Automatic identification of wind turbine models using evolutionary multiobjective optimization
Introduction
As wind energy grows across the globe and new offshore wind turbine installations encounter new operating environments, the models that inform the design and control of these multimillion-dollar machines become increasingly important. Typical multimegawatt wind turbines exhibit nonlinear behavior and are subject to wind (and sometimes wave) disturbances that are often hard to estimate. These properties make the simulation of their dynamics not only challenging but also site-dependent, because of the influence of wind, wave, and foundation characteristics. Accordingly, the first-principles models of wind turbines, such as the one embedded in the aero-hydro-elastic simulation tool FAST [1], are prone to cumulative discrepancies between prediction and reality. These models are also computationally expensive to run because of their fairly comprehensive representation of wind turbine dynamics. Although the use of engineering models is fundamental to the structural design and loads analysis process, model-based controllers preferably rely on a customized model of the real system in the field, rather than a first-principles model that may miss key elements present in the real system [2].
As an alternative to potentially inaccurate and computationally expensive first-principles models, empirical models of wind turbines are obtained from experimental data to provide a customized representation of the wind turbine. These models are usually in the form of auto-regressive moving-average (ARMAX) models [2], [3], [4], [5], neural networks [6], or fuzzy logic models [7], among others, to provide the structural flexibility for adapting the model according to the measured observations. Although these empirical models provide an effective means of estimation/prediction, they have the major drawback of lacking transparency about the physics of the process [8]. This lack of transparency obscures the knowledge of the process that is gained through their development. Ideally, the model should not only be accurate, but intelligible so that the user acquires the insight attained through the model's development. A well-formed model serves two purposes: (i) it improves knowledge of the underlying dynamics of the system; and (ii) it improves the ability of the wind turbine controller to extract power and minimize loads on the turbine.
In order to improve the intelligibility of adapted models, empirical models in the form of symbolic equations can be formulated by symbolic regression [9], [10]. In symbolic regression, the process variables, inputs, and parameters (constants) are treated as symbols and integrated as blocks to form candidate model structures. Free of restrictions from the form (structure), the search is typically conducted by genetic programming (GP) for candidate models having the best-fit outputs to the measured observations [9]. However, in the absence of a presumed model structure and guided only by the prediction error (i.e., the difference between the modeled and measured outputs), symbolic regression often yields illegible, albeit accurate, models that do not convey any of the physics of the process. The method proposed for modeling here safeguards against this potential shortcoming by two innovations. First, it uses a novel GP method known as epigenetic linear genetic programming (ELGP) that combines the flexibility of stack-based GP representations with an epigenetic encoding to allow for topological search of the candidate model structures, leading to less complex and more accurate results than traditional GP [11], [12]. Second, it uses an evolutionary multiobjective optimization (EMO) framework [13] that includes the complexity of the model as an objective in order to yield accurate models that are as intelligible as possible.
In this paper we evaluate the applicability of the proposed ELGP method in identifying wind turbine models based on experimental data collected in normal closed-loop operation from the three-bladed Controls and Advanced Research Turbine (CART3), a turbine maintained by the National Renewable Energy Laboratory (NREL). The paper is organized as follows. First, we present a brief overview of wind turbine mechanics. We then review previous system identification work. Next, the problem formulation as sought by multiobjective optimization is presented, followed by a description of the proposed ELGP method. We then detail the wind turbine identification procedure and analyze results pertaining to local and global models of the wind turbine. The paper concludes with a discussion of the intelligibility of the identified models as they inform the physics of the process.
Section snippets
Wind turbine mechanics
Identification of wind turbine models is a difficult undertaking because of the many layers of nonlinearity governing their behavior. Moreover, modern horizontal-axis wind turbines (HAWTs) are controlled using variable-speed and variable-blade pitch operation, further complicating the dynamics. Consider for instance the steady-state aerodynamic rotor torque (QR) and thrust (TR) generated by the rotor operating in freestream wind speed V, defined by:where the
Related work
Most system identification attempts at modeling wind turbines have focused on producing linear time-invariant (LTI) models via ARMAX models [2], [4] or modified forms of closed-loop subspace identification (SSID) [3], [5]. Although LTI models seem to be effective in characterizing simulated wind turbine behavior at specific operating wind speeds [2], [4], they provide only localized representation. As a remedy, SSID methods have been extended to account for the time-varying, nonlinear dynamics
Problem statement
The underlying assumption of symbolic regression is that there exists an analytical model of the system that would generate the measured observations y(tk) at the sample times tk = t1,…,tN under the input, u(t), as:where is the modeled output, ν represents measurement noise in y, x = [x1,…,xn]T is the vector of state variables, and M∗(x,u,Θ∗) denotes the correct model form embodied by the correct parameter values Θ∗, written M∗ hereafter for brevity. In the
Proposed method
In symbolic regression, the search for candidate models is conducted by GP, whereby a population of computer programs that produce models of the process are evolved. Mathematical building blocks compose the genotype of each program, which is optimized by an evolutionary algorithm. The optimization process, shown in Fig. 2, starts with randomly constructed programs that are repeatedly assessed for their fitness and selectively recombined and mutated until an adequate solution is produced.
System identification of CART3
The proposed method is evaluated in application to experimental data from the CART3 system. NREL's CART3 is instrumented with numerous sensors to make the identification of various system models possible. The nature of experimental data available from this system, including its instrumentation, data collection procedure, and control system, is described first. We then describe the settings used for ELGP, including general and problem-specific settings, followed by the types of models considered
Results
The performance of the identified local models by ELGP is summarized in Table 2. The results correspond to the final model with the maximum VAF from training. The results indicate that the accuracy of the models of Ω, ω, and P is excellent in all cases except for , where the response of the system was flat (i.e., var(Ω) < 5e−4), obscuring the dynamics. The DTM model has a better performance for this case. The other two models, MFA and MSS, produce generally accurate outputs, but not
Discussion
The local and global models obtained by ELGP using EMO demonstrate the potential for the succinct resultant models to enhance the understanding of the characteristics observed from a process. We have shown that the models are transparent enough to link process estimation to understandable model components (Eqns. (14), (15), (16)) and accurate enough to shed light on the characteristics of the closed-loop behavior of the system. Furthermore, by studying the archive we are able to see how
Conclusion
In this work we use a novel symbolic regression system in an evolutionary multiobjective optimization framework to identify compact models of a wind turbine from operating data with minor assumptions. The models are not only accurate, but succinct and intuitive, and have been shown to embody process knowledge in several instances. This method of system identification may be a promising middle ground between conducting computationally expensive physics simulations and using black-box models
Acknowledgments
The authors would like to thank Dr. van der Veen for sharing his insights into identification of wind systems. This work is partially supported by the NSF-sponsored IGERT: Offshore Wind Energy Engineering, Environmental Science, and Policy (Grant Number 1068864), as well as Grant Nos. 1017817, 1129139, and 1331283. This work was also supported by the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308 with the National Renewable Energy Laboratory. Funding for the work was
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