Hierarchical MPC–QP Control for Primary Frequency Regulation of a Wind–Storage Hybrid System with Intra-Farm Power Optimization

Abstract

High penetration of wind power increases the need for coordinated frequency support from wind farms and battery energy storage systems (BESSs). To address insufficient power coordination, state-of-charge (SOC) deviation, and suboptimal intra-farm allocation of frequency-regulation power, this paper proposes a hierarchical MPC–QP control strategy for primary frequency regulation of a wind–storage hybrid system. At the upper level, model predictive control (MPC) minimizes system frequency deviation and a composite frequency-regulation cost index subject to wind-farm power, BESS power, and SOC constraints, thereby determining optimal regulation-power references for the wind farm and BESS. At the lower level, quadratic programming (QP) minimizes wind-farm regulation-power tracking error and equivalent torque fluctuations to achieve differentiated power allocation among equivalent turbine clusters. In a representative 1 h simulation with continuously varying wind speed and random load disturbances, the proposed strategy reduces the RMS frequency deviation, maximum absolute frequency deviation, RMS SOC deviation, and cumulative regulation cost index by 24.20%, 24.53%, 93.86%, and 84.72%, respectively, compared with PI control. Compared with equal allocation, it reduces the RMS wind-farm power-tracking error and RMS equivalent torque fluctuation by 87.93% and 15.52%, respectively. Three additional stochastic realizations demonstrate consistent improvements in the selected performance indices, further supporting the effectiveness of the proposed hierarchical strategy under the investigated simulation conditions.

1. Introduction

With the large-scale integration of wind power and other renewable energy sources, the proportion of conventional synchronous generators in power systems has gradually decreased, leading to reduced system inertia and weakened primary frequency-regulation capability. Consequently, frequency stability has become an increasingly critical issue in power systems with high renewable energy penetration [1,2,3]. Modern variable-speed wind turbines can provide frequency support by releasing rotor kinetic energy or reserving active power. However, their regulation capability is strongly dependent on wind speed, rotor speed, and operating conditions, and standalone wind-turbine frequency regulation may also cause rotor-speed recovery problems and secondary frequency drops [4,5,6]. Energy storage systems, owing to their rapid response and flexible bidirectional power regulation capability, can compensate for the operating-state-dependent limitations of wind power. Therefore, coordinated wind–storage participation in primary frequency regulation has become an effective approach to enhancing the frequency support capability of renewable-dominated power systems [7,8,9].

In recent years, considerable efforts have been devoted to coordinated control and power optimization for wind–storage primary frequency regulation. Existing studies have improved coordinated frequency support by considering factors such as wind speed, wind-turbine operating conditions, and the state of charge (SOC) of energy storage systems [10]. In particular, the frequency-regulation capability of wind turbines varies considerably with operating conditions, making it necessary to adapt turbine participation and regulation characteristics to different wind-speed regions. Meanwhile, advanced coordinated control and optimization methods, including model predictive control (MPC), have been increasingly applied to wind-power and wind–storage frequency regulation [11,12,13]. MPC is particularly attractive owing to its capability for receding-horizon optimization and explicit handling of power and SOC constraints. Beyond system-level wind–storage coordination, recent studies have further investigated hierarchical frequency-regulation strategies and wind-farm active-power optimization, including intra-farm power allocation, wake-aware control, and mechanical-load mitigation [14,15,16,17,18]. Reliable wind-farm operation also requires consideration of uncertainty and abnormal turbine conditions. Recent studies have investigated wind-turbine fault diagnosis using structural analysis and uncertainty-aware interval estimation [19], as well as pitch-system fault detection based on adaptive Kalman filtering [20]. These studies highlight the importance of condition monitoring in practical wind-farm control. Related operational considerations have also motivated wind–storage coordination studies addressing asset degradation and wind-cluster effects [21,22]. To clarify the main characteristics of representative recent studies, a comparison is presented in Table 1.

Table 1. Comparison of representative recent studies and the proposed method on wind–storage frequency regulation and wind-farm power optimization.

Reference	Operating-Region Awareness	Wind–Storage Coordination	Dynamic Regulation Bounds	Intra-Farm Optimization
[12]	Not explicit	Explicit	Not explicit	Not considered
[13]	Explicit	Explicit	Not explicit	Explicit
[14]	Not explicit	Explicit	Not explicit	Explicit
[15]	Not explicit	Not considered	Not explicit	Explicit
[16]	Not explicit	Not considered	Not explicit	Explicit
[17]	Not explicit	Not considered	Not explicit	Explicit
[18]	Not explicit	Explicit	Not explicit	Explicit
Proposed	Explicit	Explicit	Explicit	Explicit

As summarized in Table 1, Cai et al. [14] proposed a hierarchical MPC strategy in which the upper level generates a total wind–storage regulation-power reference, while the lower level allocates it among wind turbines and the BESS. In contrast, the proposed upper-level MPC predicts the coupled frequency, wind-farm power, BESS power, and SOC responses and directly optimizes separate power references for the wind farm and BESS. Operating-region-dependent regulation limits are aggregated into wind-farm constraints for the upper level and retained as cluster-specific constraints for the lower level. Unlike the lower-level MPC in [14], the proposed QP allocates the wind-farm reference among equivalent turbine clusters by jointly minimizing power-tracking error and equivalent torque fluctuation. Therefore, the main contribution lies in the constraint-aware hierarchical formulation and its integration of wind–storage coordination with intra-farm power allocation, rather than in developing a new MPC or QP algorithm.

To address these issues, this paper proposes a hierarchical MPC–QP primary frequency-regulation strategy for a wind–storage hybrid system with intra-farm power optimization. The framework links full-wind-speed turbine regulation capability with system-level wind–storage coordination and turbine-level power allocation. The main contributions are as follows:

An operating-region-dependent wind-turbine frequency-regulation strategy is established over the full wind-speed range. Low-, medium-, and high-wind-speed regions are distinguished, with rotor-overspeed deloading and pitch-angle deloading adopted in the corresponding regulation regions. The resulting regulation-power limits are aggregated as wind-farm constraints for the upper-level MPC and retained as cluster-specific bounds for the lower-level QP.

An upper-level MPC-based wind–storage coordination strategy is developed. Frequency deviation and a composite regulation cost index are minimized to determine the wind-farm and BESS regulation-power references under dynamically updated wind-farm power limits and BESS power and SOC constraints.

A QP-based intra-farm power allocation method is proposed. Considering differences in cluster regulation capability, power-tracking error and equivalent torque fluctuation are jointly minimized to achieve differentiated power allocation under cluster-specific regulation-power constraints.

The hierarchical MPC–QP strategy is validated under continuously varying wind speed and random load disturbances, demonstrating improvements in frequency response, SOC regulation, power tracking, and equivalent torque fluctuation under the simulated conditions.

2. Materials and Methods

2.1. Wind–Storage Hybrid System and Hierarchical Control Framework

The wind–storage hybrid system considered in this study consists of a thermal power unit, a wind farm, a BESS, and the grid load, as shown in Figure 1. During primary frequency regulation, the thermal power unit, wind farm, and BESS jointly compensate for the active-power imbalance caused by load disturbances. The wind farm and BESS provide complementary frequency support owing to their different dynamic response characteristics.

Figure 1. Hierarchical MPC–QP control framework of the wind–storage system.

A hierarchical MPC–QP control framework is adopted to coordinate the wind farm and BESS. At the upper level, model predictive control (MPC) determines the regulation-power references of the wind farm, ΔP_{w_ref}, and the BESS, ΔP_{b_ref}, according to the system operating states. At the lower level, quadratic programming (QP) allocates the wind-farm regulation-power reference among individual wind turbines according to their operating states, thereby obtaining the turbine-level power references ΔP_w,i,ref.

The feasible regulation-power range of the wind farm varies with wind speed and turbine operating conditions. Therefore, operating-region-dependent turbine frequency-regulation modes and corresponding power limits are first determined and subsequently incorporated into the hierarchical optimization framework as power constraints. The detailed operating-region-dependent regulation strategy, upper-level MPC formulation, and lower-level QP allocation method are presented in the following sections.

2.2. Operating-Region-Dependent Wind-Turbine Frequency Regulation and Power Limits

2.2.1. Wind-Turbine Operating Regions and Frequency-Regulation Modes

The frequency-regulation capability of a DFIG varies with wind speed and operating condition [23]. To balance regulation capability and operational safety, the turbine operating range is divided into low-, medium-, and high-wind-speed regions, with different regulation modes adopted in each region. Wind-speed-aware MPC and frequency-support feasible-region modeling have also been investigated to account for variations in wind-farm regulation capability [24,25].

Low-wind-speed region: The turbine has limited regulation margin and does not participate in primary frequency regulation; frequency support is mainly provided by the BESS.

Medium-wind-speed region: Rotor-overspeed deloading is adopted for primary frequency regulation, while the BESS provides supplementary support.

High-wind-speed region: Pitch-angle deloading is adopted for primary frequency regulation, with the BESS providing auxiliary frequency support.

Based on the above operating-region classification, the regulation-power limits of the wind turbines and wind farm are further determined and used as constraints in the subsequent MPC and QP optimization.

2.2.2. Regulation-Power Limits Under Different Operating Regions

After the frequency-regulation modes are determined for different wind-speed regions, the feasible regulation-power range of each wind turbine can be further obtained according to its current operating point. Let the upper and lower regulation-power limits of the ith wind turbine be denoted by ΔP_i,max and ΔP_i,min, respectively.

(1)

Low-wind-speed region

In the low-wind-speed region, the wind turbine does not participate in primary frequency regulation. Therefore, both the upper and lower regulation-power limits are set to zero:

ΔP_i,max=ΔP_i,min = 0

(1)

(2)

Medium-wind-speed region

In the medium-wind-speed region, rotor-overspeed deloading is adopted [26]. The deloaded power is given by:

P_del,i = (1 − d)P_MPPT,i

(2)

where d is the deloading ratio, which is set to 10% in this paper. Taking the deloaded operating point as the reference, the upward regulation power is determined by the reserve margin between the MPPT power and the deloaded power, while the downward regulation power is determined by the minimum allowable output power. Thus, the corresponding regulation-power limits can be expressed as:

$$\begin{array}{l}\Delta P_{i,\max }=P_{\mathrm{MPPT},i}-P_{\mathrm{del},i}\\ \Delta P_{i,\min }=P_{\min ,i}-P_{\mathrm{del},i}\end{array}$$

(3)

(3)

High-wind-speed region

In the high-wind-speed region, pitch-angle deloading is adopted to reserve regulation power [27]. Let β_del denote the pitch angle under deloaded operation. With respect to the deloaded operating point, the upward and downward regulation-power limits can be determined from the turbine output power under different pitch angles as:

$$\begin{array}{l}\Delta P_{\beta ,\max }=P\left(v,{\beta }_0\right)-P\left(v,{\beta }_{\mathrm{del}}\right)\\ \Delta P_{\beta ,\min }=P\left(v,{\beta }_{\max }\right)-P\left(v,{\beta }_{\mathrm{del}}\right)\end{array}$$

(4)

where β₀, β_del, and β_max are the initial pitch angle, the deloaded pitch angle, and the maximum allowable pitch angle, respectively.

Figure 2 illustrates the operating characteristics and bidirectional regulation-power limits of a representative 1.5 MW DFIG under different wind-speed regions and a 10% deloading ratio. The regulation-power limits in Figure 2b are calculated based on the wind-turbine aerodynamic power characteristics. The overspeed-control limits are determined by the allowable rotor-speed range, while the pitch-control limits are obtained from the power differences at different pitch-angle operating points, with a maximum pitch-angle increment of 5°.

Figure 2. Operating characteristics and regulation-power limits of the DFIG under different wind-speed regions.

For a wind farm represented by N equivalent turbine clusters, the overall upper and lower regulation-power limits can be obtained by aggregating the feasible regulation-power ranges of all clusters:

$$\begin{array}{l}\Delta P_{\mathrm{wf},\max }={\displaystyle \sum \Delta P_{i,\max }}\\ \Delta P_{\mathrm{wf},\min }={\displaystyle \sum \Delta P_{i,\min }}\end{array}$$

(5)

2.3. Upper-Level MPC-Based Wind–Storage Coordinated Optimization

To coordinate the regulation power between the wind farm and BESS, an upper-level MPC strategy is adopted. Based on the system states and operating constraints, the optimal regulation-power references of the wind farm and BESS are obtained through receding-horizon optimization.

2.3.1. Prediction Model

To describe the dynamic responses of the wind–storage hybrid system during primary frequency regulation, a discrete state-space prediction model is established. The continuous-time model is discretized using the forward Euler method and expressed as:

$$\left\{\begin{array}{l}x\left(k+1\right)=\mathit{Ax}\left(k\right)+\mathit{Bu}\left(k\right)+\mathit{Gd}\left(k\right)\\ \begin{array}{cc}& \end{array}y\left(k\right)=\mathit{Cx}\left(k\right)\end{array}\right.$$

(6)

where x(k), u(k), d(k), and y(k) denote the state vector, control input, disturbance input, and output vector, respectively.

The state vector is defined as:

x(k) = [ΔP_g(k),ΔP_ch(k),ΔP_th(k),ΔP_b(k),ΔP_w(k),SOC(k),Δf(k)]^T

(7)

where ΔP_g, ΔP_ch, and ΔP_th denote the governor output power increment, steam-chest output power increment, and thermal-unit output power increment, respectively. ΔP_b and ΔP_w denote the regulation-power increments of the BESS and wind farm, respectively. The SOC of the BESS is included in the state vector to characterize its available energy and enable SOC-aware frequency regulation [28].

The control input and disturbance input are defined as:

$$\begin{array}{c}u\left(k\right)={\left[\Delta P_{\mathrm{b}\_\mathrm{ref}}\left(k\right),\Delta P_{\mathrm{w}\_\mathrm{ref}}\left(k\right)\right]}^T\\ d\left(k\right)=\Delta P_{\mathrm{L}}\left(k\right)\end{array}$$

(8)

The discrete state equations are given by:

$$\left\{\begin{array}{l}\Delta P_{\mathrm{g}}\left(k+1\right)=\left(1-{\displaystyle \frac{T}{T_G}}\right)\Delta P_{\mathrm{g}}\left(k\right)-{\displaystyle \frac{T}{R{T}_G}}\Delta f\left(k\right)\\ \Delta P_{\mathrm{ch}}\left(k+1\right)=\left(1-{\displaystyle \frac{T}{T_{\mathit{CH}}}}\right)\Delta P_{\mathrm{ch}}\left(k\right)+{\displaystyle \frac{T}{T_{\mathit{CH}}}}\Delta P_{\mathrm{g}}\left(k\right)\\ \Delta P_{\mathrm{th}}\left(k+1\right)=\left(1-{\displaystyle \frac{T}{T_{\mathit{RH}}}}\right)\Delta P_{\mathrm{th}}\left(k\right)+{\displaystyle \frac{F_{\mathit{HP}}T}{T_{\mathit{CH}}}}\Delta P_{\mathrm{g}}\left(k\right)+\left({\displaystyle \frac{T}{T_{\mathit{RH}}}}-{\displaystyle \frac{F_{\mathit{HP}}T}{T_{\mathit{CH}}}}\right)\Delta P_{\mathrm{ch}}\left(k\right)\\ \Delta P_{\mathrm{b}}\left(k+1\right)=\left(1-{\displaystyle \frac{T}{T_{\mathrm{b}}}}\right)\Delta P_{\mathrm{b}}\left(k\right)+{\displaystyle \frac{T}{T_{\mathrm{b}}}}\Delta P_{\mathrm{b}\_\mathrm{ref}}\left(k\right)\\ \Delta P_{\mathrm{w}}\left(k+1\right)=\left(1-{\displaystyle \frac{T}{T_{\mathrm{w}}}}\right)\Delta P_{\mathrm{w}}\left(k\right)+{\displaystyle \frac{T}{T_{\mathrm{w}}}}\Delta P_{\mathrm{w}\_\mathrm{ref}}\left(k\right)\\ \mathit{SOC}\left(k+1\right)=-{\displaystyle \frac{T}{E_{\mathrm{b}}}}\Delta P_{\mathrm{b}}\left(k\right)+\mathit{SOC}\left(k\right)\\ \Delta f\left(k+1\right)={\displaystyle \frac{T}{2H}}\left(\Delta P_{\mathrm{th}}\left(k\right)+\Delta P_{\mathrm{b}}\left(k\right)+\left.\Delta P_{\mathrm{w}}\left(k\right)-\Delta P_L\left(k\right)\right)+\left(1-{\displaystyle \frac{\mathit{DT}}{2H}}\right)\Delta f\left(k\right)\right)\end{array}\right.$$

(9)

The output vector is selected as:

$$y\left(k\right)={\left[\Delta P_{\mathrm{b}}\left(k\right),\Delta P_{\mathrm{w}}\left(k\right)\right]}^T$$

(10)

The above discrete equations form the prediction model used for the subsequent MPC-based receding-horizon optimization.

2.3.2. Objective Function

The upper-level MPC aims to suppress system frequency deviation and minimize the composite frequency-regulation cost index. The objective function is defined as:

$$\min J={\displaystyle \sum _{i=1}^{Np}\left[{\lambda }_f\Delta f_{k+i|k}^2+{\lambda }_c\left(C_{\mathrm{w}\_(k+i|k)}+C_{\mathrm{b}\_(k+i|k)}\right)\right]}$$

(11)

where N_p is the prediction horizon; λ_f and λ_c are the weighting coefficients for the frequency-deviation term and the composite frequency-regulation cost term, respectively; Δf_k+i|k is the predicted system frequency deviation at time k + i based on the information available at time k; and C_{w_k+i∣k} and C_{b_k+i∣k} denote the frequency-regulation cost indices of the wind farm and BESS, respectively.

The wind-farm frequency-regulation cost index is defined as:

$$C_{\mathrm{w}\_(k+i|k)}=a_1{\left[\Delta P_{\mathrm{w}}(k+i|k)\right]}^2$$

(12)

The BESS frequency-regulation cost index considers both the charging/discharging power and SOC deviation and is defined as:

$$C_{\mathrm{b}\_k+i|k}=a_2{\left[\Delta P_{\mathrm{b}}\left(k+i|k\right)\right]}^2+a_3{\left[\mathrm{SOC}\left(k+i|k\right)-{\mathrm{SOC}}_{\mathrm{ref}}\right]}^2$$

(13)

where a₁, a₂, and a₃ are the weighting coefficients for wind-farm regulation power, BESS charging/discharging power, and SOC deviation, respectively, and SOC_ref is the reference SOC of the BESS.

2.3.3. Upper-Level MPC Constraints

To ensure that the upper-level MPC solution satisfies the operating requirements of the wind–storage hybrid system, constraints are imposed on the wind-farm regulation power, BESS power, and SOC [29].

The wind-farm regulation-power constraint is given by:

ΔP_w,min(k) ≤ ΔP_{w_ref}(k + i|k) ≤ ΔP_w,max(k)

(14)

where ΔP_w,min(k) and ΔP_w,max(k) are the lower and upper regulation-power limits of the wind farm at time k, respectively, and ΔP_w,ref(k + i∣k) is the predicted wind-farm regulation-power reference.

At each sampling instant, the wind-farm regulation-power limits are determined from the current operating conditions and held constant over the 2 s prediction horizon. These limits are updated at the next MPC step, while possible operating-region transitions within the prediction horizon are not explicitly predicted. This approximation avoids uncertain short-term wind-speed forecasts and simplifies online optimization; however, operating-region transitions within the prediction horizon may temporarily cause discrepancies between the assumed and actual regulation-power limits.

The BESS power and SOC constraints are expressed as:

$$\begin{array}{l}-P_{\mathrm{b},\mathrm{rated}}\le \Delta P_{\mathrm{b}\text{\_}\mathrm{ref}}\left(k+i|k\right)\le P_{\mathrm{b},\mathrm{rated}}\\ {\mathrm{SOC}}_{\min }\le \mathrm{SOC}\left(k+i|k\right)\le {\mathrm{SOC}}_{\max }\end{array}$$

(15)

where P_b,rated is the rated power of the BESS, and SOC_min and SOC_max are the lower and upper SOC limits, respectively.

2.4. Lower-Level QP-Based Intra-Farm Power Allocation

After the upper-level MPC determines the total wind-farm regulation-power reference, the lower level allocates it among the equivalent turbine clusters according to their operating conditions and available regulation ranges. Considering wind-speed differences, wake effects, and the resulting heterogeneous regulation capabilities of the equivalent turbine clusters [30], QP is employed to improve wind-farm power-tracking accuracy and reduce equivalent torque fluctuations while satisfying cluster-level operating constraints. Related wind-farm dispatch studies have jointly considered power-command tracking and mechanical-load mitigation [31,32].

2.4.1. Power Allocation Objective Function

The wind farm is represented by N equivalent turbine clusters, where N = 5 in this study. The regulation-power reference of each cluster is selected as the lower-level optimization variable. At sampling instant k, the optimization variable is defined as:

$$\Delta P_{\mathrm{w}\text{\_}\mathrm{ref}}\left(k\right)={\left[\Delta P_{\mathrm{w},1,\mathrm{ref}}\left(k\right),\dots ,\Delta P_{\mathrm{w},N,\mathrm{ref}}\left(k\right)\right]}^T$$

(16)

where ΔP_w,i,ref(k) denotes the regulation-power reference of the i-th equivalent turbine cluster.

Wind-farm power allocation should consider both regulation-power tracking performance and turbine operating smoothness [33]. The lower-level QP objective function is therefore formulated as:

$$\min J={\lambda }_p{\left[e_p\left(k\right)\right]}^2+{\lambda }_T{\displaystyle \sum _{i=1}^N{\left[\Delta T_i\left(k\right)\right]}^2}$$

(17)

where λ_p and λ_T are the weighting coefficients for the power-tracking error and equivalent torque fluctuation, respectively.

The wind-farm regulation-power tracking error is defined as:

$$e_p\left(k\right)={\displaystyle \sum _{i=1}^N\Delta }{P}_{\mathrm{w},i,\mathrm{ref}}\left(k\right)-\Delta P_{\mathrm{w}\text{\_}\mathrm{ref}}\left(k\right)$$

(18)

where ΔP_{w_ref}(k) is the total wind-farm regulation-power reference generated by the upper-level MPC.

To characterize the torque fluctuation caused by rapid changes in the regulation power of each cluster, the equivalent torque fluctuation of the i-th equivalent turbine cluster is defined as:

$$\Delta T_i\left(k\right)={\displaystyle \frac{\Delta P_{\mathrm{w},i,\mathrm{ref}}\left(k\right)-\Delta P_{\mathrm{w},i,\mathrm{ref}}\left(k-1\right)}{P_{\mathrm{b},i}{\omega }_i\left(k\right)}}$$

(19)

where ω_i(k) is the equivalent rotor speed of the i-th turbine cluster at sampling instant k, and ΔP_w,i,ref(k − 1) is the regulation-power reference of the same cluster at the previous sampling instant. P_b,i denotes the per-unit rated capacity of the i-th equivalent turbine cluster on the system power base. Since each equivalent turbine cluster is rated at 30 MW and the system power base is 100 MW, P_b,i = 0.3 p.u.

2.4.2. Lower-Level QP Constraints

Constraints are imposed on cluster regulation power, rotor speed, and equivalent torque fluctuation to ensure feasible power allocation.

The regulation-power constraint of the i-th equivalent turbine cluster is given by:

ΔP_w,i,min(k) ≤ ΔP_w,i,ref(k) ≤ ΔP_w,i,max(k)

(20)

where ΔP_w,i,min(k) and ΔP_w,i,max(k) are the lower and upper regulation-power limits of the i-th equivalent turbine cluster at sampling instant k, respectively.

The rotor-speed prediction used in the lower-level QP is derived from the incremental rotor dynamics. Over the short allocation interval, the baseline mechanical and electrical powers are assumed to be approximately balanced, and variations in aerodynamic mechanical power are neglected. Therefore, the regulation-power increment is taken as the dominant factor affecting short-term rotor-speed variation, yielding:

$${\omega }_i\left(k+1\right)={\omega }_i\left(k\right)-{\displaystyle \frac{T_s}{2H_i{P}_{\mathrm{b},i}{\omega }_i\left(k\right)}}\Delta P_{\mathrm{w},i,\mathrm{ref}}\left(k\right)$$

(21)

where T_S is the sampling period, H_i is the equivalent inertia constant of the i-th turbine cluster, and P_b,i is its per-unit rated capacity on the system power base. This equation is used only for short-term rotor-speed prediction in the QP, while the complete rotor dynamics are retained in the simulation model.

The rotor-speed constraint is expressed as:

ω_i,min ≤ ω_i(k + 1) ≤ ω_i,max

(22)

where ω_i,min and ω_i,max are the lower and upper allowable rotor-speed limits of the i-th equivalent turbine cluster, respectively.

The equivalent torque fluctuation is constrained by:

−ΔT_i,max ≤ ΔT_i(k) ≤ ΔT_i,max

(23)

where ΔT_i,max is the allowable magnitude of the equivalent torque fluctuation of the i-th equivalent turbine cluster. In this study, ∆T_i,max is adopted to limit excessive instantaneous torque variations.

2.5. Hierarchical MPC–QP Control Procedure

Figure 3 shows the hierarchical MPC–QP control procedure. At each sampling instant, the regulation-power limits are updated according to the system states and wind speed. The upper-level MPC determines the wind-farm and BESS power references, while the lower-level QP allocates the wind-farm reference among the five equivalent turbine clusters.

Figure 3. Hierarchical MPC–QP control procedure for the wind–storage hybrid system.

3. Results

To verify the effectiveness of the proposed hierarchical MPC–QP primary frequency-regulation strategy, a wind–storage hybrid system is modeled in MATLAB/Simulink R2024a (The MathWorks, Inc., Natick, MA, USA). The system base power is 100 MW, and the rated capacities of the thermal power unit and wind farm are 600 MW and 150 MW, respectively. The wind farm comprises 100 DFIG-based turbines rated at 1.5 MW each, represented by five equivalent turbine clusters of 30 MW each. The BESS has a rated power of 15 MW, an energy capacity of 15 MWh, and an initial SOC of 0.5. The main simulation parameters are listed in Table 2.

Table 2. Main simulation parameters of the wind–storage hybrid system.

Symbol	Parameter	Value
R	Thermal-unit droop coefficient	0.04
Hsys	System inertia constant	4
D	Load damping coefficient	2
TG, TCH, TRH	Thermal-unit time constants	0.2 s, 0.3 s, 7 s
FHP	High-pressure turbine power fraction	0.3
N	Number of equivalent turbine clusters	5
v1,v2	Operating-region boundaries	3 m/s, 10 m/s
d	Deloading ratio	10%
Dw	Equivalent turbine rotor diameter	286.98 m
kw,CT	Jensen wake-model coefficients	0.04, 0.8
L	Equivalent cluster spacing	5Dw
Hi	Equivalent cluster inertia constant	4 s
ωmin,ωmax	Rotor-speed limits	0.2, 1.2 p.u.
ΔTi,max	Equivalent torque-fluctuation limit	0.03 p.u.
Pb,rated,Ebess	BESS rated power/capacity	15 MW, 15 MWh
SOCmin,SOCmax	BESS SOC limits	0.1, 0.9
Ts, Np	Sampling period/prediction horizon	0.1 s, 20
λf, λc	Upper-level MPC weights	0.7, 0.3
a1, a2, a3	Regulation cost coefficients	1.0, 1.5, 1.5
Kp, Ki	PI proportional/integral gains	25, 1.8
Ts,QP	Lower-level QP sampling period	0.1 s
λp, λT	Lower-level QP weights	0.7, 0.3

A 1 h simulation with varying wind speed and random load disturbances is conducted at a data sampling interval of 0.2 s. The simulation duration is selected to evaluate repeated primary frequency-regulation responses and associated BESS SOC variations under continuously changing wind-speed and load conditions. The load disturbance is generated by combining three independent Ornstein–Uhlenbeck processes (time constants: 20, 100, and 420 s; weights: 0.25, 0.55, and 0.75), followed by Gaussian smoothing (6 s), mean removal, normalization, and a 30 s cosine startup ramp. To further evaluate performance consistency, three additional stochastic realizations are generated using random seeds of 2066, 3066, and 4066 under identical system and controller settings. The MATLAB script and accompanying README for generating the stochastic wind-speed and load-disturbance inputs are provided in Supplementary File S1. Five equivalent turbine clusters with identical parameters are arranged in a single row at a spacing of five rotor diameters. The first cluster receives the free-stream wind, while downstream wind speeds are calculated using the Jensen wake model with full wake overlap and root-sum-square superposition of velocity deficits. The upper-level MPC uses YALMIP with MATLAB’s quadprog solver, while the lower-level QP employs quadprog with the interior-point-convex algorithm. Figure 4 shows the simulation inputs of the representative case (seed 1066), with the corresponding upper- and lower-level control results presented in Figure 5 and Figure 6, respectively. The statistical results across the three stochastic realizations are reported in Section 3.3.

Figure 4. Simulation test conditions.

3.1. Upper-Level Wind–Storage Coordinated Control Results

For the upper-level comparison, a PI-based wind–storage coordinated control strategy is adopted as the benchmark. The total regulation-power command generated by the PI controller is initially allocated to the wind farm and BESS at a ratio of 0.6:0.4. When the wind-farm power reference exceeds its feasible regulation limits, the excess regulation demand is transferred to the BESS, subject to its power and SOC constraints. SOC-dependent correction and integral anti-windup are incorporated to mitigate SOC deviation and prevent integrator windup under saturation. Both strategies employ the same lower-level QP allocation method under identical wind-speed and load-disturbance conditions. The corresponding results are presented in Figure 5.

Figure 5. Comparison of wind–storage coordinated control responses.

Figure 5. Comparison of wind–storage coordinated control responses.

As shown in Figure 5a, both control strategies respond to the continuously varying wind speed and random load disturbances, while the proposed MPC strategy exhibits smaller frequency deviations, particularly during periods of significant load variations. Figure 5b,c illustrate the coordinated regulation-power responses of the wind farm and BESS, with the MPC strategy producing smoother power adjustments. Figure 5d shows that the SOC under PI control deviates noticeably from its initial value, whereas the proposed MPC maintains the SOC close to 0.5 throughout the simulation.

The quantitative results for the representative case are summarized in Table 3. Compared with PI control, the proposed MPC reduces the RMS frequency deviation and maximum absolute frequency deviation by 24.20% and 24.53%, respectively. The RMS SOC deviation decreases by 93.86%, while the cumulative regulation cost index is reduced by 84.72%. These results indicate that the proposed MPC strategy improves frequency regulation performance and SOC management under the representative simulation conditions.

Table 3. Quantitative comparison of upper-level wind–storage coordinated control performance under the representative case.

Control Strategy	Δfrms/Hz	Δfmax/Hz	SOCrms	Ccost
Proposed MPC	0.0476	0.1529	0.0017	1.5579
PI Control	0.0628	0.2026	0.0277	10.1945

3.2. Intra-Farm Power Allocation Results

For the lower-level comparison, a constrained equal-allocation strategy is adopted as the benchmark. The wind-farm regulation-power reference generated by the upper-level MPC is equally divided among the participating turbine clusters, and the resulting individual power references are limited by their respective feasible regulation-power ranges, as expressed by:

$$\Delta P_{i,\mathrm{ref}}^{\mathrm{avg}}(k)=\min \left\{\Delta P_{i,\max }(k),\max \left\{\Delta P_{i,\min }(k),{\displaystyle \frac{\Delta P_{w\mathrm{f},\mathrm{ref}}(k)}{N_a(k)}}\right\}\right\}$$

(24)

where N_a(k) denotes the number of participating turbine clusters at sampling instant k. Nonparticipating clusters are assigned zero regulation power. Unlike the proposed QP strategy, equal allocation does not optimize the power distribution according to differences in cluster regulation capability. Consequently, individual power limits and turbine response dynamics may cause deviations between the total actual regulation power and the upper-level reference. Both strategies use the same upper-level MPC reference and turbine power-response dynamics. The comparison results are shown in Figure 6.

Figure 6. Intra-farm power allocation results under different strategies.

Figure 6. Intra-farm power allocation results under different strategies.

As shown in Figure 6a, the proposed QP strategy closely tracks the wind-farm regulation-power reference generated by the upper-level MPC, whereas equal allocation exhibits larger tracking deviations during certain periods. These deviations can arise from the cluster-specific power limits and turbine response dynamics discussed above. Figure 6b shows that the two strategies exhibit similar trends in equivalent torque fluctuation, while the proposed QP strategy reduces the overall fluctuation level. As illustrated in Figure 6c, the regulation power is distributed nonuniformly among the five equivalent turbine clusters according to their operating conditions and available regulation ranges, with Cluster 1 generally undertaking larger power adjustments.

The quantitative results for the representative case are summarized in Table 4. Compared with equal allocation, the proposed QP strategy reduces the RMS power-tracking error from 0.0058 to 0.0007, corresponding to a reduction of 87.93%, while the RMS equivalent torque fluctuation decreases from 0.0058 to 0.0049, representing a reduction of 15.52%. These results indicate that the proposed QP strategy improves power-tracking accuracy and reduces the overall level of equivalent torque fluctuation under the representative simulation conditions.

Table 4. Quantitative comparison of intra-farm power allocation performance under the representative case.

Strategy	ep,rms	ΔTeq,rms
Proposed QP	0.0007	0.0049
Equal Allocation	0.0058	0.0058

3.3. Statistical Evaluation Under Multiple Stochastic Realizations

To further evaluate performance consistency, three additional 1 h stochastic simulations are conducted using random seeds of 2066, 3066, and 4066. All strategies are evaluated under identical system settings and disturbance inputs within each case. Table 5 summarizes the mean, sample standard deviation (SD), and worst observed values of the main performance indices.

Table 5. Statistical comparison of control performance under three stochastic realizations.

Performance Index	Proposed	Baseline
Δfrms (mHz)	53.2 ± 7.8 (61.3)	63.0 ± 9.3 (72.6)
Δfmax (mHz)	144.9 ± 0.9 (145.5)	173.8 ± 1.0 (174.4)
SOCrms (×10−3)	1.90 ± 0.30 (2.20)	20.0 ± 7.1 (27.5)
Ccost	2.37 ± 0.54 (2.92)	9.15 ± 2.81 (10.82)
ep,rms (×10−3 p.u.)	0.347 ± 0.053 (0.407)	7.77 ± 1.37 (9.00)
ΔTeq,max (×10−3 p.u.)	16.9 ± 2.0 (19.2)	18.9 ± 2.2 (21.4)

Note: Values are presented as mean ± SD (worst observed value). SD denotes the sample standard deviation, and the worst value is the maximum observed value for each index.

As shown in Table 5, compared with PI control, the proposed MPC reduces the mean RMS frequency deviation, maximum absolute frequency deviation, RMS SOC deviation, and regulation cost index by 15.6%, 16.6%, 90.5%, and 74.1%, respectively. At the lower level, the proposed QP reduces the mean RMS power-tracking error and maximum equivalent torque fluctuation by 95.5% and 10.6%, respectively, compared with equal allocation. The proposed strategies also exhibit lower standard deviations and worst observed values for the reported performance indices. These results indicate consistent performance improvements across the three tested stochastic realizations, although further validation under more diverse operating conditions is required.

4. Discussion

The results indicate that wind–storage primary frequency regulation requires coordination not only between the wind farm and the BESS but also among equivalent turbine clusters within the wind farm. The proposed hierarchical MPC–QP strategy addresses both levels, with improvements in frequency response, SOC regulation, and intra-farm power allocation observed under the simulated conditions.

At the upper level, MPC coordinates the wind-farm and BESS regulation-power references according to the predicted system response and operating constraints. By incorporating the dynamic regulation-power limits of the wind farm together with BESS power and SOC constraints, the controller can redistribute regulation demand between the two resources as operating conditions change. This explains the improved frequency response, reduced SOC deviation, and lower regulation cost index observed in the simulations.

At the lower level, wake effects and wind-speed differences lead to unequal regulation capabilities among the equivalent turbine clusters. Equal allocation does not account for these differences, whereas the proposed QP strategy distributes the wind-farm regulation-power reference according to the available regulation ranges of the clusters while balancing power tracking and equivalent torque fluctuation. Under the simulated conditions, the proposed strategy improves power-tracking accuracy and reduces RMS equivalent torque fluctuation. Equivalent torque fluctuation serves as an indirect indicator of mechanical loading rather than a direct measure of fatigue damage. The three additional stochastic realizations further support the consistency of improvements in the selected performance indices, but do not establish robustness under unmodeled uncertainties or reductions in actual fatigue damage.

The upper- and lower-level optimizations therefore play complementary roles. The upper-level MPC determines coordinated wind–storage power references, while the lower-level QP ensures that the wind-farm reference is reasonably distributed among the turbine clusters. Nevertheless, the present study adopts equivalent turbine clusters, the Jensen wake model, and ideal communication and measurement conditions. Annual energy production and capacity factor are not evaluated in this study, as their reliable estimation requires representative long-term wind-resource data and consideration of turbine availability, deloading operation, and associated energy losses. Future work will consider parameter uncertainty, communication delays, measurement noise, storage degradation, adaptive weighting, and hardware-in-the-loop validation.

5. Conclusions

To improve wind–storage coordination and intra-farm power allocation in primary frequency regulation, this paper proposes a hierarchical MPC–QP control strategy that links operating-region-dependent wind-turbine regulation capability with system-level coordination and turbine-cluster-level power allocation. The main conclusions are as follows:

(1)

The upper-level MPC coordinates wind-farm and BESS regulation power under dynamically updated wind-farm power limits. In the representative simulation case, compared with PI control, the RMS frequency deviation and maximum absolute frequency deviation are reduced by 24.20% and 24.53%, respectively, while the RMS SOC deviation and cumulative regulation cost index decrease by 93.86% and 84.72%, respectively, indicating improved frequency regulation and SOC management under the simulated conditions.

(2)

The lower-level QP achieves differentiated power allocation among equivalent turbine clusters while balancing power-tracking accuracy and equivalent torque fluctuation. In the representative case, compared with equal allocation, the RMS power-tracking error and RMS equivalent torque fluctuation are reduced by 87.93% and 15.52%, respectively, demonstrating improved tracking accuracy and reduced RMS equivalent torque fluctuation.

(3)

Three additional stochastic realizations further demonstrate consistent improvements in the selected performance indices. However, these findings are limited to the equivalent turbine-cluster model, Jensen wake representation, and ideal communication and measurement conditions. Equivalent torque fluctuation is an indirect mechanical-load indicator rather than a direct measure of fatigue damage. Further validation using higher-fidelity models, uncertainty analysis, and hardware-in-the-loop experiments is required.