Optimal Coordinated Operation for Hydro–Wind Power System

Abstract

The intermittent and stochastic characteristics of wind power pose a higher demand on the complementarity of hydropower. Studying the optimal coordinated operation of hydro–wind power systems has become an extremely effective way to create safe and efficient systems. This paper aims to study the optimal coordinated operation of a hybrid power system based on a newly established Simulink model. The analysis of the optimal coordinated operation undergoes two simulation steps, including the optimization of the complementary mode and the optimization of capacity allocation. The method of multiple complementary indicators is adopted to enable the optimization analysis. The results from the complementary analysis show that the hydraulic tracing effect obviously mitigates operational risks and reduces power losses under adverse wind speeds. The results from the analysis of capacity allocation also show that the marginal permeation of installed wind capacity will not exceed 250 MW for a 100 MW hydropower plant under random wind speeds. These simulation results are obtained based on the consideration of some real application scenarios, which help power plants to make the optimal operation plan with a high efficiency of wind energy and high hydro flexibility.

1. Introduction

The undesired environmental impact of combustion-based electricity provides an important opportunity for reforming the energy structure [1,2]. Renewable energy sources (RESs) are progressively occupying a larger share of the energy mix [3,4]. Among the abundant RESs, the power potential of wind energy exceeds all other electricity consumptions in the world [5]. For instance, wind energy contributes up to 50% of the installed renewable capacity in Australia [6] and accounts for 40% of the domestic electricity generation in Germany [7]. On the other hand, hydropower is one of the most stable RESs, making it an ideal counterpart to regulate the power fluctuations in integrated power systems [8]. Hydropower and wind power generation have great development potential, which will maintain an upward tendency to 2030 (as shown in Figure 1) due to their huge benefits for the environmentally friendly and sustainable electricity industry [9]. Due to the intermittent and stochastic characteristics of wind power, the optimal coordinated operation of hydro–wind-based power systems becomes more important for the efficient use of energy.

In examining the literature, there are three ways to achieve the optimal coordinated operation of hydro–wind-based power systems. The first is establishing an optimization model to optimize the operating coefficients, with the optimized results revealed by the model outputs or transformed forms of these model outputs. In other words, the established model is able to simulate the dynamic operation characteristics of a power system over time. The key to this is to establish an appropriate optimization model, as such a model can reflect the coupling characteristics of each component or subsystem. For instance, Zhang et al., (2019) [10] proposed a hydro–wind–photovoltaics optimization model to calculate the optimal coefficients of coordinated operations in different seasons. Ju et al., (2022) [11] proposed a fully distributed power flow calculation method for the coordinated operation of hybrid microgrids. Simiao and Ramos (2020) [12] presented a hydro–wind–photovoltaics optimization model to optimize the cost and reliability indices. Mou et al., (2023) [13] constructed a fault diagnosis model for intelligent dispatching energy storage systems based on a deep learning algorithm. Alvarez and Ronnberg (2018) [14] established a reservoir-type hydropower equivalent model for hydro–wind–thermal power systems to optimize the coordinated operation of multiple energies. The second way to achieve the optimal coordinated operation is by designing an operational optimization method or scheduling method to maximize the benefits of operation goals. This mostly uses linear/nonlinear programming methods to establish objective functions and constraints without the need to build complex mathematical models of the various components of power systems. The coverage of objective functions and the comprehensiveness of constraints are the key points of this approach to the evaluation of optimizing operations. For instance, Meng et al., (2024) [15] proposed an optimal scheduling strategy for hybrid power systems that can combine distributionally robust optimization and Stackelberg game theory. Zhu et al., (2024) [16,17,18,19] applied a series of novel optimization methods, like the three multi-objective optimization method and the multi-objective grey wolf optimization technique, to achieve the optimal operating mode of hybrid power systems. Li et al., (2022) [20] developed a continuous-time distributed algorithm for the economic dispatch problem of a power–water network. Guo et al., (2022) [21] proposed the risk-averse day-ahead optimization scheduling of hydro–wind–photovoltaic power systems, considering the steady requirement of power delivery. Majumdar et al., (2021) [22] achieved hydro–wind–photovoltaics–thermal scheduling based on the method of Mayfly algorithm optimization. Wang et al., (2017) [23] presented an improved hybrid method based on biogeography/complex and metropolis for many-objective optimization. Zhang et al., (2021) [24] presented the long-term optimization scheduling of a hydro–wind power system. The third way is optimizing the complementary performance of fluctuations in output power or frequency. This method emphasizes in-depth excavation from a certain research perspective, which generally includes controller optimization and the optimization of operation variables or scenarios. Regarding controller optimization, Shirkhani et al., (2023) [25] summarized microgrid decentralized energy/voltage control methods. Saidi et al., (2016, and 2011) presented passivity-based direct power control of the shunt active filter [26], a method for the quality improvement of the shunt active power filter [27], and shunt active filter control based on adaptive fuzzy logic [28]. Regarding the optimization of operation variables or scenarios, Avila et al., (2021) [29] evaluated hydro–wind complementarity in medium-term planning from the viewpoint of the periodic stream flow and wind speed time series. Liu et al., (2024) [30] studied the performance, pricing, and coordination of a hybrid system from a data security perspective. Fayek and Kotsampopoulos (2021) [31] optimized the complementary performance of hybrid power systems from the viewpoint of frequency deviation. Ren et al., (2022) [32] investigated hydro–wind–photovoltaics complementarity from the viewpoint of fluctuations in power output. Tang et al., (2020) [33] optimized the sizes of renewable power plants for a hydro–wind–photovoltaics power system based on the power output complementarity.

Table 1. The summary of literature review for achieving the optimal coordinated operation.

Summary		Classification of Optimization Ways
		Way 1	Way 2	Way 3
Description		Establish an optimization model to optimize the operating coefficients.	Design an operational optimization method or scheduling method to maximize the benefits of operation goals.	Optimize the complementary performance of the fluctuations in output power or frequency.
Common characteristic		The calculation process is reasonable and the optimization results are reliable.
Differences	(i) Key step	Establish a model that reflects the coupling characteristics of each component or subsystem.	Present a linear/nonlinear programming method.	Emphasize in-depth excavation from a certain research perspective.
	(ii) Key factor that is related to the reliability of results	Model outputs.	Objective functions and constraints.	Controllers, operation variables, or operation scenarios.
	(iii) The literature involved	References [10,11,12,13,14]	References [15,16,17,18,19,20,21,22,23,24]	References [25,26,27,28,29,30,31,32,33]

is used to clearly show the differences in the above three ways of achieving the optimal coordinated operation.

Compared with the former two ways, the third way has the advantage of a small amount of calculation. In addition to this, the third way flexibly reflects the coordinated operation of hydro–wind-based power systems, since there are various complementary indicators. It is summed up through the above literature review that these complementary indicators mainly include frequency deviations, the volatility of power outputs, and other indicators with fluctuations. These complementary indicators have been applied separately in the existing literature, but several studies have comprehensively considered coordinated operations with multiple indicators. Moreover, the power loss in this paper is adopted as a newly added indicator to participate in the multi-indicator optimizing operation of a hydro–wind power system.

Based on the above considerations, this paper studies the optimal coordinated operation of a hydro–wind power system by considering multiple complementary indicators. The detailed complementary indicators involve the hydraulic tracing power, the volatility of wind power, the fluctuations in hydraulic frequency, and the power loss. By fully considering the need for actual power plants, the optimal coordinated operation of the hybrid system includes two aspects: (i) the optimal complementary mode between hydro and wind power, and (ii) the optimal capacity allocation of wind power. The optimal complementary mode is analyzed under two real application scenarios, including the on-demand hydropower scenario and the surplus hydropower scenario. Among them, the on-demand hydropower scenario means that the planned electricity produced by the stations is equivalent to the demand of the daily load. The surplus hydropower scenario is an operating condition where the planned electricity is larger than the demand of the daily load. The analysis is conducted by using the software of Matlab 2023b. The simulation results from the complementarity analysis will help the hybrid power plant to stimulate the best regulation capability. Similarly, the simulation results from the analysis of the optimal capacity allocation will contribute to the maximum utilization of wind energy.

The novelties of this paper are summarized in two aspects. (i) A relatively complete model of the hydro–wind hybrid power system is established based on their rigorous mathematical relationship. In this hybrid model, the hydropower subsystem considers the dynamic characteristics of all major hydraulic–mechanical–electrical components, and the wind power subsystem adopts Type IV wind turbines (WTs) to better adapt to changes in wind speed. This hybrid model facilitates the research of numerical simulations. (ii) Multiple complementary indicators, particularly power loss, are comprehensively considered to achieve the optimal coordinated operation of the hybrid power system, helping to improve the reliability of the evaluation results. Correspondingly, the benefits of this study are concluded as: (i) The proposed assessment method that constructs multi-type optimization indicators based on the real-time outputs of the simulation model fills a major gap in previous investigations. (ii) The applicability of the proposed method can be extended to most hybrid power systems involving hydropower, which contributes to enhancing energy efficiency by maximizing the dynamic regulation benefits of hydropower. (iii) The results obtained in this work can be effectively used by energy engineers to improve the operational stability and economic benefits of power stations.

This paper is structured as follows: Section 2 models the hydro–wind power system. Section 3 presents the methodology. Section 4 optimizes the complementary mode. Section 5 optimizes the capacity allocation to maximize the power generation. The conclusions and discussion in Section 6 summarize the paper.

2. Model

The working mechanism of the integration of hydropower and wind power systems is shown in Figure 2, and its detailed modeling is discussed in the following subsections.

2.1. Wind Power System

The wind power model adopts a Type IV wind turbine, which has a directly coupled permanent magnet synchronous generator (PMSG). The PMSG is composed of various converters.

2.1.1. Wind Turbine

The wind turbine (WT) converts the kinetic energy into mechanical power, and it is expressed as [34]:

$$P_t={\displaystyle \frac{1}{2}}\rho A{v}^3{C}_p(\lambda ,\beta ),$$

(1)

where ρ, A, and v denote the air density, swept area of blades, and wind speed (m/s), respectively. C_p is the wind power coefficient, represented by the blade pitch angle β and tip–speed ratio λ. The tip–speed ratio λ is the ratio of the blade tip linear speed to the wind speed. The longer the blade or the faster the blade speed, the higher the tip–speed ratio in the same wind speed conditions. The tip–speed ratio λ [35] is written as:

λ = Rω_t/v,

(2)

where R and ω_t are the radius of the blade (m) and rotational speed of the wind turbine (rad/s).

The relationship between the tip–speed ratio λ and the blade pitch angle β is related to the maximum wind power coefficient C_pmax, as described in Figure 3 [35].

The extraction of the maximum power from the wind turbine depends on the maximum value of the wind power coefficient C_p(λ, β), and this maximum value is defined as:

$$C_{p\max }(\lambda ,\beta )={\displaystyle \sum _{i=1}^n{\displaystyle \sum _{i=1}^n{\alpha }_{ij}{\lambda }^i{\beta }^j}}.$$

(3)

2.1.2. PMSG

The PMSG directly connects to the wind turbine, and this configuration can reduce noise, mechanical loss, and fault probability and also improve efficiency. The stator voltage in the synchronously rotating d-q frame [36,37] is expressed as:

$$\left\{\begin{array}{l}{u}_{sd}=R_s{i}_{sd}+{\displaystyle \frac{d{\psi }_{sd}}{dt}}-{\omega }_e{\psi }_{sq}\\ u_{sq}=R_s{i}_{sq}+{\displaystyle \frac{d{\psi }_{sq}}{dt}}-{\omega }_e{\psi }_{sd}\end{array}\right.,$$

(4)

where u_s, i_s, R_s, ${\psi }_s$, and ω_e are the voltage, current, stator resistance, magnetic flux, and rotor speed of the generator, respectively. Subscripts d and q refer to the d-axis and q-axis, respectively.

The magnetic flux of the stator in the d-axis and q-axis is written as:

$$\left\{\begin{array}{l}{\psi }_{sd}=L_d{i}_{sd}+{\psi }_0\\ {\psi }_{sq}=L_q{i}_{sq}\end{array}\right.,$$

(5)

where ${\psi }_0$ is the magnetic flux of the permanent magnet. L_d and L_q are the d-axis and q-axis inductances, respectively. The electromagnetic torque of the PMSG is expressed as:

T_e = 1.5n(ψ _d i_q − ψ _q i_d),

(6)

where n is the pole pair. If L_d = L_q = L, Equation (6) can be transformed into the following form:

T_e = 1.5nψ ₀i_sq.

(7)

2.1.3. Full-Size Converter

The full-size converter is composed of the diode rectifier, DC–DC boost converter, and VSC. The diode rectifier is used to track the maximum power of the wind turbines. The DC–DC boost converter avoids the failure of voltage rectification under undesired wind speeds. The boost converter adjusts the input current on the rectifier side, which reduces harmonic pollution. The mechanism of the DC–DC boost converter [38,39] is described as:

$$\left\{\begin{array}{l}{\displaystyle \frac{d{u}_{dc}}{dt}}={\displaystyle \frac{1}{C{u}_{dc}}}(u_g{i}_g+u_s{i}_s-P_l)\\ u_g{i}_g=u_{gd}{i}_{gd}+u_{gq}{i}_{gq}\\ u_s{i}_s=u_{sd}{i}_{sd}+u_{sq}{i}_{sq}\end{array}\right.,$$

(8)

where u_gd and i_gd are the d-axis voltage and current on the grid side. u_gq and i_gq are the q-axis voltage and current on the grid side. u_dc is the DC link voltage. P_l is the total active power loss.

The VSC is essentially a grid-side inverter which maintains the stability of the DC link voltage. The VSC’s equation is generally expressed as:

$$\left\{\begin{array}{l}{u}_{gd}=-R_g{i}_{gd}-L_g{\displaystyle \frac{d{i}_{gd}}{dt}}+{\omega }_g{L}_g{i}_{gq}+e_{gd}\\ u_{gq}=-R_g{i}_{gq}-L_g{\displaystyle \frac{d{i}_{gq}}{dt}}-{\omega }_g{L}_d{i}_{gd},\end{array}\right.$$

(9)

where R_g and L_g are the line resistance and inductance, respectively. ω_g and e_gd denote the synchronous angular speed of the grid and the grid voltage in the d-axis, respectively.

According to refs. [40,41], the block diagram of the full-size converter is modeled in Figure A1 (see Appendix A).

2.2. Hydropower System

The hydropower system includes a Francis hydro-turbine, synchronous generator, frequency-governing system, and pipe system. The utilization mode of water energy of Francis hydro-turbines belongs to the reaction type. The operating head of Francis hydro-turbines is in the range from 20 to 700 m, and the corresponding hydropower system is suitable to be built in the river section with slow gradients and a large discharge. Hydropower regulates intermittent wind power and maintains the reliability of the power system. Its detailed modeling is shown in the following subsections.

2.2.1. Pipe Characteristic

Considering the elastic characteristic of the pipe wall, the dynamic characteristic of the pipe [42] is described as:

$$G_D(s)={\displaystyle \frac{{\overline{h}}_q(s)}{\overline{q}(s)}}={\displaystyle \frac{b_3{s}^3+b_2{s}^2+b_{1}s+b_0}{a_2{s}^2+a_{1}s+a_0}},$$

(10)

where G_D(s) is the transfer function between the turbine head and flow, $\overline{h}q$(s) is the head variation, and $\overline{q}$(s) is the flow variation. The coefficients in Equation (10) are further expressed as:

$$\left\{\begin{array}{l}{a}_0=24h_w, a_1=6T_r{h}_f, a_2=3T_r^2{h}_w\\ b_0=-24T_r{h}_w{h}_f, b_1=-24T_r{h}_w^2, b_2=-3T_r^2{h}_w{h}_f, b_3=-T_r^3{h}_w^2,\end{array}\right.$$

(11)

where h_w is the pipe characteristic coefficient, T_r is the water hammer time constant, and h_f is the hydraulic friction loss.

Based on the Laplace transform of G_D(s) [43,44], the turbine head and flow are finally obtained as:

$$\left\{\begin{array}{l}{h}_q=1-f_p{q}^2-{\displaystyle \frac{q^2}{y^2}}\\ q=q_0+\overline{q}=a_2{x}_3+a_1{x}_2+a_0{x}_1+q_0,\end{array}\right.$$

(12)

where q₀ and y are the relative values of the initial flow and guide vane opening. The state equation of Equation (12) is described as:

$$\left\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=x_3\\ {\dot{x}}_3=-{\displaystyle \frac{b_0}{b_3}}{x}_1-{\displaystyle \frac{b_1}{b_3}}{x}_2-{\displaystyle \frac{b_2}{b_3}}{x}_3+{\displaystyle \frac{1}{b_3}}{h}_q.\end{array}\right.$$

(13)

2.2.2. Hydraulic Generator

The electromotive force equation of the synchronous generator [45] is expressed as:

$$\left\{\begin{array}{l}{\dot{E}}_0=\dot{U}+r_a\dot{I}+j{\dot{I}}_d{x}_d+j{\dot{I}}_q{x}_q\\ x_d=x_{ad}+x_{\sigma }\\ x_q=x_{aq}+x_{\sigma }\end{array}\right.,$$

(14)

where E₀ is the excitation electromotive force. U is the terminal voltage. x_d and x_q are the d-axis and q-axis synchronous reactance. x_ad and x_aq are the d-axis and q-axis armature reactance. r_a is the armature resistance. x_σ is the magnetic flux leakage.

The electromagnetic power of the synchronous generator is expressed as:

$$P_{em}=m{E}_{\sigma }I\cos {\phi }_i,$$

(15)

where m is the phase number of the stator. φ is the power factor angle. φ_i is the phase angle between the air gap voltage and the armature current. The output power of the generator is derived from the difference between the electromagnetic power (P_em) and armature copper loss (P_Cu), which is:

$$\left\{\begin{array}{l}{P}_{out}=P_{em}-P_{Cu}\\ P_{Cu}=m{I}^2{r}_a\end{array}\right..$$

(16)

The power and torque relationship is expressed as:

$${\displaystyle \frac{P_{in}}{\Omega }}={\displaystyle \frac{P_0}{\Omega }}+{\displaystyle \frac{P_{em}}{\Omega }},$$

(17)

where Ω is the mechanical rotor speed, Ω = 2πn/60. P_in and P₀ are the input power of the generator and no-load loss, respectively. Equation (17) is further represented by the electromagnetic torque, i.e.,

$$M_{in}=M_0+M,$$

(18)

where M_in is the input mechanical torque of the generator, M₀ is the no-load torque, and M is the electromagnetic torque.

Ignoring the armature copper loss P_Cu, the power angle characteristic of the generator is:

$$P_{em}=m{\displaystyle \frac{E_{0}U}{x_d}}\sin \theta +m{\displaystyle \frac{U^2}{2}}({\displaystyle \frac{1}{x_q}}-{\displaystyle \frac{1}{x_d}})\sin 2\theta ,$$

(19)

where θ is the power angle.

Similarly, the reactive power of the generator is:

$$Q_{em}=m{\displaystyle \frac{E_{0}U}{x_d}}\cos \theta -m{\displaystyle \frac{U^2}{2}}{\displaystyle \frac{x_d+x_q}{x_d{x}_q}}+m{\displaystyle \frac{U^2}{2}}{\displaystyle \frac{x_d-x_q}{x_d{x}_q}}\cos 2\theta .$$

(20)

2.2.3. Hydro-Turbine and Frequency Regulation System

Generally, hydro-turbines are defined by mechanical torque and flow. These variables are further represented by the guide vane opening, rotational speed, and turbine head. The dynamic characteristic of hydro-turbines [46] is, therefore, described as:

$$\left\{\begin{array}{l}{M}_t=M_t(y,h,n)\\ Q_t=Q_t(y,h,n),\end{array}\right.$$

(21)

where M_t, Q_t, h, y, and n are the mechanical torque, flow, head, guide vane opening, and rotational speed, respectively. Equation (21) can be further expressed by the relative values of M_t and Q_t, i.e., the relative flow q and relative mechanical torque m_t. It is expressed as:

$$\left\{\begin{array}{c}{m}_t=e_{x}x+e_{y}y+e_{h}h\\ q=e_{qx}x+e_{qy}y+e_{qh}h\end{array}\right.,$$

(22)

where e_x, e_y, and e_h are the relative deviations of the mechanical torque with respect to the rotational speed, the guide vane opening, and the turbine head, respectively. e_qx, e_qy, and e_qh are the relative deviations of the flow with respect to the rotational speed, the guide vane opening, and the turbine head, respectively.

The dynamic performance of hydro-turbines is governed by the PID controller, which is also used to achieve the desired frequency regulation. The conventional PID controller [47] is adopted as:

$${\displaystyle \frac{Y_{pid}(s)}{\Delta F(s)}}=(K_p+{\displaystyle \frac{K_i}{s}}+{\displaystyle \frac{K_{d}s}{T_n+1}}),$$

(23)

where Y_pid(s) is the Laplace transform of the guide vane opening. ΔF(s) is the Laplace transform of the frequency. T_n is the acceleration time constant. K_p, K_i, and K_d are the proportional, integral, and differential adjustment coefficients, respectively.

Based on Equation (23), the electro-hydraulic servo system is expressed as:

$${\displaystyle \frac{Y(s)}{Y_{pid}(s)}}={\displaystyle \frac{1}{T_{y}s+1}}$$

(24)

Based on the above descriptions, a block diagram of the hydro-turbine and frequency regulation system is shown in Figure A2 (see Appendix A). A description of the setting of the PID control parameters is shown in Appendix B.

2.3. Hybrid Power System

A hybrid model of the hydro–wind power system is shown in Figure A3 (see Appendix A). In this integrated model, the capacity of the single-machine wind turbine is 2 MW and the rated frequency is 60 Hz. The total installed capacity of the wind farm is 100 MW, with 50 wind turbines. The generation capacity of the hydropower system is 100 MW, and its load reserve capacity is 3 MW. The hydropower and wind power connect to the power grid through two transformers. The load demand of the power grid is 200 MW, which equals the total installed capacity of the hybrid system.

3. Methodology

Based on the established mathematical model of the hydro–wind hybrid power system in Section 2, the hydraulic power, wind power, and hydraulic frequency are output by the model. According to these outputs, four complementary indicators, including the hydraulic tracing power, volatility of wind power, fluctuations in hydraulic frequency, and power loss, are obtained to optimize the coordinated operation of the hybrid system. The definitions of these four indicators are described in

Table 2. Descriptions of complementary indicators in the optimization methodology.

Indicator	Definition
Hydraulic tracing power	This is equal to the power load minus the actual hydraulic power. The coordinated operating quality of the hybrid system is excellent when the hydraulic tracing power is closer to the actual wind power.
b. Volatility of wind power	This is equal to the difference between the maximum and minimum actual wind power.
c. Volatility of hydraulic frequency	This is equal to the ratio of the product of the hydro-turbine rotational speed and generator pole number to 60, which is generally revealed by the value and occurrence time of the maximum deviation. The hydraulic frequency cannot exceed ± 0.5 Hz, which is an accepted range for the stability of power grids.
d. Power loss	This is equal to the difference between the end-use power of a grid and the mix-produced power of a hybrid system, which is generally revealed by the mean, maximum, and peak time.

. The optimization methodology is shown in Figure 4.

4. Optimization of Complementary Mode

This section aims to maximize the total power generation by optimizing the complementary mode. The analysis is conducted based on two operating scenarios, including the on-demand hydropower scenario (Scenario 1) and the surplus hydropower scenario (Scenario 2).

Table 3. Descriptions of the studied two scenarios.

Scenario	Characteristics
Scenario 1: on-demand hydropower scenario	The planned power generation is equivalent to the load demand. The installed hydropower capacity that can participate in regulation is prone to shortages if there is a prolonged period of extreme wind speed. It is not easy to produce surplus water for the hydropower system.
Scenario 2: surplus hydropower scenario	The planned power generation is larger than the load demand. When the fluctuation in wind power is large, the installed hydropower capacity that can participate in regulation is sufficient. More surplus water is produced.

describes the characteristics of these two scenarios. Three basic types of wind speeds are considered to optimize the complementary characteristics of the hybrid system, including constant, mutational, and random winds [48,49,50,51]. A constant wind speed reflects changes in the average wind speed in the wind field, which mainly determines the rated power delivered by the wind generator to the hydro–wind power system. A mutational wind speed can simulate sudden changes in wind speed, which is generally used to assess the degree of power and voltage fluctuations under large changes in wind speed when analyzing the influence of the wind power system on the stability of the power grid. A random wind speed can simulate the characteristics of random fluctuations in wind speed, which influence the stability of the wind power output. For the sake of analysis, constant wind speed, mutational wind speed, and random wind speeds are abbreviated as wind type 1, wind type 2, and wind type 3. The waveforms of these three wind speeds are shown in Figure 5a. The transmission power loss is finally calculated to reflect the differences in power generation under the three wind speeds.

4.1. Scenario 1: On-Demand Hydropower Operation

In Scenario 1, hydropower is considered as a regulator to regulate fluctuations in wind power. Also, it is used to complement electricity shortages of the power grid. The complementary characteristic is investigated under three types of wind speeds, as shown in Figure 5a. The corresponding power structure is shown in Figure 5b.

In Figure 5a, it is observed that the hydraulic power and wind power show opposite trends under the three wind conditions. This means that hydropower has an excellent complementary effect on regulating fluctuations in wind power, since the sum of the hydraulic power and wind power is close to the power requirement. There are some major differences in the complementary characteristics, which are mainly manifested in the power variation in the magnitude and trends under the three wind speeds. The relative wind power for the constant wind fluctuates slightly between 1 and 0.986. Conversely, the relative wind power for the mutational wind displays a large fluctuation from 1 to 0.3751. From the perspective of power fluctuations, the mutational wind has a relatively larger influence on the operating safety of the power system. However, hydraulic power closely tracks fluctuations in wind power, which can help to mitigate operating risks under mutational wind conditions.

In Figure 5b, the power gap is the negative difference between the power demand of the grid and the power generation of the hybrid system, while the surplus power is the positive difference. The coordinated operation of the hybrid system shows a promising power structure in the random wind speed condition (Type 3). The main reason behind this is the relatively less surplus power under Type 3 in comparison to the large power under the constant wind (Type 1). The surplus power produced in Type 1 is due to the inappropriate and sub-optimal complementary mode of hydro and wind energies. Moreover, compared with the mutational wind (Type 2), the power gap is relatively less under Type 3. The power gap produced in Type 2 is attributed to it being the most unfavorable wind condition.

For a hybrid power system, transmission power loss is generally unavoidable, and it will reduce the amount of electricity available to users. For this reason, the end-use power consumption of the power grid is likely to be less than the mix-produced power of the hybrid system. Figure 6a shows the influence of the three wind speeds on the end-use power and mix-produced power, and Figure 6b shows the corresponding power loss obtained from the difference between the mix-produced power and end-use power.

In Figure 6, the end-use power changes slightly by around 194.5 MW for Type 1 and 3 winds, and the corresponding total power generation changes by around 201 MW. The end-use power and total power generation for Type 2 change significantly compared to the power for Type 1 and 3 winds. This means that Type 2 produces the worst wind conditions for the coordination operation of the hybrid power system, since the power loss is the largest among the three wind speeds. Moreover, it is worth noting that the end-use power under different wind speeds is always less than the power demand of the grid (i.e., 200 MW). This is caused by the unexpected power loss in the transmission process. As shown in Figure 6, the power loss continuously changes throughout the whole operating cycle (i.e., 0 s to 30 s). Compared to Type 1 and 3 winds, the power loss for Type 2 wind is relatively large and reaches a maximum of 6.83 MW. On the other hand, the mean power loss for Type 1 and 3 winds is roughly around 6 MW.

4.2. Scenario 2: Surplus Hydropower Operation

In Scenario 2, the power demand of the grid changes from 200 MW to 150 MW. The hydro generator in this scenario mainly acts as a regulator to suppress wind power fluctuations. The results of the complementary characteristics of the hydro–wind power system are shown in Figure 7a.

The hydraulic tracing power is defined as the difference between the power consumption and the actual hydraulic power. There is an excellent complementary mode if the hydraulic tracing power is approximately consistent with the variation in the actual wind power under the wind speed. As shown in Figure 7a, the complementary mode between the hydro and wind power systems is excellent in Scenario 2. The hydraulic tracing power sees small variations under Type 1 wind, while the mutational wind triggers large power fluctuations. In particular, the wind power decreases to a minimum of 6.56 MW, which obviously reduces the efficiency of the power system. However, this insufficient power capacity can be supplemented by hydropower without consuming additional reserve capacity. Type 3 wind causes the wind power to decrease from 100 MW to 77.54 MW. Such a big variation cannot influence the overall power generation capability of the power system due to the timely tracing of the hydropower. However, power loss may exist in Figure 7a, because the curve of the hydraulic tracing is almost coincidental with the curve of the wind power. For this reason, Figure 7b shows the variation trend of power loss.

Figure 7b shows the hydraulic tracing effect and power loss in Scenario 2. It reveals that the maximum power loss (4.68 MW) for Type 2 wind is larger than that for Type 1 wind (4.64 MW) and Type 3 wind (4.63 MW). The corresponding mean power loss for the Type 2 wind is the lowest (3.98 MW) due to the small amplitude of the fluctuations in the power loss. However, the mean power loss for the other two wind types is relatively small.

Compared with Scenarios 1 and 2, the wind volatility is almost similar in different wind conditions, and the hydraulic tracing effects are generally excellent for all wind types. To better compare the complementary characteristics in the two operating scenarios, the obvious differences in power loss enable this analysis. It is clearly shown that there are massive power losses produced in Scenario 1 compared to the operation in Scenario 2. The main reason behind this is the high demand for the power of the grid in Scenario 1 relative to Scenario 2. Specifically, the mean and maximum power loss for Scenario 1 are 6 MW and 6.8 MW, while for Scenario 2, they are around 4 MW and 4.6 MW, respectively. The occurrence time of the maximal power loss for these two scenarios is between 1 s and 3 s, which is caused by the power regulation capability of the hydropower at the initial coordinated operating stage.

5. Optimization of Capacity Allocation

This section is dedicated to increasing the total power generation by optimizing the capacity allocation of hydro and wind energies. From Section 4, it is obtained that Type 2 wind is the most unfavorable wind condition, since it leads to greater power loss than the other types of wind speed. Considering this, the capacity allocation under the wind type 2 wind is emphasized in this analysis. The number of WTs is 50, corresponding to the installed wind capacity of 100 MW, as studied in Section 4. The results obtained from Section 4 clearly show that 50 WTs is not the marginal permeation of the installed wind capacity when the hybrid power system begins to lose its stability. For this reason, the initial studied number of WTs starts from 51 in this section. The selected number of WTs is finally 51, 55, 60, and 61 corresponding to the installed wind capacities of 102 MW, 110 MW, 120 MW, and 122 MW. In this optimized analysis, the effect of the WTs’ number on the hydraulic suppression effect is first studied, and the power loss is then compared under the three wind types.

The most unexpected wind is Type 2 wind, as discussed earlier in Section 4. For this reason, Type 2 wind is considered in this analysis to optimize the capacity allocation. The result of the hydraulic suppression effect is shown in Figure 8, and the result of the power loss is shown in Figure 9.

Figure 8a shows the power complementarity and fluctuations in hydraulic frequency. It is observed that the wind power fluctuation gradually increases with an increase in WT numbers. Despite this, the hydraulic frequency cannot exceed ± 0.5 Hz, which is an accepted range for the power grid. Figure 8b shows the out-of-control hydraulic frequency that denotes ineffective regulation of the hydropower system at this instant. The out-of-control point (OCP) of the hydraulic frequency is the time where the hydropower system starts to lose its stability. For the power system with 61 WTs, the OCP occurs at 21.65 s, and its hydraulic frequency infinitely closes to 0.5 Hz. The wind power is out of control when its relative value decreases to 0.35 at 21.65 s.

Figure 9 shows the effect of capacity allocation on the power loss under Type 2 wind. As depicted in Figure 9, there always exists surplus power at 1.95 s with an increase in WT numbers, and the corresponding value of the surplus power shows an increasing trend from 2.2 MW to 2.8 MW. The power gap also increases when the WT number changes from 51 to 60. This is caused by the increasing fluctuation of wind power with the permeation of the installed wind capacity. The power loss continuously changes, influenced by the mutational wind in the whole production process. This reveals that the quantity of the power loss is closely related to the capacity allocation of hydro and wind energies.

Table 4. Optimization results of the capacity allocation under the constant (Type 1), mutational (Type 2), and random (Type 3) wind speeds.

Wind Type	WT Number	Wind Capacity (MW)	Complementary Characteristic		Hydraulic Frequency		Power Loss
			Wind Volatility (p.u.)	Hydraulic Tracing Effect	Maximum Deviation (Hz)	Occurrence Time (s)	Mean (MW)	Maximum (MW)	Peak Time (s)
1	51	102	0.0121	Excellent	0.123	5.17	6.82	7.00	0.95
	75	150	0.0174	Excellent	0.198	1.45	9.00	10.79	0.6
	100	200	0.0244	Good	0.2686	4.93	11.80	14.80	0.62
	125	250	0.0297	Good	0.3095	4.10	15.50	19.00	0.8
	150	300	0.0379	Good	0.432	1.12	19.00	24.00	0.76
	163	326	0.0413	Good	0.4297	1.38	21.80	27.50	0.79
	164	328	--	--	--	0	--	--	--
2	51	102	0.6352	Excellent	0.0776	23.56	6.90	7.50	0.47
	55	110	0.686	Excellent	0.0957	26.76	7.20	7.75	0.46
	60	120	0.7505	Good	0.3568	27.64	7.75	8.80	0.47
	61	122	--	--	--	21.55	--	--	--
3	51	102	0.1397	Excellent	0.2329	1.60	6.86	7.43	0.46
	75	150	0.208	Good	0.3219	15.98	9.38	11.21	0.49
	100	200	0.2744	Good	0.4761	2.62	13.50	16.56	0.49
	125	250	0.3427	Good	0.5000	2.31	19.50	23.91	0.57
	127	254	--	--	--	--	--	--	--

clearly reveals the detailed quantities of power losses under the three wind speeds. For Type 1 wind, the hydraulic frequency exceeds ± 0.5 Hz until 164 WTs, and the state of OCP is maintained around five seconds. This means that the hydropower rapidly loses the regulation effect on balancing wind power fluctuations when the installed wind capacity approaches the adjustable marginal permeation. For Type 2 and 3 winds, from the OCP perspective, the marginal permeation values of the installed wind capacity are 61 WTs and 127 WTs. From this angle, it is found that Type 1 wind has a great capability to adapt wind energy to the power grid, and the capability of Type 2 is the worst.

Moreover, owing to the hydraulic suppression effect, the complementary characteristic of the hybrid system is excellent or good before the occurrence of the OCP. The mean and maximum power loss show an increasing trend with an increase in WT numbers, and the peak loss occurs in the first second. Comparing the three wind speeds, the growth rate of the power loss is the fastest for Type 2 wind, followed by Type 3 wind and Type 1 wind. Note that the power loss is small at 60 WTs for Type 2 wind. In this situation, the critical wind permeation is determined by the considerably high hydraulic frequency. When the accepted WT number for Type 1 wind reaches 163 and the number for Type 3 wind reaches 125, the corresponding mean power loss exceeds 20 MW. In this situation, the critical wind permeation is determined by the rate between the mean power loss and the installed wind capacity. Correspondingly, the rates for these two winds are 5.12% and 5.57%, respectively. Thus, comprehensively considering the power loss, complementary characteristics, and hydraulic regulation capability, the critical wind permeation values for Type 1, 2, and 3 are 326 MW, 120 MW, and 250 MW with respect to 163, 60, and 125 WTs.

6. Conclusions and Discussion

This work studies the optimal coordinated operation of hydro–wind power systems by establishing multiple complementary indicators. The power loss is different from previous investigations, as it is a newly presented indicator in the optimization method. Apart from the power loss, the other complementary indicators include the hydraulic tracing power, the volatility of wind power, and the fluctuations in hydraulic frequency. All these four indicators are constructed from the output variables of the established model. The analysis encompasses both the optimal complementary mode and the optimal capacity allocation, with the consideration of two real application scenarios, including the on-demand hydropower scenario and the surplus hydropower scenario. The key findings from this study include the following:

(1)

The effect of wind speeds on the complementary mode: The mutational wind speed causes the worst complementary mode of the power system because there is an increasing burden of hydro regulation to cope with frequently drastic changes in wind speed. On the contrary, the complementary mode is the best under the random wind speed, since the power system makes full use of the regulation capability of hydropower in this situation.

(2)

The effect of application scenarios on the complementary mode: The most distinctive feature of complementarity for both real application scenarios lies in the power loss. The mean power loss for the on-demand hydropower scenario is 6 MW, while that for the surplus hydropower scenario is around 4 MW. Thus, the on-demand hydropower scenario is a better choice for the economic operation of power stations from the perspective of power loss.

(3)

The necessity for multiple indicators: When the hybrid power system is close to instability with the increasing wind power permeation, the marginal permeation of the installed wind capacity cannot be determined by a single indicator. For instance, the system loses stability when the hydraulic power and wind power still have good complementary tracking effects. In this situation, the indicators of power loss and hydraulic frequency are able to reveal this instability.

(4)

The optimal capacity allocation: Based on the comprehensive evaluation of multiple indicators, the marginal permeations of the installed wind capacity for a 100 MW hydropower system are approximatively 326 MW, 120 MW, and 250 MW under constant, mutational, and random wind speeds.

The optimal coordinated operation in this paper is conducted based on complementary indicators. For a deeper investigation, future work will try to establish an optimization method represented by objective functions and constraints to study the optimal coordinated operation of hybrid power systems, and it will also be compared with some modern algorithms like the Mountain Gazelle Optimizer in ref. [52].