With the continuous growth of global energy demand and the tightening of environmental protection regulations, hydropower—characterized by its clean, renewable, and sustainable nature—has increasingly become a vital component of strategic infrastructure in modern power systems (Bladh, 2012). To enhance energy utilization efficiency and optimize the configuration of the AC power grid, hydropower transmission via modular multilevel converter-based high-voltage direct current (MMC-HVDC) systems has gained considerable attention. In addition, with the continuous expansion of hydropower development capacity, numerous high-head and large-capacity hydropower units have been commissioned, posing new challenges to the reliable operation of hydropower transmission systems. To accurately describe the dynamic characteristics of high-head hydropower transmission via MMC-HVDC systems, an electromagnetic transient (EMT) simulation model is required rather than a simplified phasor-domain model. Phasor-domain models, although computationally efficient and suitable for long-term electromechanical studies, neglect high-frequency switching harmonics, fast control dynamics, and electromagnetic interactions that are prominent in MMC-based HVDC systems—particularly during fault transients and rapid control actions. The EMT model, by resolving sub-millisecond time steps, captures detailed converter switching behavior, control-loop dynamics, and the coupling between electrical and mechanical subsystems, which are critical for accurately assessing transient stability and dynamic performance under severe disturbances. Although EMT simulations incur substantially higher computational costs compared to phasor-domain models, the computational burden in this study is mitigated through Sobol sensitivity analysis for parameter space reduction and the use of an Adaptive Cuckoo Search (ACS) algorithm optimized for faster convergence. As a dominant part of the EMT model, the parameters in the model have a significant impact on the accuracy of the simulated response (Barros et al., 2003). Consequently, conducting high-precision identification of these parameters is not only theoretically significant for constructing accurate hydro-mechanical-electrical coupled models, but also of substantial practical value in enabling frequency regulation and peak shaving (Alvarez, 2020), fault prediction (Quintana and Van Cutsem, 1988), and stability analysis (Zarco and Exposito, 2000).

The main technical bottlenecks in parameter identification stem from the following complex characteristics: The foremost challenge lies in the inherently multi-physical coupling of unit dynamics, wherein dynamic responses are governed by the interplay among hydraulic, mechanical, and electrical domains (Zarco and Exposito, 2000), compounded by variations in operating conditions and coordinated control strategies (Rakpenthai et al., 2012). In high-head hydropower systems, such coupling is further complicated by phenomena like the water hammer effect in long penstocks, where rapid load changes induce pressure waves that interact with both the turbine and the electrical control system. These factors introduce significant uncertainty into model-based inverse parameter estimation. Furthermore, the control systems of hydropower units exhibit strong nonlinearity and hysteresis (Guo et al., 2014), for example, the nonlinear dead zones and rate-dependent hysteresis in turbine-governor servomotors, making traditional frequency-domain-based linear system identification approaches inadequate for capturing the unit’s dynamic stiffness matrix, thereby compromising identification fidelity (Rakpenthai et al., 2012; Petra et al., 2017). In addition, random disturbances in operational environments—such as non-stationary hydrological inputs and stochastic power grid load fluctuations—result in multi-source, non-stationary noise contamination of measured data (Zeng and Teng, 2011), greatly reducing the signal-to-noise ratio of input–output data pairs. A particularly critical challenge is the time-varying drift of system parameters, which poses a serious threat to model robustness (Milojević et al., 2018). Studies suggest the need for time-varying parameter identification frameworks with dynamic tracking capabilities (Mukherjee et al., 2020). By integrating online adaptive algorithms, continuous rolling updates to model parameters can be achieved, thereby enhancing identification accuracy over the entire lifecycle of the unit (Regulski et al., 2015). Addressing these issues necessitates the development of high-performance parameter identification methodologies.

To this end, researchers have explored a range of algorithms for parameter identification in hydro-turbine generator units, achieving notable progress. The majority of these approaches focus on improving global search capabilities, yet their time and space complexity remain largely unexplored.

For instance, an improved particle swarm optimization (PSO) algorithm introduced in Fang et al. (2011) leverages adaptive learning factors to enhance global search capability, significantly improving the tuning of PID controllers. However, the time complexity of this algorithm remains high, which limits its scalability for large systems. Another study (Liu et al., 2010) combined PSO with the uniform design method to optimize turbine governor parameters, demonstrating superior performance in complex systems. This approach has proven effective in improving parameter accuracy but struggles with higher computational demands, especially in systems with a large number of parameters.

Recent hybrid optimization strategies attempt to address these limitations. The hybrid moth-flame-PSO (HMFPSO) approach in Shaikh et al. (2023) integrates exploration-exploitation balancing mechanisms, achieving faster convergence in transmission line parameter estimation. While effective for mid-scale systems, its O(N²) complexity and sensitivity to control parameters hinder deployment in real-time large-grid scenarios. Similarly, grey wolf optimization (GWO) in Shaikh et al. (2021) reduces parameter dependencies but exhibits premature convergence when handling non-convex landscapes in three-phase power system.

Genetic Algorithms (GAs), as well-established global optimizers, have also found wide application in this domain. The method proposed in Gao et al. (2009), based on an enhanced GA incorporating chaotic mutation, achieved improved global search capability and was successfully applied to fluid transient process modeling. However, the time complexity for this method is O(N^2), which may limit its real-time application in large-scale systems. Similarly (Jiang et al., 2006), highlighted the GA’s effectiveness in PID parameter optimization, particularly in enhancing system stability and responsiveness. For multimodal optimization problems in hydro units, the Bacterial Foraging Optimization Algorithm (BFOA)—an algorithm inspired by bacterial foraging behavior—has demonstrated strong performance. In Kou et al. (2010), BFOA was used to identify turbine governor parameters, showing high robustness in handling complex scenarios, but its performance deteriorates when applied to large-scale systems due to its exponential time complexity.

Recently, Gravitational Search Algorithms (GSA) and their improved variants (IGSA) have emerged as efficient solutions for parameter identification. IGSA, as proposed in Chen et al. (2014a), integrates PSO’s velocity update mechanism with chaotic mutation, resulting in accelerated convergence and improved global search. The time complexity of IGSA is O(N log N), similar to PSO, yet it has shown faster convergence in practice. To address system uncertainties (Chen et al., 2017), proposed three novel identification approaches using distinct parameter observers based on system stability theorems, as well as an Ant Lion Optimizer (ALO)-based method. In this work, the ALO-based method demonstrated superior accuracy compared to PSO and GA-based methods, with the added benefit of reducing computational time, making it more efficient in real-time applications.

In turbine control system modeling (Zhang et al., 2018), presented a hybrid approach combining white-box mapping with a radial basis function (RBF) neural network. This method, particularly effective under data-scarce conditions, minimized reliance on large-scale testing, demonstrating practical viability. Finally (Chen et al., 2014b), focused on designing fractional-order PID controllers for turbine governor systems and employed a chaotic NSGA-II algorithm to optimize controller parameters. Results showed that fractional-order PID controllers outperformed traditional PID controllers in terms of control precision and response time, although the optimization process exhibits O(MN²) complexity, primarily dominated by its non-dominated sorting mechanism.

Collectively, these studies reveal three persistent challenges: (1) the complexity-accuracy trade-off in population-based algorithms, (2) poor generalization of hybrid strategies across varying power system topologies, and (3) limited theoretical guarantees for convergence in non-convex landscapes. This work addresses these gaps through a computationally constrained co-evolutionary framework, systematically optimizing time complexity while maintaining solution robustness for large-scale hydro-turbine systems. To address this critical gap, this work systematically reviews and analyzes existing PI algorithms for hydro-turbine units, focusing on identifying their inherent computational complexities and scalability limitations. We then propose a novel parameter identification method that integrates Sobol sensitivity analysis and an Adaptive Cuckoo Search (ACS) algorithm. The Sobol sensitivity analysis is used to calculate the sensitivity index of candidate parameters, from which dominant parameters with high sensitivity are selected for identification. Subsequently, ACS is employed to identify these dominant parameters efficiently. Finally, the accuracy of the proposed method is validated through a case study conducted on the CloudPSS simulation platform, demonstrating its effectiveness in reducing computational burden while maintaining high accuracy for high-head hydropower MMC-HVDC EMT model identification.

The remainder of the manuscript is presented in five sections. In Section 2, the EMT model of high-head hydropower transmission MMC-HVDC system is constructed. Section 3 introduces the screening method of dominant parameters according to the Sobol sensitivity analysis. The traditional CSA method and the ACS method are dis-cussed in Section 4. In Section 5, a simulation case is conducted. Section 6 concludes the manuscript.

There are a large number of parameters in the established hydropower MMC-HVDC system model. They have different impacts on the dynamic response of the system. If the comprehensive parameter identification is directly carried out, it is easy to lead to the problems of complex identification process, large amount of calculation and poor convergence of identification results. Therefore, it is necessary to select the dominant parameters based on the sensitivity analysis before the parameter identification, which can significantly improve the efficiency and accuracy of identification.

As a well-known sensitivity analysis method, the Sobol global sensitivity analysis is adopted in this manuscript to determine the dominant parameters. It is a variance-based decomposition technique, quantifies the influence of individual parameters and their interactions on system dynamics by orthogonally decomposing the out-put variance of a computational model. Compared with alternative global sensitivity methods such as the Morris method or the Fourier Amplitude Sensitivity Test (FAST), Sobol analysis offers higher accuracy in quantifying both first-order effects and higher-order interaction effects. This capability is particularly critical in the context of high-head hydropower MMC-HVDC EMT models, where strong multi-physical coupling (e.g., between hydraulic transients, mechanical inertia, and converter control loops) can lead to significant parameter interaction effects that simpler screening methods may overlook. Although Sobol analysis typically incurs higher computational costs than Morris or FAST, this study mitigates the computational burden by first constraining the candidate parameter set to those with potential physical significance, and then applying parallelized simulations within the CloudPSS environment to accelerate the variance decomposition process. This work applies this methodology to dominant parameter identification in power system EMT models, where the impedance response Y = f(X) is modeled as a function of a d-dimensional uncertain parameter vector X=(x ₁,x ₂, … ,x _d)∈Φ _d, with Φ _d denoting the parameter domain.

According to Sobol’s decomposition theorem, when f(X) satisfies square-integrability, the model output can be uniquely expressed as shown in Equation 19. Y = f 0 + ∑ i = 1 d f i x i + ∑ 1 ≤ i < j ≤ d f i j x i , x j + ⋯ + f 1 , 2 , … , d x 1 , x 2 , … , x d (19) where f ₀ represents the constant term, f _i(x _i) captures the independent effect of x _i, and f _ij(x _i,x _j) quantifies pairwise interactions. These components satisfy the orthogonality condition given in Equation 20. ∫ Φ f i 1 , … , i s x i 1 , … , x i s d x k = 0 , ∀ k ∈ i 1 , … , i s (20)

The total variance decomposition is given in Equation 21. Var Y = ∑ i = 1 d V i + ∑ 1 ≤ i < j ≤ d V i j + ⋯ + V 1 , 2 , … , d (21) where V i = Var E Y | x i denotes the first-order contribution of x _i, and V i j = Var E Y | x i , x j − V i − V j characterizes pairwise interactions, and higher-order terms account for multi-parameter synergies.

The first-order Sobol index S _i and total Sobol index S _Ti are defined in Equation 22: S i = V i Var Y , S T i = E x − i Var x i Y | x − i Var Y (22) where S _i represents the individual influence of x _i and S _Ti incorporates all higher-order interactions involving x _i (Spall, 2003).

For numerical implementation, an enhanced Monte Carlo sampling scheme is adopted:

Generate an N×2d sample matrix, splitting it into submatrices A (first d columns) and B (last d columns).

This algorithm achieves unbiased estimation of parameter sensitivity through N model evaluations, demonstrating particular efficacy for high-dimensional nonlinear systems like power system EMT models. In this manuscript, the X denotes the parameters requiring identification within the hydropower MMC-HVDC system, primarily the PI controller settings; Y corresponds to the active and reactive power observed during a phase-to-phase short circuit. Based on the Sobol index analysis, parameters associated with larger S _Ti values are selected as dominant and subsequently identified using the proposed parameter identification method.

A parameter identification method for a high-head hydropower flexible DC transmission system, based on Sobol sensitivity analysis and an adaptive cuckoo search algorithm, is presented in this manuscript. Our theoretical framework and simulation studies demonstrate that the proposed method can effectively achieve accurate parameter identification for the system model. The simulation indicates that the introduced approach can obtain the better optimization results when compared with other optimization algorithms. Moreover, the simulation results verify that the identified parameters remain accurate under different operational conditions. The proposed method enables precise modeling of the high-head hydropower flexible DC system, ensuring that simulation outcomes faithfully reflect the behavior of the actual physical system. This contributes to improved stability and reliability in flexible DC grid-connected hydropower operations. Future research will focus on extending the proposed approach to larger-scale systems and more complex parameter sets, assessing its performance in higher-dimensional search spaces and under diverse optimization landscapes. This may include applying the method to different power system component models, incorporating distributed or parallel computing strategies to enhance efficiency, and evaluating its robustness in scenarios with stronger non-convexity or lower signal-to-noise ratios. Additionally, further investigation into real-time stability analysis methods based on the identified parameters, especially under fluctuating hydraulic conditions, would provide added value to the robustness of system operation.