The cold front propagation model is structured into two primary components: pre-observation and the propagation model. In the pre-observation phase, multi-point monitoring employing Kalman filtering processes wind speed and direction data to extract gust core parameters. Subsequently, the propagation model then computes the temporal sequence of gust impacts on the turbine for each inflow layer, utilizing these core parameters.

The Blade Element Momentum (BEM) theory is commonly employed to calculate the rotor loads of wind turbines subjected to non-uniform inflow conditions, such as those generated by cold fronts [24]. In this approach, each blade is discretized radially into multiple independent segments, termed blade elements. For each element the lift forces dL and drag forces dD are computed based on the AOA determined by the relative inflow velocity. Subsequently, these aerodynamic forces are resolved to determine key rotor performance and structural parameters, including the axial force dFaxi and tangential force dFtan, which can be calculated as follows, (9){dFaxi=dLcosφ+dDsinφdFtan=dLsinφ−dDcosφ,where φ is the inflow angle.

Building upon the previous description, the calculations for the key parameters are detailed as follows. The torque exerted on the rotor is generated by the tangential forces acting on the blade elements. Therefore, the torque dTt,i can be obtained by integrating the contributions of the tangential forces across the element, as given by, (10)dTt,i=dFtan∑i=1lri,where ri is the radius corresponding to the ith blade element, l is the total number of blade elements, rhub is the blade root radius.

The blade root load (bending moment) Mroot, which characterizes the structural load at the root, arises from the combined effects of tangential and axial forces distributed along the blade, and is computed as, (11)Mroot=∫rhubRr×dFdr=∫rhubR(00r)×(dFtandFaxi0)dr,where R is the rotor radius.

Finally, the maximum shear stress τmax on each blade element is determined based on the shear force and the cross-sectional area, using the expression, (12){V(r)=Fri(r)=∫rriR(dFaxidr)2+(dFtandr)2drτmax(r)=32V(r)A(r),where Fri is the total force at the selected spanwise location, rri is the radius of the selected point, A is the cross-sectional area at the selected point.

The total axial force FFA and lateral force FSS acting on the nacelle are determined based on the axial and tangential forces of the blade elements calculated from the wind turbine rotor load model, (13){FFA=∑j=1ldFaxi(φm,j)+∑j=1ldFaxi(φm+120,j)+∑j=1ldFaxi(φm+240,j)FSS=cos⁡(φm)∑j=1ldFtan(φm,j)+cos⁡(φm+120)∑j=1ldFtan(φm+120,j)+cos⁡(φm+240)∑j=1ldFtan(φm+240,j),where φm is the corresponding blade rotation angle.

The tower is simplified as a two-degree-of-freedom system with a lumped mass at the top, which represents the fore-aft (FA) and side-to-side (SS) motions. The mass matrix Mmass, damping matrix C, stiffness matrix K, time-varying excitation force vector F(t)=[FFA(t);FSS(t)], and displacement vector x are defined, (14){Mmassx¨+Cx˙+Kx=F(t)x=[xFA;xSS],where xFA and xSS are the FA and SS displacements, respectively.

The fundamental frequency f of the tower is calculated from the mass m and stiffness k. (15)f=12πkm.

Finally, the solution is derived from the state-space equations, and the time-domain response is transformed to the frequency domain using the Fast Fourier Transform to identify the dominant frequency components of the vibration.

In the non-uniform inflow environment induced by cold fronts, unlike yaw-controlled turbines under uniform inflow conditions, the wind speed and direction experienced by the rotor vary with height in a gradient manner. This non-uniform inflow introduces both wind speed gradients and wind direction gradients. Consequently, in addition to conventional blade root load optimization, it is essential to optimize the maximum shear force on the blades to reduce the ultimate stress on the materials.

Furthermore, cold fronts accompanied by gusts can cause rapid changes in wind direction. As a result, the yaw servo system of wind turbines often fails to respond in real time, leading to a passive yaw state. This state, in turn, causes in a sharp decline in active power. Therefore, increasing the average rotor torque has become a key optimization objectives to mitigate the rate of power decrease.

Currently, optimization processes in this field commonly employ algorithms such as PSO. For multi-objective optimization problems, multiple objectives are typically integrated into a single comprehensive objective function by assigning weights to each target. In this study, based on the expert scoring method, weights k1, k2 and k3 are assigned to the blade root load Mroot, shear stress τmax, and torque T, respectively. Multi-objective optimization is then achieved through a comprehensive evaluation value P, (16)maxβP(β;un,θn,h,ϕt)s.t.β∈[−45,45]P=k1Mroot+k2τmax+k3T,where h is the corresponding height plane, ϕt is the yaw orientation of the wind turbine, β is the pitch angle.

Optimization is performed using the PSO algorithm with the following parameters: Cognitive coefficient c1 is 1.2, social coefficient c2 is 1.2, inertial weight ω decreasing linearly from 0.9 to 0.4, and a maximum iteration count of 200. The torque weight is 0.2, shear force weight is 0.15, blade root moment weight is 0.65, and blade pitch rate is 7∘/s. It should be noted that the multi-objective PSO method based on weight allocation introduced here is primarily used for comparative analysis with the proposed adaptive AOA difference IPC strategy to objectively evaluate the performance advantages of the proposed method. This conventional approach is not the focus of innovation in this study but serves as a benchmark for performance validation.

IPC optimizes performance by independently adjusting the pitch angle of each blade to maintain the AOA at its aerodynamically optimal design position. However, under non-uniform inflow induced by cold fronts or during yaw misalignment, significant variations occur in the actual inflow angles of different blade elements along the span. In such scenarios, a comprehensive blade AOA for each sector can be calculated based on the contribution ratio of each blade element to the blade root bending moment and its actual inflow angle, (17){W=∑i=1Nciriφbn=1W∑i=1Nciriφi,where W is the total blade torque parameter, calculated from the chord length ci of the ith blade element and the radius ri at which the element is located, φbn is the CIA of sector n, φi is the actual inflow angle of the corresponding blade element.

The optimization results of the PSO algorithm for the wind turbine under a yaw orientation of 30° and a gust condition of 60° are compared with the CIA, as shown in Fig. 4.

Fig. 4 shows that the CIA and the PSO-optimized pitch angle exhibit a generally consistent overall trend, indicating that the CIA influences the variation of the optimized AOA.

As shown in Fig. 5, which presents the torque data Mz, total loads data on FA Px and SS Py of the turbine under the same conditions as in Fig. 4, both the loads and torque exhibit cyclic variations correlated with the rotational period of the rotor. This result indicates that sector-bound cyclic individual pitch data for the blades can be derived through sector-based CIA–AOA differential individual pitch control.

Using the CIA as a reference, the required pitch adjustment can be determined by calculating the difference between the CIA within a sector and the optimal AOA. Adopting a method similar to Eq. (16), the optimal AOA αon is obtained by calculating the optimal AOA for each blade element. Based on the CIA and the optimal AOA for each sector, the corresponding optimal pitch angle βcn is then derived, (18)βcn=φbn−αon.

As shown in Fig. 6, although the variation trend of the optimized angle exhibits a certain correlation with the CIA, differences in the final optimized angle and corresponding results arise due to the need for independent optimization across sectors, each with different sensitivities to the three optimization objectives. These discrepancies primarily stem from the algorithm settings: within a search range of 20° and a maximum of 200 iterations, it cannot be guaranteed that particles in all sectors converge uniformly. Additionally, constraints such as the pitch rate limit and the torque-dominant weighting of the optimization objectives collectively contribute to deviations in peak regions. In the 180° to 360° interval, however, the amplitude of the AOA varies less, leading to closer alignment between the two angles and demonstrating better consistency. It should be noted that the optimization process was conducted in sectors spaced at 20° intervals from 0° to 340°. Since 0° is equivalent to 360°, the decline observed between 340° and 360° in the figure is not directly caused by the optimization process.

Owing to the sectional structure of a cold front, the gust front at different heights of the wind turbine rotor can be approximated as straight lines. Consequently, for a turbine subjected to a cold front, multiple coverage states are defined based on the percentage of rotor height affected by the cold front.

As illustrated in Fig. 7, the rotor is divided into four equal segments, yielding seven distinct coverage states. In Fig. 7, blue represents the pre-cold-front wind field, yellow denotes the mixing zone, and red indicates the cold pool region.

First, the effectiveness of the optimization results and the adaptive AOA difference IPC strategy is compared. Evaluations are conducted across all seven coverage states, with results benchmarked against the baseline case without pitch control, as summarized below:

As shown in Table 3, the results of optimizing the three objectives with corresponding weights using PSO are presented. Among these, the Type 4 (full coverage) state achieves the optimal load optimization performance. This outcome arises because, under full coverage, the wind speed and direction gradients across the rotor are smooth and continuous without abrupt changes. In contrast, other coverage states consist of a mixture of the mixing zone, gradient wind, and gust components, resulting in uneven wind speed and direction gradients along the height axis, which diminishes the optimization effect. Under the same conditions, using pitch data derived directly from the CIA and the optimal AOA as input yields the results shown in Table 4.

The results optimized using the AOA difference approach place greater emphasis on improving load performance. As shown in Table 4, compared to the PSO optimization results, the blade root load optimization improved by an average of 64.84%, and the shear stress optimization improved by an average of 66.35%. However, the torque optimization results decreased by an average of 43.37%. This phenomenon occurs because the PSO optimization method, based on a weighted multi-objective function, optimizes each target to varying degrees according to the assigned weights. Overall, the adaptive AOA difference control strategy achieves strong comprehensive optimization performance and demonstrates superior effectiveness in load optimization.

To further compare and validate the adaptive AOA difference control strategy, a two-degree-of-freedom nacelle dynamic model for FA and SS motions was developed, incorporating tower vibration characteristics. The vibration responses under three control schemes were analyzed: fixed 0° pitch angle, PSO-optimized pitch angle, and AOA difference-based pitch angle.

Fig. 8a,b displays the vibration response without pitch angle control in FA and SS, where the red and green dashed lines denote the rotor rotational frequency and its third harmonic, respectively. The vibration energy in the FA direction concentrates at the third harmonic, a phenomenon attributed to the excitation of fundamental tower vibration caused by the coupling between non-uniform inflow and wind turbine blades. In contrast, the SS vibration is dominated by the second harmonic, resulting from inflow imbalance between upper and lower sectors due to wind speed and its gradient.

Fig. 9 illustrates the vibration response under PSO-optimized control. Compared to the uncontrolled case, the non-smooth pitch angle transitions in the PSO strategy (as shown in Fig. 4) induce significant fluctuations in the load curves (Fig. 9a). Consequently, the FA and SS accelerations in Fig. 9b exhibit higher amplitudes at both the third and second harmonics relative to the uncontrolled scenario. Additionally, amplitudes from the third to the eighth harmonic in the SS frequency domain generally increase. These findings underscore a critical drawback of PSO-based control: while it reduces rotor fatigue loads, its abrupt control actions exacerbate tower vibrations, potentially compromising structural integrity in practical applications. This limitation highlights the need for smoother control transitions to minimize adverse effects on tower fatigue performance.

Fig. 10 presents the vibration response under the AOA difference control strategy. Although this method also causes fluctuations in the load curves (Fig. 10a), its variations are smoother compared to PSO control. In the frequency domain (Fig. 10b), the amplitude of the third harmonic in the FA direction is lower than under PSO control, with only a minor peak at the second harmonic; responses in other frequency bands closely resemble the uncontrolled case. For SS vibration, the peak at the second harmonic is reduced relative to the uncontrolled scenario, and the amplitude at the third harmonic is also lower than in PSO control. A slight increase occurs at the fourth harmonic compared to PSO control, while components at the fifth harmonic and above are nearly eliminated. This smoother response demonstrates the superior adaptability of AOA difference control, as it effectively mitigates vibration magnitudes and attenuates high-frequency components without introducing excessive fluctuations. Such characteristics are vital for real-world wind turbines, where dynamic environmental conditions demand robust and stable control strategies.

To quantitatively assess the control strategies, Table 5 summarizes the maximum and harmonic acceleration responses extracted from the frequency domain analyses in Figs. 7–9. The data reveal that the PSO-optimized control, which aims to balance multiple objectives such as reducing fatigue loads and enhancing torque, leads to a 37.17% increase in FA acceleration and a dramatic 211.63% rise in third harmonic SS acceleration compared to the normal case. This vibration amplification stems from the inclusion of torque in the weighting setup of the PSO optimization. To enhance torque, the PSO strategy increases energy extraction from the wind. However, due to the sector-by-sector optimization process, the control actions become non-smooth, resulting in fluctuating energy absorption and consequently elevated vibrations. In practice, adjusting these weights for different scenarios would require repeated optimizations and corrections based on evolving baseline data, significantly increasing operational complexity, although this aspect is not directly reflected in the presented figures. Moreover, incorporating vibration metrics into PSO optimization introduces significant computational challenges. Since vibration analysis relies on time-domain simulations that require iterative computational steps, adding this as an objective expands dimensionality and drastically slows optimization speed, whereas other objectives like torque or load calculations can be resolved more efficiently in several steps.

In contrast, the AOA difference control shows only a 6.63% increase in FA acceleration and a 3.71% reduction in SS acceleration, with a minimal 8.03% increase in third harmonic component, demonstrating a more balanced performance that effectively suppresses vibrations without imposing excessive computational burdens. This highlights that while PSO control may achieve data-level trade-offs, its practical utility is limited by vibration penalties and optimization inefficiencies, whereas the AOA strategy offers a more adaptable and scalable solution for real-world wind turbine deployments.

In summary, during cold front transit, dual gradients in wind speed and direction create non-uniform inflow across the rotor, exciting not only the third harmonic (related to the blade count) but also the second and fourth harmonics due to inflow imbalance. Direct application of PSO-optimized control may amplify vibration amplitudes from the first to the fifth harmonics, whereas the AOA difference control achieves comparable performance while suppressing vibration magnitudes and attenuating high-frequency components, demonstrating enhanced adaptability. From a practical perspective, the AOA difference control strategy offers significant benefits for wind turbine operations, including potential extensions in component lifespan and reductions in maintenance costs through minimized structural vibrations. However, critical challenges remain, such as the need for accurate sensor data and computational efficiency in real-time implementation. Future work should focus on integrating this strategy with other adaptive controls to further enhance reliability and scalability in diverse wind conditions.