As explained in Section 3.1.1.4, model validation is an important step in SI. The first model as- sessment is computed on the training data and, if the results are not satisfactory with the training data, the model cannot be accepted. In this case, the problem may be an incorrect selected model, which is not able to describe the complexity of the process [205], or the identification algorithm
may provide a parameter vector too far from the optimal solution. In a case where the performance achieved on the training data is acceptable, the real validation assessment comes when the model is tested on new data, that is, when the model has to predict the outcome of an experiment which was not used for the model parameter identification [278]. In order to deliver the training and validation steps, the available data is divided into separate training and validation data sets [205] [221]. It is important to remember that a model, which fits the training data well, is not necessary a good model on different experimental data (i.e. overfitting problem). Common methods utilised for model validation are the 1-step (ahead) prediction test and the multi-step (ahead) prediction test [221]:
• 1-step (ahead) prediction test. The model output prediction, ˆy, is calculate by utilising only the process measurements u and y (no predicted past values of the output, ˆy, are used), as shown in Fig. 3.28(a). At each time step, the prediction error e(k) = y(k)− ˆy(k) is computed. 1-step predic- tion tests are often not sufficient to show model inadequacy and even models with low accuracy may generate almost perfect 1-step predictions [221].
• Multi-step (ahead) prediction test. The mathematical model is initialised by a few known mea- sured output values and, successively, the model output is calculated by the previous model pre- dicted output, ˆy, and by the given measured input, u, as shown in Fig. 3.28(b). Therefore, at each time step the output error e(k) quickly accumulates.
u(k − nd) u(k − nd− 1) u(k − nd− nb) y(k − 1) y(k − na) Model ˆy(k) u(k − nd) u(k − nd− 1) u(k − nd− nb) ˆy(k − 1) ˆy(k − na) Model ˆy(k) (a) (b) Predicted output Measured output Measured
input Measuredinput
Figure 3.28: (a) 1-step prediction (b) Multi-step prediction.
Different error metrics can be utilised to compare the model prediction, ˆy(k), with the mea- sured signal, y(k) (either for 1-step or multi-step predictions):
• The mean squared error [279]. As explained in Section 3.3, the MSE is defined as: MSE = 1 N N
∑
k=1|y(k) − ˆy(k)| 2, (3.93)where the constant N1 is commonly replaced by 12 or 2N1 [205]. In this section, the constant N1 is utilised to maximise the compatibility with the other error metrics introduced. The main disad- vantage is that the MSE is not normalised with respect to the magnitude of y(k); therefore, it is difficult to compare different fitting results.
• The mean absolute percentage error (MAPE). In order to obtain a metric normalised with respect to the magnitude of y(k), the MAPE is defined [279] as:
MAPE =100 N N
∑
k=1 |y(k) − ˆy(k)| |y(k)| (3.94)However, a drawback of the MAPE is that it can give a distorted picture of the error, in the case where there are zero (or nearly-zero) values in the measured signal, y(k). Since, in the context of wave energy, it is common that y(k) oscillates around the zero value, there is a good possibility of obtaining a distorted picture of the error. For example, Fig. 3.29 shows the presence of spikes of the quantity |y(k) − ˆy(k)|/|y(k)|, when the measured displacement, y(k), crosses the zero value. • The normalised root mean-squared error (NRMSE), defined [8] [12] as:
NRMSE = ky(k) − ˆy(k) k2 k y(k) k2 = q 1 N∑Nk=1|y(k) − ˆy(k)|2 q 1 N∑Nk=1|y(k)|2 , (3.95)
has the advantage of being normalised with respect to the magnitude of y(k) and, at the same time, the presence of nearly-zero values in the measured signal y(k) does not alter the picture of the error. 1680 1700 1720 1740 1760 1780 1800 1820 −0.05 0 0.05 k Displacement (m) Data Prediction 1680 1700 1720 1740 1760 1780 1800 1820 0 50 100 |y (k )− ˆy( k )| |y (k )| k (a) (b)
Figure 3.29: (a) Time evolution of the body displacement provided by the experimental data and predicted by the model. (b) Time evolution of the quantity |y(k) − ˆy(k)|/|y(k)|.
3.5 Summary and discussion
This chapter has introduced the fundamental aspects of SI, a discipline well known in a variety of engineering subjects; in this thesis, SI is applied for WECs modelling. Section 3.1 shows that SI is based on an iterative sequence of four steps (experiment design and data gathering, model order and structure selection, fitting criterion and identification algorithm selection, and model validation). At the end of the SI procedure, an identified model is provided. It is crucial to stress that a model is identified on some training data, but the quality of the model has to be evaluated on some fresh data (validation data); indeed, a model not able to generalize on new data is a relatively worthless model, but just provides an imperfect copy of the original training data, with no useful
interpolation value. Usually, the use of SI to resolve an engineering problem does not lead to an ideal unique answer, but provides a variety of possible solutions, depending on decisions taken during the SI procedure, which could be all correct and reasonable. Furthermore, the definition of a ‘best’ model may change from application to application; the best model usually meaning the best compromise for the specific application. The ideal compromise sought in this thesis is that between a high model prediction accuracy (increased with the introduction of nonlinearity) and a low computational requirement. In Section 3.2, by starting from Cummins’ equation, different CT model structures have been obtained, and utilised as a guideline for the construction of DT model structures, obtaining grey-box models such as FBO and Hammerstein model structures. Furthermore, DT black-box model structures are proposed, such as ARX, KGP and ANN models. In Section 3.3, the use of linear and nonlinear optimization for the identification of the parameters of WEC model structures is explained, underlining the crucial difference between a model with a nonlinear input/output relationship and a model nonlinear in the parameters. Indeed, it is pos- sible to have models with a nonlinear input/output relationship but being linear in the parameters (such as Hammerstein and KGP models), which have the relevant advantage of requiring only linear optimization for parameter identification. On the other hand, ANNs are nonlinear both in the input/output relationship and in the parameters, with the drawback of requiring a nonlinear optimization. In Section 3.4, the model validation concept is introduced, and the importance of the model testing on new data is stressed. The important SI step, regarding experiment design and data gathering, is introduced in Section 3.1.1.1 and will be comprehensively illustrated in Chapter 4, in the specific case of wave tank experiment design for WEC model identification.
Chapter
4
Wave tank experiment design for WEC
model identification
4.1 Introduction
In Section 3.1.1.1, the importance of utilising informative data for system identification, especially in the presence of nonlinear systems, was introduced. This chapter focuses on an experiment de- sign in wave tanks (for both NWT and RWT), in order to generate data for WEC model identifica- tion. In recent years, a significant variety of different WECs have been studied in RWTs, in order to analyse their hydrodynamic characteristics. Often, the experiments are based on a WEC subject to incoming waves, which can be regular monochromatic waves or irregular waves with a specific wave spectrum (wave spectra are usually characterised by a significant wave height and a peak period). The data generated by these kinds of experiments can be utilised for WEC model identifi- cation, but the possibility of designing an experiment, specifically for WEC model identification, can render the generated data more informative and effective, in order to obtain an accurate iden- tified model. It is important to stress that, by extending the duration of an experiment, the amount of information contained in the data usually increases but, at the same time, there are increasing disadvantages, for both NWT and RWT cases. Indeed, in the case of a CFD-NWT, the amount of computation time can become unsustainable (the computation time could be up to 1000 times the simulation time, as suggested in Section 2.3) whereas, in the case of a RWT, a set of long tank experiments corresponds to an increase of the facility renting costs. In the wave energy commu- nity, it is usual to use monochromatic waves in order to study the WEC hydrodynamic behaviour at a specific wave frequency and amplitude; therefore, in order to cover the whole input range in both frequency and amplitude (in this case the input is the FSE), the number of experiments quickly becomes significantly large, leading to a considerable experimental time to collect all the required information. An interesting alternative is the use of ‘time-shrunk’ input signals, charac- terised by a high concentration of information. These particular signals contain the whole variety of frequencies and amplitudes, necessary to excite the system over the whole range of operation, in a more compacted time frame. In the case where the mathematical model structure, utilised to model the process, is available before the experiment realization, the experiment design can be even more optimised. Indeed, it is usual to describe a complex system as the interconnection of smaller sub-systems, each one having a specific input and output. Therefore, the design of a set of experiments, each one designed to excite a different sub-block, leads to the direct simula- tion/measure of the input and output of each sub-system, providing the data for the identification of each sub-block.
This chapter is laid out as follows: Section 4.2 describes the design of excitation signals for data generation, with Section 4.2.1 explaining the main excitation signal characteristics (spectral content, amplitude range and amplitude distribution) and Section 4.2.2 illustrating a variety of pos-
sible input excitation signals. In Section 4.3, different typologies of WT experiments are shown, with Section 4.3.2 describing the WT identification experiments (free decay, input wave, input force and prescribed motion experiments) and Section 4.3.1 explaining the WT preliminary ex- periments (free decay and input force preliminary experiments). In Section 4.4, a methodology to compare 2D NWT data with 3D BEM (in this case WAMIT) data is presented. Finally, a summary and discussion are presented in Section 4.5.