effectively reduce the impact of weight uncertainty on random decision results in wind–photovoltaic–hydropower systems decision-making, resulting in more distinct probability ranking outcomes and providing decision-makers with clearer information.

In contrast to the deterministic VIKOR model, the SMAA–VIKOR model, which considers the uncertainty of indicator weights, provides varying probabilities for each scheme across different rankings. Scheme A2 has the highest probability of ranking first at 0.69 but also holds probabilities for other rankings, such as 0.13 for ranking fourth. The scheme with the worst ranking acceptability index, A12, has a probability of 0.017 to rank first. Schemes A1, A4, and A7, which rank in the middle, have probabilities of 0.227, 0.211, and 0.069 to rank first, and probabilities of 0.084, 0.020, and 0.117 to rank sixth, respectively. Unlike the deterministic VIKOR model, which provides only one ranking for each scheme, the SMAA–VIKOR model does not require decision-makers to provide decision information in advance before multi-attribute decision-making. This eliminates the difficulty decision-makers face in determining wind-hydro system scheduling plans at the initial stages of decision-making. The SMAA–VIKOR model, which does not require decision groups to provide indicator weight information, has certain advantages and can offer decision-makers more useful information during the decision-making process.

The complementary characteristics of wind, photovoltaic, and hydro power generation exist at different time scales, and the uncertainty of prediction presents a complex coupling relationship. Conducting research on the prediction and uncertainty analysis of basin wind, photovoltaic, and hydro power generation is of great significance for elucidating the coupling laws of complementary relationship and prediction uncertainty. This study focuses on the deterministic input used in the long-term scheduling of basin wind, photovoltaic, and hydro power systems, without involving the study of long-term prediction and uncertainty of wind and photovoltaic energy. Future research needs to start from the long-term prediction uncertainty of basin wind, photovoltaic, and hydro power, introduce prediction models for the coupling relationship of long-term prediction uncertainty in basin wind, photovoltaic, and hydro power generation, and use them to scientifically formulate long-term scheduling plans for basin wind, photovoltaic, and hydro power systems.

W.L., Y.Z., Y.L. conceptualized the study; W.L., Y.Z., X.X. performed the methodology; W.L., X.G., R.M., Y.Z. did formal analysis and investigation; W.L., Y.Z. wrote and prepared the original draft; W.L., X.G., R.M, Y.L., Y.Z. wrote, reviewed, and edited the article; Y.L., Y.Z., W.L. acquired the funds; W.L., Y.Z. collected resources; Y.L. supervised the article.

This work was supported by the National Key Research and Development Program of China (Grant No. 2022YFC3202300); the National Natural Science Foundation of China (Grant No. 52209032); the China Postdoctoral Science Foundation Funded Project (Grant No. 2021M702313).

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.