Short-Term Optimal Operation Method for Hydro–Wind–Thermal Systems Considering Wind Power Uncertainty

Abstract

Wind curtailment, caused by wind power uncertainty, has become a prominent issue with the large-scale grid connection of wind power. To fully account for the uncertainty of wind power output, a short-term hydro-wind-thermal operation method based on a wind power confidence interval is proposed. By utilizing the flexible start-stop and efficient ramp-up of cascade hydropower plants to smooth fluctuations in wind power output, a multi-objective optimal scheduling model that minimizes the cost of power generation and maximizes the consumption of clean energy is constructed. To reduce the solution’s complexity, we chunk the model according to the energy type using a hierarchical solution. The overall solution framework, which integrates a nonparametric method, a heuristic algorithm, and an improved particle swarm algorithm, is constructed to solve the model rapidly. The simulation results of a regional power grid show that the proposed method can attain an efficient solution in 83.5 seconds. Furthermore, the proposed method achieves an additional 455,600 kWh of hydropower and a reduction of ¥233,300 in the cost of coal consumption. These findings suggest that the proposed method is a good reference for the short-term operation of a hydro-wind-thermal combination in large-scale wind power access areas.

1. Introduction

With the continuous depletion of fossil energy sources and environmental pollution, renewable energy, such as wind, nuclear, and solar energy [1,2,3], has been vigorously developed to alleviate these problems. Wind power has developed rapidly because of its low generation cost and wide distribution. Africa, with a weak development base, is forming a new African energy pattern based on clean energy due to the wide distribution of hydropower resources, wind energy resources, photovoltaic resources, and the abundance of technological development to solve the dilemma of traditional energy shortage and achieve sustainable development in Africa. In 2023, 117 GW of new wind power capacity was installed globally, reaching an all-time high. However, randomness, uncertainty, and intermittency render wind energy challenging to consume in a power system in the same way as conventional energy sources [4,5,6,7]. As a result, wind curtailment occurs frequently as power systems struggle to consume the high proportion of wind power being integrated into the grid [8,9]. Establishing a multi-energy complementary system is an important approach to increasing the flexibility of energy systems, thereby reducing wind curtailment [10,11].

Multi-energy complementary systems leverage the complementary nature of wind power and conventional energy sources, such as pumped storage [12,13,14], hydropower [15,16], thermal power [17,18,19], and energy storage [20,21,22], to mitigate the impact of wind power uncertainty on the grid’s operational safety and to improve the grid’s ability to absorb wind power. Some scholars focus on the power supply characteristics of wind power, regard uncertain power sources such as wind power as uncertain variables, and establish a multi-power source combined optimal scheduling model to achieve the purpose of reducing the peak-valley difference and smoothing the fluctuation of wind power to mitigate the impact of large-scale wind power access on the safe and stable operation of the power system [23]. The literature [24,25] established an interval optimal scheduling model for the coordination of wind power and thermal power, which does not need the uncertain distribution information of wind power and only needs the upper and lower limits of wind power output. While considering the uncertainty of wind power, the literature [26] used probability density functions (PDFs) to represent the wind speed and applied Gaussian, Weibull, and log-normal PDFs to generate realistic probability distributions for wind power outputs. The literature [27] used Copula to represent the uncertainty of wind and photovoltaic power and simulate typical operating scenarios through the Monte Carlo method to obtain samples. The solution is simple, but using thermal power unit output to regulate the fluctuation of wind power sacrifices the lifespan of the thermal power unit and the operating efficiency. The literature [28] used the Weibull probability density function (PDF) to establish the wind speed model and an optimal scheduling model for wind energy storage. However, energy storage is relatively expensive. The literature [29] expresses the possible wind power output using an uncertainty set, and the optimization objective is a min-max-min function. However, this study does not consider the effect of the water head change on the generation power. Studies of multi-energy complementary scheduling that consider the uncertainty of wind power have achieved good results, but the inherent stochasticity of wind power cannot be eliminated as when more wind is integrated beyond the hydro-regulating ability, thermal units are forced to regulate frequently, which affects their efficiency and service life and seriously affects the stable operation of the power system [30]. Considering the ongoing commissioning of cascade hydropower plant clusters and the large-scale integration of wind power, which leads to the mismatches between supply and demand and wind curtailment in many regions such as Northwest China, Europe, and Africa, the complementary scheduling of cascade hydropower plants and wind power is necessary [31,32]. In the literature [33], an output interval with binary variables was introduced to describe the wind power uncertainty. Pumped storage is introduced to cope with wind fluctuations. The model achieves relatively good results, but not all systems have pumped storage equipment. The literature [34] adopted the Monte Carlo method to simulate the wind speed prediction error, and the literature [35] adopted the Latin Hypercube Sampling method to simulate the wind power output error; the literature [36] used a multivariate Gaussian kernel density function to fit constant forecast error and the estimated density function to generate one-day wind power deviation scenarios to describe the wind power uncertainty and scheduled the ramping flexibility of the hydrothermal system to match it. However, the scenario method requires generating many scenarios, which increases the model-solving time. With the complementary operation of large-scale wind power and cascade hydropower, while thermal power is still the dominant energy source of the power system in most of China, improving the operational efficiency of the multi-energy power system, reducing the cost of power generation, and extending the lifetime of thermal power units while meeting the conditions for safe and stable operation of the power system are issues that are difficult to solve at present.

Therefore, based on the actual demands of power grid short-term operation, this paper establishes a hydro-wind-thermal coordination optimization model considering the confidence interval of wind power, which makes full use of the complementary characteristics of power sources and the rapid response, flexible start-up, and shut-down characteristics and storage capacity of cascade hydropower stations. The confidence interval is introduced to describe the wind power output process, thus addressing the impact of wind power output errors on the power system balance. Moreover, fluctuations in thermal power output and generation cost are reduced by the peak shaving and storage capacity of the cascade hydropower plants, which also increases the service life of thermal power units. The overall solution framework combines the nonparametric method, heuristic algorithm, and Improved Particle Swarm Optimization (IPSO) algorithm to achieve efficient coordinated hierarchical optimization of multiple energy sources.

• The main objectives of this study are as follows: To make full use of the complementary characteristics of the hydro-wind-thermal system in short-term optimal operation in order to achieve more hydropower generation and reduce the cost of thermal power generation.
• To fully take into account the uncertainty of wind power, to express the wind power output with a confidence interval, and to reduce the wind curtailment through the regulating ability of the cascade hydropower plants.
• To improve the efficiency and results of the short-term optimal operation model using the proposed method.

2. Methodology

The complementary hydro-wind-thermal power scheduling is a highly complex mixed integer nonlinear programming problem [37,38,39], which is difficult to solve when the system scale is large. In this paper, the concept of hierarchical solving is adopted to decompose the larger-scale problem into several smaller-scale problems to reduce the complexity [40]. The model is decomposed into three scheduling layers: hydropower, wind power, and thermal power. The optimal scheduling strategies in different layers are developed according to the characteristics of the different power sources and the grid’s demand.

The overall solution idea is shown in Figure 1. First, the confidence interval method is used to describe the fluctuations in wind power. Then, according to the wind power output in the confidence interval, the output process of the cascade hydropower stations is optimized. By tapping the sufficient flexibility of hydropower, the fluctuation of wind power is dealt with to ensure that the wind power in the confidence interval is consumed while providing more peak shaving power so that the net load process left for thermal power is as smooth as possible. Finally, the thermal power is optimized. In the solution process, hydropower is solved by the IPSO algorithm, and thermal power is solved by a heuristic algorithm to determine the optimal output under the unit combination. Wind power is described by a confidence interval and adjusted by hydropower and thermal power so that the wind power in the confidence interval is fully consumed. The optimization calculation of hydropower and thermal power is considered in the subsequent section.

2.1. Objective Function

The overall objective of the hydro-wind-thermal optimal scheduling model is to reduce the cost of power generation and improve the level of renewable energy utilization to guarantee the power supply. The objective function is as follows:

$$\left\{\begin{array}{l}{f}_1=\min (F^{th}+F^{hp}+F^{wp})\\ f_2=\min {\displaystyle \sum _{t=1}^T(\Delta p_t^{wp}+\Delta P_t^{hp})\times }T\end{array}\right.$$

(1)

where $f_1$ represents the total cost of power generation, $f_2$ represents the sum of wind and water abandonment, $F^{th}$, $F^{hp}$, and $F^{wp}$ represent the costs of coal, hydropower, and wind power generation, respectively, and $\Delta p_t^{wp}$ and $\Delta P_t^{hp}$ represent total wind power and hydropower abandonment, respectively.

The variable costs of wind farms and hydropower plants are ignored here because they are minimal after construction. The model focuses on optimizing the output of the cascade hydropower plants under the condition of guaranteeing the full consumption of wind power within the confidence interval and seeks to reduce the thermal power generation under the condition of minimum water abandonment to improve the overall benefits of the system. The objective function above can be transformed into:

$$\left\{\begin{array}{l}{F}^{th}=\min {\displaystyle \sum _{t=1}^{T=96}{\displaystyle \sum _{h=1}^{N^{th}}{\displaystyle \sum _{j=1}^{N_h^{th}}({a_{h,j}{P}_{h,j,t}^{th}}^2+b_{h,j}{P}_{h,j,t}^{th}+c_{j,i})}}}\\ P^{hp}=\max {\displaystyle \sum _{t=1}^T{\displaystyle \sum _{m=1}^{N^{hp}}{\displaystyle \sum _{i=1}^{N_m^{hp}}{P}_{m,i,t}^{hp}}}}\end{array}\right.$$

(2)

where $a_{h,j}$, $b_{h,j}$, and $c_{h,j}$ are the coal combustion coefficients of the thermal power station $h$ unit $j$, $P_{h,j,t}^{th}$ is the output of the thermal power station $h$ unit $j$ at time period $t$, $N^{th}$ is the total number of thermal power stations in operation, $N_h^{th}$ is the number of thermal power station units $h$, $N^{hp}$ is the number of hydropower stations, $m$ is the number of hydropower stations, $N_m^{hp}$ is the number of units in a hydropower station $m$,$P^{hp}$ is the total output of the cascade hydropower station in the scheduling period (which should be as large as possible), $t$ is the time period number, $t=1,2,\mathrm{\dots },T$ ($T$ is the number of total time periods), and $P_{m,i,t}^{hp}$ is the output of the hydropower station $m$ at time period $t$.

2.2. Constraints

2.2.1. Power System Balance Constraints

Ensuring the balance between supply and demand in the power system, i.e., ensuring that the total output of hydropower, wind power, and thermal power is equal to the load demand of the grid.

$${\displaystyle \sum _{m=1}^{N^{hp}}{\displaystyle \sum _{i=1}^{N_m^{hp}}{P}_{m,i,t}^{hp}}}+{\displaystyle \sum _{h=1}^{N^{th}}{\displaystyle \sum _{j=1}^{N_h^{th}}{P}_{h,j,t}^{th}}}+p_t^{wp}-P_t^{ld}=0$$

(3)

where $P_t^{ld}$ is the system forecast load demand at the time period $t$, and ${\displaystyle \sum _{m=1}^{N^{hp}}{\displaystyle \sum _{i=1}^{N_m^{hp}}{P}_{m,i,t}^{hp}}}$,${\displaystyle \sum _{h=1}^{N^{th}}{\displaystyle \sum _{j=1}^{N_h^{th}}{P}_{h,j,t}^{th}}}$, and $p_t^{wp}$ are the predicted outputs of hydropower, thermal power, and wind power, respectively.

2.2.2. Cascade Hydropower Plant Constraints

• Hydraulic constraints

There are close hydraulic connections among the cascade hydropower stations, and the inflow of the downstream stations is affected by the outflow of the upstream stations as well as the interval flow. The outflows are also related to navigation and ecological water use, so the hydraulic constraints of the cascade hydropower stations are as follows.

$$\left\{\begin{array}{l}{V}_{m,t+1}^{hp}=V_{m,t}^{hp}+(Q_{m+1,t-\tau }^{out}+Q_{m,t}^{int}-Q_{m,t}^{out})\Delta t\\ \underset{¯}{Q_{m,t}^{out}}\le Q_{m,t}^{out}\le \overline{Q_{m,t}^{out}}\\ \underset{¯}{Z_{m,t}}\le Z_{m,t}\le \overline{Z_{m,t}}\end{array}\right.$$

(4)

where $\Delta t$ is the length of the time period, $V_{m,t}^{hp}$ is the reservoir capacity of hydropower station $m$ at the beginning of time period $t$, $Q_{m+1,t-\tau }^{hp}$ is the outflow from the upstream power station of hydropower station $m$ at time period $t-\tau$ ($\tau$ is the lag time of the cascade hydropower station), $Q_{m,t}^{int}$ and $Q_{m,t}^{out}$ are the interval flows and outflows of hydropower station $m$ at time period $t$, $\overline{Q_{m,t}^{out}}$ and $\underset{¯}{Q_{m,t}^{out}}$ are the upper and lower limits of the outflow from hydropower station $m$, and $\overline{Z_{m,t}}$ and $\underset{¯}{Z_{m,t}}$ are the upper and lower limits of the water level of hydropower station $m$, respectively.

2.

Operational constraints

The operational constraints of hydropower stations include power generation flow constraints, maximum output constraints, and the unit’s minimum start-stop constraint.

$$\left\{\begin{array}{l}{q}_{m,i}^{\min }\le q_{m,i}\le q_{m,i}^{\max }\\ \max \left\{P_{m,i,t}^{hp}\right\}\le P_{m,i,\max }^{hp}\\ (B_{m,i,t}^{hp}-B_{m,i,t-1}^{hp})(G_{ON,m,i,t-1}^{hp}-G_{ON,m,i}^{hp})\ge 0\\ (B_{m,i,t}^{hp}-B_{m,i,t-1}^{hp})(G_{OF,m,i,t-1}^{hp}-G_{OF,m,i}^{hp})\ge 0\end{array}\right.$$

(5)

where $q_{m,i}^{\max }$ and $q_{m,i}^{\min }$ represent the maximum and minimum outflow capacities, respectively, of the hydropower station $m$ unit $i$, $B_{m,i,t}^{hp}$ is a 0–1 variable, which indicates the operation state of the hydropower station $m$ unit $i$ during the period $t$. The start-up is 0, and the shutdown is 1. $G_{ON,m,i,t}^{hp}$ and $G_{OF,m,i,t}^{hp}$ indicate the start-up and shutdown durations, respectively, of the power station $m$ unit $i$ from the last unit operational status change to time period $t$, and $G_{OF,m,i}^{hp}$ and $G_{ON,m,i}^{hp}$ represent the minimum start-up and shutdown times, respectively, of the hydropower station $m$ unit $i$.

2.2.3. Thermal Unit Constraints

• Output constraint

$${\displaystyle \sum _{h=1}^{N^{th}}{\displaystyle \sum _{j=1}^{N_h^{th}}{B}_{h,j,t}^{th}{P}_{h,j,\min }^{th}}}\le {\displaystyle \sum _{h=1}^{N^{th}}{\displaystyle \sum _{j=1}^{N_h^{th}}{P}_{h,j,t}^{th}}}\le {\displaystyle \sum _{h=1}^{N^{th}}{\displaystyle \sum _{j=1}^{N_h^{th}}{B}_{h,j,t}^{th}{P}_{h,j,\max }^{th}}}$$

(6)

where the 0–1 variable $B_{h,j,t}^{th}$ represents the start-stop change in the thermal power plant $h$ unit $j$ during a period of time $t$, when the start-stop state changes, it is 1; otherwise, it is 0. $P_{h,j,\min }^{th}$ and $P_{h,j,\max }^{th}$ indicate the minimum and maximum outputs of the thermal power plant $h$ unit $j$.

2.

Start-stop constraint

Currently, large thermal power units, which dominate the power system, usually take one to two days to start up from a cold state to grid integration, so this paper does not consider intraday startup and shutdown of thermal power units in dispatch. This means that once the power generation status of a thermal power unit has been determined, it will not change within the day.

$$B_{h,j,0}^{th}=B_{h,j,1}^{th}=,\mathrm{\dots },=B_{h,j,t}^{th}=,\mathrm{\dots },=B_{h,j,T}^{th}$$

(7)

3.

Climbing constraints

$$\left\{\begin{array}{l}{p}_{h,j,t+1}^{th}-p_{h,j,t}^{th}\le \overline{{\Delta }_{h,j}}\\ p_{h,j,t-1}^{th}-p_{h,j,t}^{th}\le \underset{¯}{{\Delta }_{h,j}}\end{array}\right.$$

(8)

where $\overline{{\Delta }_{h,j}}$ and $\underset{¯}{{\Delta }_{h,j}}$ denote the maximum upward and downward climbing speeds, respectively.

2.2.4. System Backup Constraints

The system backup capacity requirement is determined here by the wind power output and system load forecast error. Assuming that the wind power output forecast error interval is $[\overline{p_t^{pred}},\underset{¯}{p_t^{pred}}]$ and that the system load forecast error is $[\underset{¯}{{\xi }_t},\overline{{\xi }_t}]$, the system backup capacity demand is as follows:

$$\left\{\begin{array}{c}{R}_t^{up}=\max \{0,\overline{{\xi }_t}-\underset{¯}{p_t^{pred}}\}\\ R_t^{down}=\max \{0,\overline{p_t^{pred}}-\underset{¯}{{\xi }_t}\}\end{array}\right.$$

(9)

where $R_t^{up}$ and $R_t^{down}$ represent the positive and negative spinning reserves at the time $t$, respectively.

2.3. Improved PSO Algorithm for the Hydropower Scheduling Layer

The PSO algorithm is often used to solve the scheduling problem of cascade hydropower stations [41,42,43]. The process of applying the PSO algorithm to solve the power distribution of plants is as follows: the maximum output at the end of the scheduling period is taken as the fitness function, and the output at each time period of cascade hydropower stations is taken as the decision variable. The flight trajectory of a single particle is the output change value of each period, and the position of the particle represents the output of each period. The optimization iteration process is as follows:

$$v_i(k+1)=\eta v_i(k)+c_1{\varphi }_1(p_i(k)-y_i(k))+c_2{\varphi }_2(p_g(k)-y_i(k))$$

(10)

$$y_i(k+1)=y_i(k)+v_i(k+1)$$

(11)

where $i$ is the particle number, $k$ is the number of iterations, $v_i(k)$ is the velocity of the particle $i$ after $k$ iterations, $y_i(k)$ is the position of the particle $i$ after $k$ iterations, $\eta$ denotes the inertial coefficient, $c_1$ and $c_2$ are positive adjustable parameters, $p_i(k)$ represents the individual extreme value of particle $i$, ${\varphi }_1$ and ${\varphi }_2$ are random numbers between [0, 1], and $p_g(k)$ denotes the global extremum after $k$ iterations.

However, the PSO algorithm is subject to deficiencies when subjected to the initial solution, inertia weights, and elitist strategy, including premature maturity, slow convergence speed, and a proclivity to fall into a locally optimal solution. In this paper, the PSO algorithm is improved in the initial population, inertia weight, and stock quality to obtain higher solving efficiency and better-solving results.

2.3.1. Tent Mapping Improves the Quality of the Initial Population

Chaos phenomena are common in nonlinear systems, which can traverse all possible states in a specific interval. The chaotic idea is used to initialize the initial population of the PSO, which can effectively improve the diversity of the initial population and improve the search efficiency of the algorithm. The tent map, a common method in the chaotic optimization algorithm, has the advantages of fast pseudo-random speed and uniform sequence distribution [44,45,46]. In this paper, the tent map is used to initialize the population.

$$x_{n+1}=\left\{\begin{array}{cc}{\displaystyle \frac{x_n}{a}}\hfill & x_n\in [0,a)\\ {\displaystyle \frac{(1-x_n)}{(1-a)}}\hfill & x_n\in [a,1]\end{array}\right.$$

(12)

where $x_n$ denotes the sequence generated by iteration $n$ and where $a$ is the forward adjustable parameter. A tent mapping is a chaotic mapping within the range allowed by the parameters ($a\ne 1/2$ and $a\ne x_1$). After the chaotic sequence is generated, the chaotic variables need to be mapped into the original feasible region according to $y_n(1)=\underset{¯}{y}+x_n(\overline{y}-\underset{¯}{y})$ ($\overline{y}$ and $\underset{¯}{y}$ denote the upper and lower limits of the original optimization variables, respectively, and $y_n(1)$ denotes the initial position of the particle $n$).

2.3.2. Improved Inertia Weight $\eta$

During the computation, as the number of iterations increases, decreasing $\eta$ helps to improve the convergence of the algorithm by reducing the speed of the particles, so linearizing the reduction of $\eta$ has a better optimization performance [47,48,49,50]. However, $\eta$ cannot linearly reflect the nonlinear characteristics of the search process, and the ideal optimization effect cannot be obtained [51]. In this paper, a perturbation factor $\gamma$ is randomly added to the linearized $\eta$ in the calculation process to prevent the PSO algorithm from falling into local convergence.

$$\left\{\begin{array}{c}{\eta }_{i,n+1}={\eta }_{\max }-d\times n+\gamma \\ \gamma =0.1\cos {\displaystyle \frac{\pi n}{2}}\end{array}\right.$$

(13)

where $d$ is the slope of the decrease in $\eta$.

2.3.3. Improved Stock Quality

To increase the number of elite particles and eliminate the particles with lower fitness, we introduce the idea of a crossover in the search process such that in each iteration, two particles with poor fitness are eliminated. To ensure the consistency of the number of particles, we randomly select two particles with the smallest fitness among the remaining particles from the five particles with higher fitness as new particles. Roulette is used to randomly select the crossover position, and the crossover method is a single-point crossover. The cross diagram is illustrated in Figure 2, where the color depth represents the difference of particles, the dotted frame represents the crossing point, and the arrow represents the crossing process:

2.4. Wind Power Operation Layer Considering Confidence Intervals

Wind power output has strong uncertainty. After large-scale wind power is connected to the grid, the reliability of the power system is severely affected. Wind power prediction intervals can describe the uncertainty in wind power prediction [52,53,54]. In this work, according to the comparison between the actual value and the predicted value of the wind power output, the distribution function of the wind power output prediction error is established. The nonparametric method is used to calculate the confidence interval of the wind power output at a given confidence level, and the detailed steps are described in the literature [55]. An appropriate confidence interval is critical to the reliability and economy of the complementary operation of wind power and hydropower. A high confidence level indicates a low economy and high system stability, whereas a low confidence level indicates a high economy and low system stability.

2.5. Thermal Power Scheduling Layer Based on a Heuristic Algorithm

The heuristic algorithm is used to search for the optimal start-up mode of the thermal power units. The specific steps are as follows:

• Calculate the coal consumption cost of thermal power units, prioritize the start-up of units with low coal consumption cost per unit of electricity, and establish a sequence table for the start-up of units in different time periods. The cost of coal consumption per unit of electricity is as follows:

$${\beta }_{h,j,t}=a_{h,j}{P}_{h,j}^{\ast }+b_{h,j}+{\displaystyle \frac{P_{h,j}^{\ast }}{c_{h,j}}}$$

(14)

$P_{h,j}^{\ast }$ can be calculated according to Equation (15):

$$P_{h,j}^{\ast }=\left\{\begin{array}{cc}{P}_{h,j}^{\min }\hfill & \sqrt{{\displaystyle \frac{a_{h,j}}{c_{h,j}}}}<P_{h,j,\min }^{th}\hfill \\ \sqrt{{\displaystyle \frac{a_{h,j}}{c_{h,j}}}}\hfill & P_{h,j,\min }^{th}\le \sqrt{{\displaystyle \frac{a_{h,j}}{c_{h,j}}}}\le P_{h,j,\max }^{th}\hfill \\ P_{h,j,\max }^{th}\hfill & \sqrt{{\displaystyle \frac{a_{h,j}}{c_{h,j}}}}>P_{h,j,\max }^{th}\hfill \end{array}\right.$$

(15)

where ${\beta }_{h,j,t}$ is the cost of coal consumption per unit of electricity in thermal power plant $h$ unit $j$, $P_{h,j,\min }^{th}$ and $P_{h,j,\max }^{th}$ denote the minimum and maximum outputs of the thermal power unit, respectively, and $t$ is the operating time of the unit.

2.

Based on the setting of the operating time of the units, exclude the units in the start-up sequence table that are under maintenance, are still shut down, or have not met the start-up time requirement.

3.

According to the start-up sequence table, select the unit with the lowest coal consumption cost per unit of electricity and add up the maximum output of each unit one by one until the maximum net load demand is met.

4.

Sum the minimum output of each unit in step 3 to verify whether it meets the minimum load constraint of the grid. Otherwise, additional thermal power units are turned on according to the start-up sequence table and reverify if step 3 is valid until the constraint is met.

5.

Calculate whether the thermal power plant climbing constraint is satisfied in each time period. Otherwise, reselect thermal power units to perform 1–4 until the requirement is satisfied.

2.6. Overall Solution Process

The overall solution of this study is presented in Figure 3, and its detailed process is outlined below:

• For the wind power operation layer, according to the wind power output prediction results and the wind power output prediction error distribution function, the nonparametric method is used to calculate the confidence interval of the wind power output under the given confidence level $\alpha$, and the obtained confidence interval is set as $[\underset{¯}{p_{t,\alpha }^{pred}},\overline{p_{t,\alpha }^{pred}}]$.
• Calculate the minimum output required to smooth out wind power fluctuations based on the wind power output confidence interval $p_t^{py}=\overline{p_{t,\alpha }^{pred}}-\underset{¯}{p_{t,\alpha }^{pred}}$.
• The minimum output required to smooth out wind power fluctuations is allocated to the hydro and thermal scheduling layers.
• Calculate whether the output of cascade hydropower stations meets Formula (16) under the premise of water level constraints and flow constraints.

  $${\displaystyle \sum _{m=1}^{N^{hp}}{P}_{m,t}^{hp}\ge p_t^{py}}$$

  (16)
• If Formula (16) is satisfied, hydropower output can stabilize wind power fluctuations. During the peak load period, hydropower should be increased as much as possible to reduce the cost of thermal power generation.
• If Equation (16) is not satisfied, hydropower cannot completely smooth out wind power fluctuations, and to ensure that wind power is consumed within the confidence interval, the thermal power units need to assume an output of ${\zeta }_t$ under the premise of satisfying various constraints. Under the condition of ensuring that ${\zeta }_t$ is as smooth as possible, verify that the following Equation is valid:

  $${\displaystyle \sum _{m=1}^{N^{hp}}{P}_{m,t}^{hp}+{\zeta }_t\ge p_t^{py}}$$

  (17)

(a)

If it is valid, then the load of the cascade hydropower plant is allocated according to Section 2.3, and the thermal units are scheduled according to Section 2.5.

(b)

If it is not valid, then increase ${\zeta }_t$ in smaller steps $\Delta \epsilon$ until Equation (17) is satisfied under the premise of satisfying the various constraints. The load of the cascade hydropower plants is allocated according to Section 2.3, and the thermal units are scheduled according to Section 2.5.

3. Case Analysis

3.1. Engineering Context

The effectiveness of the proposed method is demonstrated by using the example of a typical day in winter for a power grid in a certain region of China. In this region, wind power output fluctuates greatly in winter, and hydropower output is small, so the results of scheduling on a typical winter day can better reflect the impact of wind power uncertainty on the power system. The power sources in this region mainly include hydropower, wind power, thermal power, and nuclear power. Since nuclear power is not involved in regulation, the calculation directly subtracts the residual load of nuclear power from the total load as the total load during the scheduling period. The example includes a group of wind farms with an installed capacity of 2238 MW, a second-stage hydropower plant with an installed capacity of 2520 MW 17 km away from each other, and a group of thermal power plants with a total installed capacity of 13,500 MW in 29 units, the scheduling period is one day, with a time interval of 15 minutes.

The actual system load and wind power output data during the scheduling period are shown in Figure 4, which reveals that the wind power output has an “anti-peaking” effect. The wind power output has the characteristic of “anti-peaking”, which increases the difficulty of scheduling, and Figure 5 shows the process of wind power output with a confidence interval of 90%.

3.2. Parameters of Calculation

The parameters of the cascade hydropower plants are shown in

Table 1. Operation parameters of cascade hydropower stations.

Condition Settings	Water Level at the Beginning of the Day (m)	Interval Flow (m3/s)	Installed Capacity (MW)	Hydraulic Relationship	Reservoir Type
Hydropower station 1	773.06	310	1200	upstream	Annual regulation
Hydropower station 2	641.13	0	1320	downstream	Runoff type

. Power Station 1 and Power Station 2 are relatively close, so the interval flow of Power Station 2 is set to 0. The maximum number of iterations of IPSO is 100, the number of particles is 50, $c_1=c_2=2.05$, and $\eta$ = 0.05. All the simulation programs are compiled in the Java language, and the running environment is a DELL computer with 4 cores of CPUs, a main frequency of 2.8 GHz, a RAM of 8 GB, and a hard disk of 1 TB.

3.3. Analysis of the Calculation Results

3.3.1. Analysis of Different Solution Methods

In order to ascertain the benefits of the IPSO algorithm, the data from typical daily cascade hydropower stations in the dry season are employed to evaluate the calculation results of the standard PSO algorithm in comparison with the IPSO algorithm. The initial population size in the particle swarm algorithm is set to 50, with $c_1=c_2=2.05$.

The convergence curves of the PSO algorithm and IPSO algorithm are shown in Figure 6. With the PSO algorithm, the cascade hydropower stations fall into the local optimum after 25 iterations, and the water consumption of the cascade hydropower stations is 3.598 × 10⁷ m³. With the IPSO algorithm, the water consumption of the cascade hydropower stations after 22 iterations is 3.589 × 10⁷ m³, and the water consumption of the cascade hydropower stations is reduced by 9 × 10⁴ m³. The improved algorithm has a faster convergence speed and a more optimal result compared to the conventional method.

3.3.2. Analysis of the Model Calculation Results

In this work, 20 simulations were carried out, and the calculation results were relatively stable, with a difference of 0.53% between the optimal and worst values of the thermal power generation cost. The longest calculation time was 83.5 seconds, which meets the timeliness requirements of short-term optimal dispatching. A comparison between the actual scheduling process (before optimization) and the optimal result (after optimization) is shown in

Table 2. Power generation results before and after optimization (10 MWh).

Power Generation	Hydropower Station 1	Hydropower Station 2	Cascade Hydropower Stations	Wind Power	Thermal Power
After optimization	1105.54	1497.71	2603.25	2135.51	11,100.71
Before optimization	1105.53	1452.16	2557.69	2135.51	11,145.95

. It shows that the full consumption of wind power within the confidence interval is achieved before and after optimization, but the hydropower increases by 455.6 MWh after optimization, which results in more generation of clean energy and less generation of thermal power.

Before and after optimization, the output of the cascade hydropower station is shown in Figure 7. Before and after the optimization of the cascade hydropower station, the hydropower output of each time period can provide enough “flexibility” to smooth the fluctuation of wind power, which means that the power output of the cascade hydropower station in any time period meets the requirements of $P_t^{hp}\ge p_t^{py}$; however, the hydropower output is too large in the early scheduling period, less water is available for scheduling at the later period of time for the cascade hydropower stations, the peaking capacity of the cascade hydropower stations is insufficient, and more pressure is exerted on thermal power peaking. After optimization, the hydropower output smooths out wind power fluctuation and has a peaking effect on the load by optimizing the water allocation, thus smoothing out the thermal power output and increasing the hydropower generation by 455.6 thousand kWh.

The start-stop sequence of the thermal power units determined by the heuristic algorithm is shown in

Table 3. List of an economic sequence of thermal power units determined by a heuristic algorithm.

Thermal Unit Number	Installed Capacity (MW)	Minimum Output (MW)	Climbing Ability (MW/h)	Unit Quantity	a 1 (10−3 ¥/MW)	b 1 (10−3 ¥/MW)	c 1 (¥)	β 2 (¥∙MW−1)
1	1000	500	400	2	8.91	171.94	14,698.53	195.56
2	660	330	260	4	12.58	185.73	10,074.75	209.27
3	600	300	240	3	14.28	190.11	9268.77	214.15
4	350	170	140	2	32.10	199.94	6172.11	228.79
5	300	160	320	6	23.83	206.30	5705.49	232.46
6	215	100	80	4	48.29	229.99	4602.57	247.66
7	150	50	60	6	65.82	236.56	3315.83	268.52

¹ Coal consumption cost coefficient per unit of electricity. ² The cost of coal consumption per unit of electricity.

. A comparison of the results before and after thermal power output optimization is shown in

Table 4. Comparison of thermal power output before and after optimization.

Project	Peak-to-Valley Difference (MW)	Standard Deviation (MW)	Unit Start-Up Number	Coal Consumption Cost (104 ¥)
Before optimization	3108.27	1017.18	11	2284.25
After optimization	2433.20	800.27	8	2260.92

. Before optimization, the peak-to-valley difference is 3,108.27 MW, the standard deviation is 1,017.18 MW, the cost of coal consumption is ¥228,410, and the number of operating units is 11. After optimization, due to the greater peak shaving output of hydropower, the peak-to-valley difference in the thermal power decreases by 675.07 MW, the standard deviation decreases by 216.91 MW, and the optimized output of thermal power units is smoother. Due to the reduction in the peak-to-valley difference in thermal power, the number of thermal power units can be reduced by 3, and the cost of coal consumption can be reduced by ¥233,300. A comparison of the output process before and after optimization is shown in Figure 8.

Figure 9 shows the power generation of each power source before and after optimization. Thermal power accounts for a relatively large proportion of the power system, and wind power and hydropower account for a relatively small proportion. After optimization, the flexibility of hydropower is fully utilized. While smoothing out wind power fluctuations, hydropower reduces the peak-to-valley difference in thermal loads by increasing output during peak load periods (40–52, 64–88) and decreasing output during valley load periods, as well as reducing the cost of thermal power generation. The short-term complementary optimal operation of hydropower, wind power, and thermal power can fully exploit the flexibility of the cascade hydropower plant and reduce the impact of random fluctuations of wind power on the power grid. This strategy achieves the purpose of generating more wind power and hydropower with less thermal power and obtains better optimal scheduling results.

4. Conclusions

This study proposed an optimization model with a hierarchical solving method to improve the solving efficiency and runtime effectiveness of a hydro-wind-thermal power system. First, fluctuations in wind power are described by a confidence interval. Second, the output process of cascade hydropower stations is optimized using the IPSO algorithm according to the wind power output within the confidence interval. Finally, the optimal thermal power output under the unit combination is solved by a heuristic algorithm. Simulations were conducted on data from power grids in a certain region of China during a typical day in winter. The main findings are as follows.

• The proposed optimal scheduling method fully considers the impact of wind power uncertainty, describes wind power output through confidence intervals, smooths thermal power fluctuations by using cascade hydropower output, and optimizes the thermal power unit output process. Due to the greater peak shaving output of hydropower, the peak-to-valley difference in thermal power decreases by 675.07 MW, the standard deviation decreases by 216.91 MW, and the optimized output of thermal power units is smoother. Under the premise of safe operation of the power system and maximum consumption of clean energy, the proposed method achieves an additional 455,600 kWh of hydropower and a reduction of ¥2.333 × 10⁵ in the cost of coal consumption. Ultimately, more wind power and hydropower are generated, less thermal power is generated, and the impact of wind power fluctuations on the power grid is reduced.
• Aiming at the slow solving speed and poor timeliness of the conventional hybrid energy system optimal scheduling model, a hierarchical solving strategy is proposed, which decomposes the complex system and combines different solving methods at each layer, reducing the problem complexity and improving the solving efficiency at the same time. The simulation results show that the proposed method can attain an efficient solution in 83.5 seconds.
• Optimizing the scheduling of multiple power sources on the power supply side is an effective means of improving the consumption of intermittent energy sources such as wind power, and the proposed hydro-wind-thermal complementary operation strategy can provide a reference for grid power generation scheduling after large-scale wind power is connected to the grid with cascade hydropower plants.
• The method proposed in this study is applicable not only to China but also to regions rich in hydropower and wind power resources, such as Africa. It also provides a reference for how to ensure the safe and efficient operation of the power system after renewable energy, such as photovoltaic resources, is massively connected to the power grid.
• However, in areas that are rich in wind power and other clean energies but not rich in hydropower, the application may be less effective. Meanwhile, the proposed method does not take the meteorological factors into account. Therefore, multi-energy complementary scheduling and meteorological factors will be considered in future research.