The development of renewable energy becomes increasingly important because of exhaustion of fossil energy. Hydropower is one of the most important ways to generate electricity from renewable energy because of its relatively stable output among them. However, hydropower project development with some inherent characteristics is highly susceptible to social and natural environments, which complicates the investment process. For this purpose, this paper proposes a feasible comprehensive optimization model of portfolio investment from the perspective of sustainable development, describing the tradeoff relationship between economic, social, and ecological factors. As a hybrid uncertain NPhard optimization problem, there are three critical challenges: (1) achieving comprehensive balance between economy, society, and ecology; (2) identifying available multiple conflicting objectives and reasonable constraints; (3) analysing the hybrid uncertain environment. Therefore, a practical problemoriented multiobjective decision analysis model is established. Then, a multiobjective adaptive particle swarm optimization algorithm is designed to solve the model. Finally, a case study is carried out to verify the practicality of the model and the effectiveness of the improved algorithm. The result demonstrates that the model can be applied as a useful decisionmaking tool for decisionmakers in sustainable hydropower project development.
Energy is the key strategic resource and basic industry of the national economy [
Current status of hydropower development in the world.
Unfortunately, hydropower investment, especially in China, has already been caught in a dilemma [
Till now, a lot of research has been done on the hydropower investment problem (HIP) [
At the same time, due to the influences of market, social, natural environmental variation, policy changes, and other factors, the uncertainty in the decisionmaking of hydropower project has been widely recognized [
Starting from the research of Zhang et al. [
Especially, the integrated MODM approach in this paper consists of a multiple objective programming model, a hybrid uncertain parameter transformation process based on chanceconstrained programming (CCP), and a multiobjective adaptive particle swarm optimization (MOAPSO). In detail, the multiobjective programming model is based on the decisionmaking objectives of the decisionmakers and is transformed into an exact mathematical model on account of the actual situation. The uncertain parameter transformation process involves identifying uncertain parameters and converting them into random variables and fuzzy variables and then processing the two uncertain variables with the CCP model. Finally, the transformed mathematical model calculated by MOAPSO, and the Paretooptimization solution for hydropower investment optimization is obtained. This model is proposed for hydropower enterprise to achieve comprehensive targets. Overall, the framework of the study can be shown as in Figure
Framework of hydropower investment portfolio optimization.
From the above discussion, this study mainly makes an effort to achieve the main contributions as follows:
Establishing a multiobjective optimization model that better reflects the actual situation, which can dig into the optimal scheme of the portfolio scheme in hydropower projects with Chinese characteristics.
Achieving an overall balance of economic, immigration, and ecology (EIE) aspects under the hybrid uncertain environment. Then, the random variable and fuzzy variable are used to describe the electricity demand and variable costs in the model.
Taking the installed capacity and quantity of the power station as a decision variable, it can better reflect the actual situation of the investment decision of the hydropower station.
Developing a searching algorithm for sustainable optimization scheme of hydropower project based on MODM.
This study explores HIP in sustainable energy through overall consideration of the EIE objective of portfolio scheme. This paper might enrich the orientation of traditional HIP, and the current methods might be enriched by the proposed model. In order to guarantee the convenience of the model in practical applications, the use of contrastive case and the intelligent algorithm can assist decisionmakers to acquire an accurate and effective conclusion.
The rest of the article is structured as follows. Section
This study aims to coordinate these mutually conflicting but closely related objectives, i.e., try to seek balance among economy, social, and ecology. The mathematical model of this paper is established on the basis of the development company’s acquisition rights. Therefore, the acquisition cost of the development right of the basin is not considered in this paper.
The hydropower station encompasses a series of aggregated costs such as fixed operating cost, initial investment cost, and variable operating cost including the salary of employees and reservoir maintenance fees, as well as environmental cost and immigration cost. In this paper, the initial cost of the hydropower station construction is considered. Once the hydropower station is completed, the fixed and variable costs will be required for operation, which is the same as the environmental costs that is generated by the electricity production. According to the classification and design standard of water conservancy and hydropower hub project in China, the hydropower station can be divided into five categories according to installed capacity (i.e., Type I, Type II, Type III, Type IV, and Type V). The classification criteria are shown in Table
Hydropower station installed scale division standard.
Type  I  II  III  IV  V 

Installed capacity range (MW)  [1200, 
[300, 1200)  [50, 300)  [10, 50)  (0, 10) 
In addition, through the investigation of the investment cost of hydropower stations, as shown in Figure
Costs for different capacities of hydropower stations.
After consulting the investment cost data of the similar basins, such as Jinsha River, Yalong River, Nanya River, and Donggu River, the costs required for different types of hydropower stations can be calculated. In summary, costs for different capacities of hydropower stations are shown in the Table
Costs for different capacities of hydropower stations.
Hydropower station scale  Type I  Type II  Type III  Type IV  Type V 

Cost per kilowatt (yuan)  6996.22  14270.05  14270.71  10000.00  10000.00 
From the above table, it can be found that the development costs of hydropower stations with different installed capacities are quite different. Furthermore, the funds that hydropower developers can use to construct hydropower plants are limited. And due to the irreversibility of investment, developers prefer to use the funds in the construction and development stage as much as possible. Therefore, the primary goal of developers is to pay attention to the development costs of hydropower station in the early stage of construction, that is to say, to minimize the initial investment quota of hydropower station, which can be described as follows:
The longterm goal of a company’s operations is profitability, which means investment efficiency. The annual investment returns of the hydropower station construction is constrained by annual fixed operating cost, annual pumping cost, annual carbon emission cost, and annual variable operating cost. Among those, the annual variable management cost in this paper includes the cost of payroll and employee benefits where the cost is closely related to employee numbers. For the investment decisionmaking of hydropower combination optimization, decisionmakers fail to give an exact number of employee numbers of the hydropower station before the power station operation begins. So it is difficult to illustrate these problem parameters as vague values, therefore becoming an uncertainty factor in the decisionmaking stage. There are some studies that have mentioned this uncertainty in power station optimization [
The construction of hydropower stations involves the relocation of residents around the original water areas. There are different policies for different regions, so the immigrant cost is related to the geographical conditions of the basin itself and the number of immigrants that cannot be generalized. Moreover, the migration costs of upstream and downstream power stations are quite different. In this paper, the relatively costs involved in similar watersheds during the survey are considered as follows to realize the minimum:
Nowadays, environmental issues have drawn worldwide attention. The environmental factors must be considered in project construction. In this paper, the impact on the environment in hydropower projects mainly considers carbon dioxide emissions. Hydropower is a clean energy source that generates extremely low carbon emissions during power generation. If more hydropower can be introduced into the power system, environmental problems can be alleviated to some extent. Therefore, the environmental benefit in the sustainable optimization of hydropower investment decisions is to minimize carbon emissions. In this paper, the environmental benefits of hydropower are obtained by comparing the carbon emissions from hydropower with those from standard coalfired power generation.
The demand for hydropower can be predicted by historical data that come from the Chinese Yearbook 2001–2017. As is shown in Figure
Electricity consumption of the whole society from 2001 to 2017.
Therefore, in the current power market, which has been vigorously developing RE power generation, the demand for hydropower should adjust accordingly. That is to say, the amount of hydropower must meet the demand for realtime electricity consumption to a certain extent. Furthermore, electricity demand is a common uncertain factor, which affects the operation of hydropower station. Due to the changes of natural and social environment, such as weather and economic development especially for longterm decision, it is difficult to accurately predict. Nevertheless, those factors significantly affect investment and operation of hydropower station. Unfortunately, it cannot be directly acquired for the annual electricity demand during the operation of hydropower station at the investment decision stage. Many researches have referred to the uncertainty of electricity demand [
Thus, the total power generation of the hydropower station developed by the company should be greater than the load demand allocated by the power grid company during the planning period, as shown below:
Similarly, for the upper limit demand forecast, the powergenerating capacity of the entire generator set during the planning period should be less than or equal to the designed installed capacity. Here, assume that the utilization of hydropower generating units is constant, the installed capacity constraints of the units are as follows:
Hydropower station scale is defined as follows: (1) Type I Hydropower Station, (2) Type II Hydropower Station, (3) Type III Hydropower Station, (4) Type IV Hydropower Station, and (5) Type V Hydropower Station (
The hydroelectric stations are interrelated and mutually constrained in a watershed. Especially for cascade hydropower stations, there is a hydraulic connection between them to form an overall coordination of hydraulic power. Considering the impact of water flow on power generation, the generation capacity should be less than or equal to the amount of hydropower available in the basin during the planning period. In this study, the water flow connection between the hydropower stations is assumed to meet the following conditions: (1) Water head connection: The cascade hydropower station is intermittently connected where the downstream water level of the upstream hydropower station is determined by the lower drain flow. Besides, the head changes of the adjacent two cascade hydropower stations are irrelevant. (2) Water flow relation: The distance between two adjacent cascade hydropower stations is moderate. To do so, the water flow changes of the two power stations can occur simultaneously. Further, the interval flow is increased in the drainage ditch of the upstream power station, which finally constitutes the inflow flow of the downstream hydropower station. (3) In this paper, the water flow time lag of the upstream drain flow and the water loss are not considered. The power generation limit of the hydroelectricity set is as follows:
Since most of the funds needed for hydropower development rely on external financing, the investment decisions of renewable energy companies such as hydropower are significantly affected by external financing constraints [
The main focus of hydropower investment is expected return and investment risk. According to the meanvariance model proposed by Markowitz [
Based on the above description, a sustainable multiobjective optimization decision model is developed, which takes into account the economic, ecological, and immigrant hydropower portfolio system. And it is used to determine the balance among economic benefits, migration problems, and environmental impacts under electricity demand and capital constraints in order to achieve the sustainable development of hydropower. To achieve economic, immigrant, and environmental balance, the optimization decision model includes four objectives: minimizing initial investment costs, maximizing operational benefits, minimizing immigration costs, and maximizing the benefits of hydropower. In this model, the fuzzy theory is used to describe the change management cost of hydropower operation stage and the stochastic theory is used to describe the demand of electricity because these parameters are difficult to determine accurately. The model considers the optimal decision from the perspective of the developer. And the decision variables are composed of the installed quantity
As described in Murto [
Step 1: collect historical data on electricity consumption
Step 2: use multivariate time series prediction model to establish prediction equation
Step 3: use prediction equation to forecast the electricity demand in the each of the previous
Step 4: assume the annual forecasting error obeys the normal distribution
Step 5: forecast the electricity demand
For the study, since there are not enough data to comprehensively analyse influencing factors, resulting in uncertainty, inaccuracy, or ambiguity [
Step 1: collect the variable management cost data of hydropower stations of Nanya River, Donggu River, and Jinsha River and divide the data into three groups
Step 2: Let the minimum value of each set become the lower bound of each group fuzzy number
Step 3: Let the maximum value of each set become the upper bound of each group fuzzy number
Step 4: Let the average value of each set become the intermediate coefficient of each group fuzzy number
The CCP method is a powerful competing tool proposed by Charnes and Cooper [
Since the objective equation (
A similar method can be used to deal with equation (
In multiobjective decisionmaking problem, there are conflicts and incomparable phenomena among multiple objectives. The main purpose of multiobjective optimization is to seek Paretooptimal solutions (nondominated solutions) to optimize the tradeoff between multiple goals [
Step 1. Particle swarm initialization and solution representation:
Step 2. Feasibility checking and decoding method: as hydropower stations investment scale should meet the demand of electricity and logical constraints, it is necessary to inspect and discard the infeasible particle matter. Then, the particlerepresented solution is decoded into a solution in a general way, called the hydropower stations investment scale.
Step 3. Particle evaluation: the random position of the
Step 4. Multiobjective method: the PAES, test program, and selection consists of the multiobjective method, which is used to calculate
Step 5. Updating inertia weight: using the equation (
where
Step 6. Updating velocity and position of each particle by using the following equations:
where
Step 7. The algorithm termination checking:if the stopping criterion of the algorithm is satisfied (i.e., iterationmax), then the MOAPSO procedure is ended to acquire the optimal solution and the process is terminated. Otherwise, the algorithm will continue.
MOAPSO algorithm flow chart.
In this paper, four suitable indicators are used to assess the quality of the Paretooptimal solution set based on the study of [
The distance function
The function
The range of the Paretooptimal front is considered in the function
The set convergence
where
In other words, if
In this paper, the Nanya River tributary (NY) and Donggu River tributary (DG) of the Dadu River basin were studied and the computational experiments were carried out. As shown in Figure
Location of Nanya River and Donggu River.
The data used in this paper were mainly obtained from Sichuan statistical yearbook and local field researches. The random electricity demand is obtained by the proposed method in the part of uncertainty treatment, which is based on historical data from Guodian Sichuan Power Generation Corporation and investigation of the basin of Nanya River and Donggu River. The electricity demand fitting function is obtained by the predictive model, and the coefficient of determination
Comparison of predicted and historical values and distribution of error values.
In this paper, two tributaries are used to make calculation experiment, where
It is assumed that fixed operation and management cost
Parameters for the investment cost of the hydropower station.
Index  Parameter  









187.45  6996.22  (2.15, 2.41, 2.68)  0.013  610.2 

205.82  6996.22  (2.15, 2.41, 2.68)  0.013  610.2  




378.39  14270.05  (2.15, 2.41, 2.68)  0.013  610.2 

415.85  14270.05  (2.15, 2.41, 2.68)  0.013  610.2  




378.41  14270.71  (7.87, 8.27, 8.67)  0.013  610.2 

415.87  14270.71  (7.87, 8.27, 8.67)  0.013  610.2  




266.3  10000  (3.22, 3.31, 3.40)  0.013  610.2 

292.57  10000  (3.22, 3.31, 3.40)  0.013  610.2  




266.3  10000  (3.22, 3.31, 3.40)  0.013  610.2 

292.57  10000  (3.22, 3.31, 3.40)  0.013  610.2 
Parameters of power generation and investment.
Parameters 









Value  5000  0.000997  0.00082  0.02  4.35  0.20847  0.8  0.008 
Parameters of river basin generation investment.
Index  Parameters  






2.335  6.535  8.529 

1.145  3.0  8.666 
The improved algorithm is calculated by software MATLAB7.0, which runs on 3.50GHz Intel Core with 4.00 GB memory. The algorithmic parameters that is adapted for the case problem are set as follows: iteration_max
The confidence level reflects the degree of acceptance of risk by decisionmakers. Table
Nondominating solution target value interval under different confidence levels.
Interval of target value 





Investment cost (10^{11} yuan)  [2.117, 3.420]  [2.234, 3.093]  [2.185, 3.140]  [2.128, 2.936] 
Investment benefit(10^{10} yuan)  [1.123, 1.882]  [1.177, 1.826]  [1.066, 1.717]  [1.192, 1.649] 
Immigrant cost (10^{9} yuan)  [0.909, 1.342]  [0.881, 1.292]  [0.901, 1.235]  [0.893, 1.215] 
Environment efficient(10^{7} kg)  [1.660, 2.451]  [1.741, 2.360]  [1.647, 2.255]  [1.631, 2.220] 
When the confidence level
Nondominating hydropower stations investment number.
Hydropower  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  

Station type  NY  DG  NY  DG  NY  DG  NY  DG  NY  DG  NY  DG  NY  DG  NY  DG  NY  DG  NY  DG  NY  DG  NY  DG  NY  DG  NY  DG  NY  NY 
I  7  5  6  2  5  2  8  0  7  6  6  2  7  2  8  1  5  1  9  3  5  3  5  3  7  1  7  5  6  2 
II  3  4  8  10  10  0  1  6  3  0  1  9  9  6  8  8  4  9  8  5  6  4  0  3  6  5  2  2  7  7 
III  6  6  10  4  9  8  9  7  3  7  3  5  0  6  2  10  9  2  1  0  10  3  5  8  0  3  6  6  4  7 
IV  2  3  7  8  8  1  7  7  2  8  6  1  6  9  6  9  2  9  9  2  6  6  9  2  1  1  5  6  8  1 
V  6  2  9  6  7  3  6  10  7  3  1  0  2  1  8  4  3  8  7  9  2  4  6  7  5  8  2  3  9  10 
Nondominating hydropower stations install capacity (MW).
Hydropower station type  1  2  3  4  5  

NY  DG  NY  DG  NY  DG  NY  DG  NY  DG  
I  2431  1275  2261  1798  1308  1786  1936  1410  2490  1608 
II  1076  810.2  377  392  910.3  831  967  1030.0  1180  664.1 
III  285.1  143.4  99.6  102.3  112.8  236.2  233.2  180.3  281.7  130.7 
IV  24.44  15.76  30.19  47.83  22.47  49.23  26.85  47.64  29.05  17.00 
V  4.004  9.976  9.011  9.051  9.550  5.208  1.655  1.965  0.378  7.412 


Hydropower station type  6  7  8  9  10  
NY  DG  NY  DG  NY  DG  NY  DG  NY  DG  


I  2389  2141  1794  1481  1761  2394  1886  2367  2158  2018 
II  837.7  462.6  867.1  798  1097.0  390  761.0  778.6  398.4  776.8 
III  268.7  68.4  295.9  297.3  99.7  100.5  270.3  155.7  191.8  288.6 
IV  26.45  21.44  32.95  47.56  26.89  25.04  19.64  45.80  35.10  16.90 
V  1.029  2.533  9.968  2.504  7.671  7.476  0.926  2.546  3.578  0.907 


Hydropower station type  11  12  13  14  15  
NY  DG  NY  DG  NY  DG  NY  DG  NY  DG  


I  2312  2166  2267  2064  1814  1420  1378  1441  1929  1965 
II  802.3  468.5  923.7  595.8  460.7  1067.9  832.8  311.7  1071  624.5 
III  296.0  162.9  255.9  290.7  243.0  259.3  216.8  231.3  165.3  54.84 
IV  21.33  34.49  24.50  41.12  15.05  11.89  42.22  38.57  36.57  25.11 
V  0.441  8.527  8.822  6.065  0.506  9.365  6.180  3.349  1.939  0.127 
Nondominating values of four objective functions.
Although there are random and fuzzy variables in the established model, the theory of random and fuzzy is used to convert them into equivalent values in this paper; therefore, the decision results will not be affected.
In this paper, a comprehensive optimization model of hydropower investment portfolio is established, which combines objectives with constraints. Then, the multiobjective decisionmaking model is adopted to determine the optimal solution set for the proposed model and provides a more effective Paretosolution set. For instance, if the decisionmaker pursues the maximization of the return on investment, the nondominating solution 5 can be chosen. According to Figure
The fuzzy stochastic programming approach specifically takes the uncertainty environment into account, and the risk acceptance degree of decisionmakers is considered by the model to make the model more realistic. For example, if the decisionmaking is risk averse, the values of
The quality of the multiobjective optimization solution is more complicated than single objective optimization. Four performance indexes and the number of solutions are discussed in order to describe the validity of Paretooptimal solutions more deeply. Figure
Assessment result of nondominating solution performance.
The standard multiobjective PSO and improved algorithm are compared in this paper. In the proposed MOAPSO, the solution represented by the particle associates the PSO particle with the solution of the problem and combines the hybrid update mechanism with the updation of inertia weight and velocity acceleration constants to successfully improve the particle search ability. As is shown is Table
Performance comparison of MOAPSO and basic PSO.
Algorithm type  Iteration 




Solution amount 

MOAPSO  150  1.54 × 10^{−7}  0.780  4.99 × 10^{−8}  1  15 
PSO  150  1.90 × 10^{−7}  0.773  2.89 × 10^{−8}  1  12 
In order to improve hydropower investment efficiency and maximize comprehensive benefits, this paper studies the portfolio optimization problem in hydropower station construction and development under hybrid uncertain environment. Under the orientation of the goal of optimizing the comprehensive benefit of investment, a comprehensive model was proposed to obtain the optimal combination of hydropower investment scale. Furthermore, a MODM model for hydropower investment portfolio optimization was initially established. As random and fuzzy parameters exist in the established model, a CCP approach that considers the risk preference of decisionmakers is introduced, which makes the model better applied to the actual uncertain decisionmaking process. Subsequently, based on random and fuzzy theory, it is equivalently converted into a crisp model. Furthermore, the MOAPSO algorithm is designed to solve the problem. At the same time, combined with the advanced Paretooptimal solution judgment criteria, the quality of particle solutions is further discussed. Finally, an example is given to illustrate the existence of the problem. The results of the example verify the value of the model and verify the solution and the efficiency of the algorithm and parameters. Based on this process, the following conclusions can be drawn:
The proposed comprehensive optimization model is theoretically effective and reasonable and can be used to improve the comprehensive benefits of hydropower investment portfolios.
Considering the uncertain environment in the model, a CCP model describing the acceptor’s acceptance is established, which can make the hydropower combined optimization model closer to the actual situation.
The result of the optimization model is calculated by MOAPSO. The MOAPOS algorithm and the standard PSO algorithm are evaluated and compared by using four evaluation indicators. The result shows that MOAPSO can be used to solve HIP problems effectively, which can provide nondominated solution sets with faster convergence and uniform distribution.
Through the established model and the use of MOAPSO, the four objectives of minimum initial investment, maximum investment returns, minimum resettlement costs, and maximum environmental benefits are contradictory. That is to say, with the reduction of migration and initial investment costs, investment benefits and environmental benefits are reduced, and vice versa.
This paper is original. Nevertheless, there are still some fields that need further study. First, a more detailed and full description of the objective functions such as resettlement cost is necessary. Secondly, the proposed model in this paper only involves the construction and operation phases, so the costs and benefits of the demolition phase of the hydropower station planning period can be incorporated into the investment decision for a more comprehensive investment. Finally, in order to get a better and more efficient solution with shorter computation time and less memory, some alternative methods and algorithms can be explored, such as genetic algorithm, ant colony algorithm, and coevolution algorithm. These areas are fully significant and equally worthy of research.
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was funded by the Humanities Social and Sciences Research Funds of Education Ministry (Grant no. 15XJC630001), the Key Funds of Sichuan Social Science Research Institution System Science and Enterprise Development Research (Grant no. Xq18B06), the Foundation of Chengdu Science and Technology (Project no. 2017RK0000274ZF).