The point prediction quality is closely related to the model that explains the dynamic of the observed process. Sometimes the model can be obtained by simple algebraic equations but, in the majority of the physical systems, the relevant reality is too hard to model with simple ordinary differential or difference equations. This is the case of systems with nonlinear or nonstationary behaviour which require more complex models. The discrete time-series problem, obtained by sampling the solar radiation, can be framed in this type of situation. By observing the collected data it is possible to distinguish multiple regimes. Additionally, due to atmospheric disturbances such as clouds, the temporal structure between samples is complex and is best described by nonlinear models. This paper reports the solar radiation prediction by using hybrid model that combines support vector regression paradigm and Markov chains. The hybrid model performance is compared with the one obtained by using other methods like autoregressive (AR) filters, Markov AR models, and artificial neural networks. The results obtained suggests an increasing prediction performance of the hybrid model regarding both the prediction error and dynamic behaviour.
Often the output observation of a stochastic process can not be associated with any exogenous excitation variable. These inabilities are due to several factors either because they are not known or because they can not be measured. In those circumstances, it is assumed that the process generates the observations, independently, without any outside intervention. A certain observer records the process response, usually in a regular time interval. The ultimate goal is to discover the process internal mechanism that generates the series of observations. There are an infinite number of possible mechanisms able to generate the sequence of observed values. Thus, in addition to the mechanism, or model which describes the dynamics of the process, it is necessary, in quantitative terms, to establish the quality of each of these models. The model, from all the possibilities, that exhibits the best performance, regarding the defined quality assessment function, will be the one who best describes the dynamic nature of the time-series generating mechanism.
Autoregressive models, which only define linear relationships between past and present observations, represent one of the first attempts to explain the operating mechanism of stochastic processes [
The remaining of the paper is organized as follows: Section
The aim of this work is to predict, as close as possible, the solar radiation dynamics during the day. This knowledge will be used, within a control loop, in order to improve the indoor temperature regulation of an agricultural building. The present problem setup includes a polyethylene cover quonset type greenhouse located at the north of Portugal. This greenhouse has a floor area of 210 m2 and is equipped with several actuators and sensors. The installed actuators are a ventilator, with a flow rate of 38000
The experimental setup: a greenhouse with floor area of 210 m2 (a). Indoor temperature, relative humidity, soil temperature, and CO2 are measured. In the outside, wind speed, outside air temperature, relative humidity, and solar radiation are measured by a weather station located in the greenhouse vicinity (b).
The temperature inside the greenhouse is kept at the reference level by controlling the average power delivered to the ventilator and heating systems. The decision about which actuator is on or off and the fraction of power supplied is computed by an embedded controller. The controller software implements a model predictive control (MPC) strategy. In abstract, model predictive control comprises a collection of control methods having in common that the controller is based on the future predictions of the system behaviour, using a mathematical model of the plant [
Basic strategy of a model based predictive controller. In a given time instant
At each sampling instant, a set of
The control effort component can be added to the main objective function, in order to minimize the actuators wearing out. The expanded objective function has, generally, the following formulation:
At each sampling instant,
This type of control strategy requires a plant model in order to obtain the predicted output value,
The greenhouse indoor temperature,
This model is used, within a predictive control strategy, in order to infer the future value of the greenhouse indoor temperature [
Taking into consideration the format of expression (
Let
Usually the function
The maximum prediction horizon taken is sixty steps ahead and the model performance is inferred taking into consideration two indexes: the average of the root-mean squared (RMS) prediction error and the percentage of change in direction (PCD). The latter is a qualitative index representing the model ability to predict the tendency. This figure of merit is very important in the context of air temperature regulation under a model predictive controller (MPC), since the heating and ventilation requirements will be computed taking into account if a heat load change is expected in the near future. The computation of both figures of merit follows:
In the next section the prediction results, regarding the use of an autoregressive (AR) model, are presented. For the sake of representativeness, only one day of radiation is used. Nevertheless the values obtained are coherent with those obtained using other days with the same dynamics. Section
This section demonstrates the inability of AR models to provide close predictions for the solar radiation in a typical day. The parameters of a 10th order model are estimated, in a particular time instant, using the previous 80 samples after removing the linear trends. Figure
In (a), the solar radiation measured at the 8 of July of 2001. In (b), the sixty-step ahead prediction of the solar radiation, from the point identified in (a), using a 10th order AR model.
It can be seen that the model is unable to describe the future behaviour of the solar radiation signal. Indeed there is an early divergence between the measured and predicted signals: the model predicts an increase in radiation while, in reality, the measured signal decreases in energy. Even neglecting the discrepancy on the dynamic tendency, there is also a large difference between the prediction and measured mean values (around 500 W/m2). To demonstrate the results persistence, another point is selected over the same radiation pattern. The new chosen point and the respective prediction result are illustrated in Figure
In (a), the solar radiation measured at the 8 of July of 2001. In (b), the solar radiation sixty-step ahead prediction, from the point identified in (a), using a 10th order AR model.
Figure
In (a), the solar radiation measured at the 8 of July of 2001. In (b), the solar radiation sixty-step ahead prediction, from the point identified in (a), using a 10th order AR model.
Once again the model was unable to generate acceptable results after the first (2-3) predictions. Moreover, the tendency is wrong even for the early predictions. The presented results allow us to conclude that the application of simple AR model is unsuccessful in providing good enough predictions for the solar radiation within the defined time horizon. For this reason, alternative models are tested. In Section
The use of artificial neural networks for time-series prediction is not a new subject. Indeed, it is one of the most prolific modelling techniques when the nonlinear relationship between samples must be explored. In the context of solar radiation prediction, ANN models have been already used with some success [
This section presents the results concerning the use of two different ANN strategies regarding the sixty-step ahead solar radiation prediction. The first is a feedforward neural network and the latter a set of four feedforward networks each one predicting a filtered version of the data.
The ordinary feedforward neural network has a single hidden layer with five neurons each one with sigmoidal activation functions. The output layer is composed of a single neuron with linear activation function and the embedded input dimension was of 10th order. The training fase, carried out by the Levenberg-Marquardt algorithm, has used the solar radiation, collected in the past day, as estimation data.
The second nonlinear model devised was composed of four ordinary feedforward ANN. Each of them has one hidden layer with two neurons with sigmoidal activation function. The output neuron has a linear activation function and the lag-space for each network was of order five. The choice for that ANN architecture is justified in [
Regarding the solar radiation of a particular day, illustrated in Figure
In (a), the solar radiation pattern used as test and, at (b), his decomposition using a filter bank. The signals
Each of the four ANN is tuned to predict one of the four decomposed signals,
Table
Results concerning the sixty-step ahead solar radiation prediction over validation data.
AR(10) | ANN | ANN-WD | |
---|---|---|---|
|
126 | 123 |
|
|
68 | 69 |
|
Computational load | < |
28% | 100% |
The obtained results suggest a slight increase in the ANN-WD model prediction capability. This improvement regards both prediction performance indexes. However, this model requires higher computational power. It is important to note that these results are obtained by the best fitted models. After several training runs with different initial solutions, the model with best prediction performance is used. The same applies to the conventional ANN. Due to the training method sensibility to the initial solution, Section
Support vector regression (SVR) models are a class of computational paradigms derived from support vector machines theory [
In this section, the SVR with two different kernel functions is used to predict the solar radiation within the temporal range of sixty steps ahead [
Here two consecutive days of solar radiation data are used. The first one is for parameter estimation and the second one is for prediction. The models used are as follows. A linear AR filter with ten poles: the filter coefficients are obtained by a least squares procedure using the estimation data. A feedforward ANN with one hidden layer with five sigmoidal neurons: the input space has dimension ten. The training procedure is carried out by using the Levenberg-Marquardt algorithm. A neurowavelet (ANN-WD) structure is like the one described in Section Two SVR models with different kernel functions: they are a linear kernel (SVR-LK) and a Gaussian kernel (SVR-GK).
The prediction results, regarding the above enumerated models, are presented in Table
Sixty-step ahead prediction results, for five different models, concerning a day with some cloud disturbance.
AR(10) | ANN | ANN-WD | SVR-LK | SVR-GK | |
---|---|---|---|---|---|
|
41.2 | 40.9 | 39.1 | 40.5 |
|
|
23.4 | 23.3 |
|
23.1 | 24.2 |
Sixty-step ahead prediction for one day of solar radiation using five different methods: a linear AR model, two distinct artificial neural networks models (ANN and ANN-WD), and two support vector regression models with different kernel functions (SVR-LK and SVR-GK). The vertical axis units are W/m2 and horizontal axis is the time expressed in minutes.
The graphic in the left high corner of Figure
The graph located at Figure
The last two graphs in Figure
These results allow us to conclude that, in average, the nonlinear models provide best predictions, in both performances indexes, when compared to a simple linear AR model. However their complexity is, by far, higher than that of the linear models and their performance depends on the proper choice of some training tuning parameters.
A final note on the nonlinear models used: regarding the ANN, good model weights were difficult to obtain. Several tests were performed until coefficients that would lead to models with good performance were achieved. On the other hand the values for the two tuning parameters of the SVR were obtained using the heuristics published in [
So far, the techniques used consider a single model to describe the overall signal. However, there are evidences of different dynamic regimes within the data. There are obvious dynamic differences between the ascent and descent part of the day. For this reason Sections
Often it is observed that a certain model does not have the same approximation quality in all the zones of a signal produced by some unknown stochastic process. It would be possible to define alternative models that best approximate the process dynamics over particular areas. Hence it will be useful to have a set of available models, one for each dynamics, and an automatic selecting mechanisms. This mechanism will choose, among the models available, the one that best represents the process for a particular operating point. Whenever the regime changes, a new model replaces the previous one.
For this computational paradigm, aside from the common difficulties associated with system identification problems, now there is an additional problem which is how to decide when the process regime changes. Eventually, an outer stochastic process may change the active model among the inner universe of stochastic ones. As in conventional system identification, where the model type and model order must be inferred by means of observed input/output values, the present regime knowledge, over a multiple set, must also be obtain by means of the same data. It is assumed that a stochastic process, internal to the system to model, is responsible for changing the regime. This process is not directly observable and its action can only be estimated from the only available information: present and past values of process output.
One way to emulate the regime generator mechanics is to use a Markov chain, in which the active model, at a given moment, reflects the state where the modelling process stands. Assuming a stochastic unknown process that generates the solar radiation only as a function of time then, only by direct observation of the system output data, it is possible to see that the system behaves in quite distinct forms. As an example, the different dynamics between the incident solar radiations is clear during the day and during the night. During the “night” regime, the model should switch to a constant zero while, at “day,” the model should follow the fluctuations of solar radiation. Notice that, during the “day” regime, it is possible to define different subregimes. For example, the “increasing regime” that corresponds to the upward trend reflecting the radiation from the sunrise until Sun’s zenith and the “decreasing regime” that corresponds to the downward behaviour from the zenith to the sunset.
In this context, this section presents the use of a multiple regime model composed by a set of autoregressive models, with different coefficient values, connected by a Markov chain which implements the model switching mechanism. A model, with this type of structure, was first presented by Hamilton and designated by Markov autoregressive model (MARM) [
The variable
A set of experiments was conducted, regarding the use of this type of model, for one- and sixty-step ahead prediction of the solar radiation. Several structures and models with distinct regimes were tested. Tables
Solar radiation prediction results for a
Filter order ( |
2 | 3 | 4 | 5 | ||||
---|---|---|---|---|---|---|---|---|
Prediction horizon/min | 1 | 60 | 1 | 60 | 1 | 60 | 1 | 60 |
|
|
160 | 8.1 |
|
8.1 | 160 | 8.3 | 150 |
|
82 |
|
83 | 62.8 |
|
64 | 84 | 62.4 |
Solar radiation prediction results for a
Filter order ( |
2 | 3 | 4 | 5 | ||||
---|---|---|---|---|---|---|---|---|
Prediction horizon/min | 1 | 60 | 1 | 60 | 1 | 60 | 1 | 60 |
|
9.1 | 195 | 8.9 | 130 |
|
|
|
170 |
|
84 | 60.8 | 85 | 70 |
|
|
85 | 61 |
Solar radiation prediction results for a
Filter order ( |
2 | 3 | 4 | 5 | ||||
---|---|---|---|---|---|---|---|---|
Prediction horizon/min | 1 | 60 | 1 | 60 | 1 | 60 | 1 | 60 |
|
8.8 | 219.9 |
|
420 | 9.1 | 219.8 | 10 |
|
|
81.3 | 63.1 |
|
|
85 | 65.3 | 84 | 59.5 |
Solar radiation prediction results for a
Filter order |
2 | 3 | 4 | 5 | ||||
---|---|---|---|---|---|---|---|---|
Prediction horizon/min | 1 | 60 | 1 | 60 | 1 | 60 | 1 | 60 |
|
9.7 | 290 | 8.6 | 230 |
|
|
9.8 |
|
|
80.5 | 57.9 | 81.2 | 59.1 | 82 | 57.9 |
|
|
From the results presented one observes that the best model structure, in both performance indexes, has three hidden states and four poles. This statement is valid for the two prediction time horizons tested: a short term forecast of one step ahead and a long term forecast of sixty steps ahead. Figures
One-step ahead prediction for one day of solar radiation using a MARM with
Sixty-step ahead solar radiation prediction for one day using a MARM with
Comparing the results documented in Table
From the obtained results it seems that the direction for a good solar radiation model follows a multimodel strategy. Also, as already shown, nonlinear models usually describe, in a more efficient way, the temporal sample dependence of the signal. However for the MARM strategy, even if it is a model suited for nonstationary time-series, it assumes only linear dependence over the sample space. Therefore, Section
The previous section showed that the use of Markov autoregressive models could improve the model prediction ability. However, the MARM assumes that, within each operating regime, the dependence between observations is linear. For this reason, in this section, an alternative switching model strategy using nonlinear functions is presented. This strategy uses multiple support vector machine regression models and a Markov chain as a decision mechanism. All the SVR models considered have a Gaussian kernel function since, as demonstrated in Section
The process of defining the support vectors involves the partition of the training data into four segments. Each segment represents a particular part of the day: the absence of radiation, during the ascending part of the day, the descending part of the day and around the peak of radiation, where the first derivative has a lower value. Subsequently four SVR models are fitted, one for each regime.
The active SVR, in a particular time instant, is defined by a four-state Markov chain with a Bakis topology [
Topology of the hybrid SVR/Markov. Four states decide which of the present SVR to be used according to the historical data.
The prediction is made under two different hierarchical levels. The first involves the prediction of the state sequence and the second the solar radiation prediction made by the model pointed out by the state sequence. This type of strategy was applied to predict the solar radiation for a particular day already presented in above sections. The performance results concerning the prediction horizon of sixty steps ahead are presented in Table
Results concerning the sixty-step ahead solar radiation prediction over validation data.
AR(10) | ANN | ANN-WD | MARM | SVR/Markov | |
---|---|---|---|---|---|
|
126 | 123 | 118 | 120 |
|
|
68 | 69 | 70 | 71 |
|
Figure
Sixty-step ahead solar radiation prediction for one day using a hybrid SVR/Markov with four states. (b), (c), and (d) represent zooming images of (a).
The obtained results, even if just for a particular day, show clearly that the hybrid SVR/Markov model was able to give a best performance, in both defined indexes, when compared to the other tested methods. Section
In order to be able to present a graphical output concerning the obtained results, the earlier sections lie in the application of several model techniques for only a particular day. Even if the spectral content of the chosen day represents those of a regular day, in this section the above mentioned computational models were used for solar radiation over a larger data set. Whenever required, the model parameters estimations were obtained, using the training methodology discussed in the preceding sections, over the previous day.
The obtained results are presented in Tables
Obtained average prediction values for ten days selected from April to May of 2000 (
Model | AR(10) | ANN | ANN-WD | SVR-GK | MARM | SVR/Markov | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Horizon/min | 1 | 60 | 1 | 60 | 1 | 60 | 1 | 60 | 1 | 60 | 1 | 60 |
|
8.35 | 113.1 | 8.2 | 112.6 | 8.2 | 108.1 | 8.1 | 107.6 | 8.31 | 109.9 |
|
|
|
87.7 | 73.3 | 88.1 | 74.7 | 88.4 | 75.7 | 88.7 | 76.1 | 88.1 | 75.2 |
|
|
Obtained average prediction values for ten days selected from July to August of 2000 (
Model | AR(10) | ANN | ANN-WD | SVR-GK | MARM | SVR/Markov | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Horizon/min | 1 | 60 | 1 | 60 | 1 | 60 | 1 | 60 | 1 | 60 | 1 | 60 |
|
8.43 | 103.8 | 8.3 | 103.5 | 8.3 | 100.6 | 8.2 | 100.2 | 8.39 | 101.1 |
|
|
|
87.5 | 72.1 | 87 | 72.6 | 88 | 74.5 | 87.8 | 75.1 | 87.6 | 73.8 |
|
|
Obtained average prediction values for ten days selected from October to November of 2000 (
Model | AR(10) | ANN | ANN-WD | SVR-GK | MARM | SVR/Markov | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Horizon/min | 1 | 60 | 1 | 60 | 1 | 60 | 1 | 60 | 1 | 60 | 1 | 60 |
|
2.22 | 32.4 | 2.18 | 31.6 | 2.12 | 30.83 | 2.1 | 30.2 | 2.19 | 31.1 |
|
|
|
91 | 82.4 | 91.4 | 84.1 | 91.8 | 87.2 | 92.6 | 86.5 | 91.9 | 83.3 |
|
|
Obtained average prediction values for ten days selected from January to February of 2000 (
Model | AR(10) | ANN | ANN-WD | SVR-GK | MARM | SVR/Markov | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Horizon/min | 1 | 60 | 1 | 60 | 1 | 60 | 1 | 60 | 1 | 60 | 1 | 60 |
|
2.69 | 53.8 | 2.68 | 54.1 | 2.6 | 52.4 | 2.5 | 52.3 | 2.51 | 52.7 |
|
|
|
87 | 77 | 86.5 | 76.8 | 87.4 | 78.8 | 88.1 | 78.1 | 88 | 78.8 |
|
|
In the first row of Tables
Regarding the dynamic behaviour results, presented in the second row of Tables
In Section
In this work the solar radiation long term prediction, represented as a time-series, was carried out by a set of linear and nonlinear models. The close knowledge of future values of this variable will be fundamental to improve the greenhouse indoor temperature controller performance.
The solar radiation prediction, within this framework, is a very complex problem in part not only due to the low correlation between samples, especially in days with high radiation variability, but also due to the nonstationary behaviour of the generating process. Moreover, due to the large prediction horizon addressed, small model changes can result in large prediction errors. This is due to the feedback nature of the prediction process.
For this reason, a set of models was tested: linear autoregressive, nonlinear autoregressive, and support vector models. In addition an alternative computational paradigm, evolving SVR and Markov chains, was proposed. All the models were applied to large data sets concerning the solar radiation collected in several days distributed along the year.
The results presented indicated that the best tested model was the one that combines support vector regression and Markov chains. The relevance of this model becomes conspicuous for increasing prediction horizons. The advantage of this model is its capability to handle processes with several operating regimes and nonlinear behaviour. Even if the SVR/Markov results are better than those obtained by other strategies, it is believed that an additional increase in performance can be achieved by using more sophisticated model training methods. For now the SVR and the Markov chain training are handled separately. Therefore it is proposed, as a trend for future research, to change the training method of this model. An alternative way would be to transfer the technique that Hamilton has used for training MARM and adapt it to this new model. That is, both the parameters chain and support vector can be evolved simultaneously. In this way one could eliminate the need for segmentation and expert assessment of the data. In addition, in the current problem, the number of system states has been defined empirically. So another research direction is the development of a technique that enables the automatic selection of the best number of hidden regimes without a full-factorial experiment.
The authors declare that there is no conflict of interests regarding the publication of this paper.