Reservoir Inflow Prediction System Based on Interval Type-2 Fuzzy Logic ^†

Abstract

Due to its fast start and stop, purity, and reliability, hydropower is becoming more important in the overall power dispatch strategy in grids with a high proportion of wind and solar power generation. Therefore, we propose an interval type-2 fuzzy logic-based rainfall classification and fuzzy neural network model to build a 48 h reservoir inflow forecasting system, addressing the challenges of renewable energy instability and extreme weather in hydropower operations.

1. Introduction

In recent years, as global awareness of carbon reduction has grown, countries have been actively developing renewable energy sources such as wind and solar power. However, wind power is inherently unstable due to fluctuating wind speeds, and solar power is affected by shading from clouds, both of which challenge grid stability. Hydropower, with its rapid start–stop capability and clean, reliable characteristics, has become increasingly important in power dispatch strategies for grids with high shares of wind and solar energy. At the same time, the rising frequency of extreme weather events and short-duration intense rainfall has posed significant challenges to reservoir operations. These operations must meet multiple objectives, including power dispatch, domestic use, and agricultural irrigation.

To ensure reliable power dispatch, efficient hydropower generation, and optimal water resource utilization, while avoiding flood discharges caused by sudden surges in water levels, it is critical to accurately forecast reservoir water levels, considering the upstream reservoir release decisions that affect downstream hydropower plants. These complex, interconnected challenges involve multi-reservoir inflow forecasting and hydropower generation decisions, making traditional experience-based manual operations insufficient and suboptimal. Consequently, it is essential to design a comprehensive system that integrates inflow forecasting and optimal hydropower scheduling across a basin with multiple reservoirs. Recent advancements in artificial intelligence—such as forecasting techniques, fuzzy logic, and neural networks—offer promising solutions to these issues.

Fuzzy logic, in particular, has proven effective in handling uncertainty in individual concepts through linguistic representation and is widely applied in AI-related fields. Originating from Professor Zadeh’s fuzzy set theory [1] proposed in 1965, type-2 fuzzy logic [2] was introduced in 1975 to address uncertainties arising from individual differences. A specific case, interval type-2 fuzzy logic [3], represents the degree of membership as an interval, making it more suitable for capturing uncertainties in various scenarios. Research has shown that interval type-2 fuzzy logic performs better than traditional fuzzy logic in dealing with input uncertainty [4,5], such as in short-term electricity load forecasting. It is thus well-suited to model the uncertainty in rainfall forecasts for reservoir catchments.

Forecasting models are generally classified into statistical and physical models. Physical models rely on known physical parameters, requiring extensive data and computational resources, which limits their practicality in real-time renewable energy forecasting and power scheduling. Statistical models, conversely, learn patterns from large datasets to produce predictions, with heavy computation during training but minimal computational requirements during deployment. Among these, intelligent forecasting models, particularly neural networks [6,7], have gained attention for their strong learning capabilities. Neural networks can model complex nonlinear relationships without the need for explicit mathematical formulations and are widely used in hydrological and forecasting applications [8,9,10,11,12,13,14].

Multi-reservoir optimization decision-making is considered a multi-objective problem, often involving time-of-use electricity pricing, maximizing power generation, and regulatory constraints such as water level curves and reservoir capacities. Key constraints include power-to-water ratio curves, storage volumes, generation heads, and maximum discharge rates. Traditional optimization techniques such as linear programming [15,16], nonlinear programming [17,18], and dynamic programming [19,20] have been widely applied to reservoir operation problems. For instance, reference [21] described a system architecture for ten reservoir states that has been proposed as a benchmark for algorithm evaluation. However, due to the nonlinear and complex nature of real-world water resource systems, these traditional methods often struggle with issues like local optima and performance in high-dimensional spaces. To address these limitations, heuristic algorithms [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38] have been widely explored. While they require longer processing times, heuristic methods often converge closer to the global optimum. In recent years, they have been widely applied in multi-reservoir hydropower scheduling.

The genetic algorithm (GA) has been applied in the optimization of power generation decision-making for multiple reservoirs [22,23]. Reference [24] proposed a multi-objective evolutionary group hybridization method with which to evaluate multi-reservoir operation. Algorithms such as particle swarm optimization [25,26,27,28], ant colony optimization [29], simulated annealing [30,31], the bat algorithm [32], and the firefly algorithm [33] have been proposed for solving such problems. However, most of these optimization approaches overlook the risks posed by sudden inflow increases due to heavy rainfall, which is particularly critical in island climates in Taiwan.

Addressing these challenges requires integrating flood control considerations into power generation decisions, transforming the problem into a trade-off among multiple objectives. This further necessitates an understanding of how reservoir water levels fluctuate due to generated decisions and short-term rainfall inflow variations [39,40,41]. Reference [41] compared recurrent neural networks, long short-term memory (LSTM), and bi-directional LSTM in predicting inflow trends. For example, reference [41] used LSTM to model the relationship between upstream rainfall and inflow to Taiwan’s Shimen Reservoir based on meteorological forecasts. While many studies focus either on hydropower optimization or inflow forecasting, few integrate both aspects simultaneously. Reference [42] used global forecast systems and weather forecast and heuristic algorithms for 1- to 16-day power generation scheduling at the Detroit and Pensacola dams in Oregon, USA. However, daily scheduling is not ideal for island regions like Taiwan, which experiences intense and rapid rainfall during typhoons.

Therefore, this study aimed to develop an intelligent reservoir inflow forecasting model for the upstream reservoir in a river basin. Specifically, using historical rainfall, forecast rainfall, and historical inflow data, we developed a rainfall scenario classification method based on interval type-2 fuzzy logic. We also developed a fuzzy neural network prediction model capable of adapting to different rainfall scenarios, enabling a 48 h, hourly reservoir inflow forecasting system under multiple scenarios.

2. Method

A 48 h hourly reservoir inflow forecasting system was developed in this study based on historical inflow data, historical rainfall data, and rainfall forecast values. The overall system architecture is divided into two main components: rainfall scenario classification and reservoir inflow forecasting. The first component uses interval type-2 fuzzy logic to classify rainfall scenarios, while the second component applies a fuzzy neural network to generate 48 h hourly inflow forecasts based on the classified rainfall scenarios.

The overall flow volume of a river basin is highly correlated with the rainfall over its catchment area. Generally, the most upstream reservoir is connected to the largest number of catchments, and its inflow directly impacts the availability of water resources throughout the basin. Moreover, the release decisions made at the most upstream reservoir have a direct influence on the runoff conditions of the entire basin. To enable water resource planning in response to varying rainfall patterns caused by different climate conditions, it is essential to accurately forecast the inflow of the most upstream reservoir. Such forecasts not only provide better insight into future water resource trends but also support subsequent forecasting and hydropower scheduling for other reservoirs within the basin. Therefore, we designed the upstream reservoir forecasting model, including the use of interval type-2 fuzzy systems, heuristic algorithms, fuzzy neural networks, and evaluation methods, followed by an analysis of results from a real-world case study.

2.1. Interval Type-2 Fuzzy Logic

The interval type-2 fuzzy logic system [43,44,45] accounts for uncertainties among individual elements. The system consists of five main components: a fuzzifier, inference engine, rule base, type reducer, and defuzzifier. First, the crisp input is transformed into a type-2 fuzzy input using type-2 membership functions. This input is then processed using the fuzzy rule base and inference engine to generate a type-2 fuzzy output. The output is subsequently type-reduced into a type-1 fuzzy set. Finally, defuzzification is applied to convert the fuzzy value into a crisp output. The following sections provide a detailed explanation.

When the interval type-2 fuzzy system has z inputs ${(x}_1,x_2,\dots x_z)$, one output y, and M rules, the l-th rule can be described as follows:

$$R^l:If x_1 is {\tilde{A}}_1^l and x_2 is {\tilde{A}}_2^l and\dots x_z is {\tilde{A}}_z^l\hspace{1em}Then y is B^l$$

(1)

where ${\tilde{A}}_i^l$ denotes type-2 fuzzy sets, and $B^l=\left[\begin{array}{cc}{b}^l-d^l& b^l+d^l\end{array}\right]$ is an interval set, with $b^l$ and $d^l$ denoting the center point and the extend value. Each fuzzy rule is inferred as follows:

$$R^l:{\tilde{A}}_1^l\times {\tilde{A}}_2^l\dots {\tilde{A}}_z^l\to B^l={\tilde{F}}^l\to B^l$$

(2)

where ${\tilde{F}}^l={\tilde{A}}_1^l\times {\tilde{A}}_2^l\dots {\tilde{A}}_z^l$ describes ${\tilde{A}}_i^l$ as a type-2 fuzzy membership function, with ${\overline{\mu }}_{{\tilde{A}}_i^l}$ and ${\underset{¯}{\mu }}_{{\tilde{A}}_i^l}$ as upper-bound and lower-bound membership functions. Assuming a singleton fuzzifier, the input and the antecedent interaction result is $F^l=\left[\begin{array}{cc}{\underset{¯}{F}}^l& {\overline{F}}^l\end{array}\right]$, where ${\overline{F}}^l={\prod }_{i=1}^z{\overline{u}}_{{\tilde{A}}_i^l}(x_i)$ and ${\overline{F}}^l={\prod }_{i=1}^z{\overline{u}}_{{\tilde{A}}_i^l}(x_i)$, and output $y^l=\left[\begin{array}{cc}{y}_l^l& y_r^l\end{array}\right]$. All M rules are aggregated to produce an output $y=\left[\begin{array}{cc}{y}_l& y_r\end{array}\right]$, where $y_l$ and $y_r$ can be obtained through a type-reduction algorithm. One commonly used method is the Karnik–Mendel (KM) algorithm [46]. According to the KM algorithm, $y_l$ and $y_r$ are obtained using Equation (3):

$$y=\left[\begin{array}{cc}{y}_l& y_r\end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{{\sum }_{i=1}^L{\overline{F}}^i{y}_l^i+{\sum }_{i=L+1}^M{\underset{¯}{F}}^i{y}_l^i}{{\sum }_{i=1}^L{\overline{F}}^i+{\sum }_{i=L+1}^M{\underset{¯}{F}}^i}}& {\displaystyle \frac{{\sum }_{i=1}^R{\overline{F}}^i{y}_r^i+{\sum }_{i=R+1}^M{\underset{¯}{F}}^i{y}_r^i}{{\sum }_{i=1}^R{\overline{F}}^i+{\sum }_{i=R+1}^M{\underset{¯}{F}}^i}}\end{array}\right]$$

(3)

where L and R are the switching points calculated with the KM algorithm. Finally, the defuzzification step then converts the type-reduced result into a crisp output value $y=(y_l+y_r)/2$. Due to its ability to handle uncertainty among individual instances, the interval type-2 fuzzy logic system is well-suited for iterative multi-step forecasting problems, where inputs at different lead times exhibit similar individual uncertainty. The parameters of the membership functions and fuzzy rules are learned using a heuristic algorithm.

For the rainfall scenario classification method used in this study, observational data were input into the interval type-2 fuzzy logic system, which produces a scenario evaluation index ranging from 0 to 1. This index reflects the intensity level of the rainfall, while a scenario switching threshold is used to classify the rainfall scenarios. Assume that $I$ is the set of observational input data, and $C$ is the set of scenario evaluation indices produced by the interval type-2 fuzzy logic. The evaluation set $C$ is expressed as follows:

$$C=\mathrm{IT}2\mathrm{FL}\left(I\right)$$

(4)

Next, a classification result is defined as follows:

$$class\left(C\left(i\right)\right)=\left\{\begin{array}{c}\begin{array}{cc}{a}_1& ,C\left(i\right)<s\end{array}\\ \begin{array}{cc}{a}_2& ,C\left(i\right)\ge s\end{array}\end{array}\right.$$

(5)

where i refers to the i-th data point in set C, $s$ is the scenario-switching threshold, and $a_1$ and $a_2$ represent different categories. The switching threshold $s$ is obtained through a genetic algorithm.

2.2. Heuristic Algorithm

There are various heuristic learning methods. We introduced three commonly used heuristic algorithms: GA [23,24], the electromagnetism-like algorithm [35], and the bacterial foraging algorithm [36,37,38,39]. The genetic algorithm consists of three main operations: reproduction, crossover, and mutation. Reproduction involves selecting better individuals from the population for replication. Crossover refers to the recombination of chromosomes between individuals. Mutation allows individuals to randomly mutate with a certain probability to increase the chance of escaping local optima.

In this study, GA was employed to train the model. The chromosome encoding used is shown in Figure 1, where double-precision chromosomes are constructed. The parameters encoded in each chromosome include the center of each semantic variable $m_1\dots m_n$ in the membership function, the consequent membership function selection parameter $R_1\dots R_r$ for each rule, the weight of each rule $w_1\dots w_r$, and the rainfall scenario switching threshold $s$.

The cost function for GA is defined assuming a threshold 50 m³/s to separate reservoir inflow levels during the training of the interval type-2 fuzzy logic model. The correct classification outcome is defined as follows:

$$inflow\left(X\left(i\right)\right)=\left\{\begin{array}{cc}{a}_1& ,X\left(i\right)<50\\ a_2& ,X\left(i\right)\ge 50\end{array}\right.$$

(6)

Let X denote the observed reservoir inflow value one hour ahead. The classification result generated by the interval type-2 fuzzy logic model is described as follows:

$$A_c(class\left(C\left(i\right)\right),inflow\left(X\left(i\right)\right))=\left\{\begin{array}{cc}1& ,\mathrm{if} \mathrm{the} \mathrm{classification} \mathrm{is} \mathrm{correct} \\ 0& ,\mathrm{if} \mathrm{the} \mathrm{classification} \mathrm{is} \mathrm{incorrect}\end{array}\right.$$

(7)

Therefore, the cost function for GA is defined as follows:

$$CF={\left({\displaystyle \frac{N}{\alpha +{\sum }_{i=1}^N{A}_c(class\left(C\left(i\right)\right),inflow\left(X\left(i\right)\right))}}\right)}^2$$

(8)

where $\alpha$ is a user-defined parameter used to avoid division by zero, and $N$ is the total number of data points. As shown in Equation (8), the goal of the model is to minimize the cost function to achieve the optimal solution.

2.3. Fuzzy Neural Network (FNN)

An FNN combines the strengths of fuzzy logic and neural networks, making it a powerful tool for data modeling. Fuzzy logic excels at handling uncertainty and vagueness in individual inputs, while neural networks are proficient at learning from data and identifying the relationships between inputs and outputs. FNN incorporates the uncertainty of input data through fuzzy logic and then optimizes the model using the strong learning capabilities of neural networks. First, input data are transformed into fuzzy sets that capture the uncertainty of the input variables. These fuzzy sets are then fed into the neural network, where the structure and weighted connections implement fuzzy rule formulation, fuzzy inference, and final defuzzification to produce crisp outputs.

The training process for fuzzy rules and membership function parameters is similar to that of traditional neural networks. Due to its strong performance in handling fuzzy data and complex nonlinear problems, FNN offers greater interpretability compared to conventional neural networks. Therefore, in this study, FNN was used to design the upstream reservoir inflow forecasting model. The crisp inputs include historical inflow, historical rainfall, and forecasted rainfall, while the crisp output refers to the reservoir inflow values for K time steps ahead. Reservoir water levels can be estimated from the storage volume ratio. In this case, K = 48, and the unit is in hours.

2.4. Assessment Method

Reference [47] introduces various forecasting error analysis methods. Among them, the most important indicators are root mean square error (RMSE) and mean absolute error (MAE). MAE represents the average error of the analyzed data and describes the model’s accuracy. RMSE, conversely, has the effect of amplifying errors, making it more sensitive to outliers. The definitions of MAE and RMSE are as follows:

$$\mathrm{MAE}={\displaystyle \frac{1}{N}}\sum _{i=1}^N\left|y_i-{\widehat{y}}_i\right|$$

(9)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N(y_i-{\widehat{y}}_i{)}^2}$$

(10)

where N denotes the total number of data samples, $y_i$ is the actual observed values, and ${\widehat{y}}_i$ is the predicted values. These two metrics quantitatively evaluate the model’s performance and provide a clear comparison of prediction quality.

3. Results

In this study, historical reservoir inflow data from the Techi Reservoir, combined rainfall data from Techi Reservoir and its upstream catchment, and meteorological forecast data were used. The training dataset spans 1 January 2015 to 1 January 2016, while the testing dataset spans 1 January 2016 to 1 January 2017. The genetic algorithm’s parameters used for training are shown in

Table 1. GA parameters.

Parameter	Value
Population size	30
Number of iterations	100
Crossover rate	0.6
Mutation	0.1
Evolutionary method	roulette wheel selection

.

A 48 h ahead forecasting error analysis was conducted using the trained models. The results are presented in Figure 2. From Figure 2, it is observed that RMSE is significantly higher than MAE. This may be attributed to the fact that although there were multiple typhoons in 2015, none resulted in extreme rainfall events (>1500 m³/s), leading to insufficient training samples for heavy rainfall. This lack of large rainfall training data made it difficult for the model to accurately predict such extreme inflows. To address this issue, incorporating more effective observational training data could improve the model’s predictive capability under heavy rainfall conditions.

4. Conclusions

We developed a reservoir inflow forecasting method that uses interval type-2 fuzzy logic for rainfall scenario classification. The classified rainfall scenarios—distinguishing between heavy rainfall (strong rainfall) and light/no rainfall (non-extreme conditions)—were then fed into a fuzzy neural network for inflow forecasting. The proposed hybrid model is suitable for island-type climates with frequent and intense rainfall events, such as Taiwan, and is capable of providing hourly reservoir inflow forecasts for the next 48 h.