comparisons between different hybrid statistical models for accurate forecasting of photovoltaic...

UNIVERSITA’ DEL SALENTOFacoltà di Ingegneria

Corso di Laurea Magistrale in Ingegneria MeccanicaA.A. 2012/2013

Tesi di Laurea in:IMPIANTI TERMOTECNICI

COMPARISONS BETWEEN DIFFERENT HYBRID STATISTICAL MODELS FOR ACCURATE FORECASTING OF PHOTOVOLTAIC SYSTEMS POWER

Relatori:

Correlatore:

Laureando: Andrea Cretì

1

Prof. Ing. Paolo M. CongedoProf. Ing. Maria Grazia De GiorgiIng. Maria Malvoni

UNIVERSITA’ DEL SALENTO - Facoltà di Ingegneria

COMPARISONS BETWEEN DIFFERENT HYBRID STATISTICAL MODELS FOR ACCURATE FORECSTING OF PHOTOVOLTAIC SYSTEMS POWER

Laureando: Andrea Cretì – A.A. 2012/2013 Pag. 2

SUMMARY

The B.E.A.M.S. Project – 7° Framework Programme;

Photovoltaic Power Plants - PV plant in Campus «Ecotekne» in Monteroni di Lecce (LE);

Acquisition and Storage System;

Electric Time Series Forecasting;

Forecasting Model I: Elman Back-Propagation Neural Network ;

Forecasting Model II: Support Vector Machines (SVMs) and Least Square SVMs (LS-SVMs);

Forecasting Model III: Least Square SVM with Wavelet Transform (WLS-SVM)

Final Comparison between Model I, II and III;

Conclusions;

Future Work Raccomendation.



THE «BEAMS» PROJECT – 7° FRAMEWORK PROGRAMME





GENERAL SPECIFICATION OF THE PV PLANT

PV MODULE SPECIFICATION

Type Mono-crystalline silicon

Nominal power (Pn) 320 Wp

Maximum power voltage (Vpm) 54.70 V

Maximum power current (Ipm) 5.86 A

Open circuit voltage (Voc) 64.80 V

Short circuit current (Isc) 6.24 A

Weight 18.6 Kg

Net [gross] module surface 1.57 m2 [1.63 m2]

PV MODULE SPECIFICATION

Type Mono-crystalline silicon

Nominal power (Pn) 960 kWp

Maximum power voltage (Vpm) 3000

Weight 4710 m2 [4892 m2]

Net [gross] module surface 1.57 m2 [1.63 m2]

PV1 PV system

Nominal power of PV system 353.3 kWp

Azimuth -10°

Tilt 3°

Total number of modules 1104

Net [gross] modules' surface 1733.3 m2 [1799.5 m2]

PV2 PV system

Nominal power of PV system 606.7 kWp

Azimuth -10°

Tilt 15°

Total number of modules 1896

Net [gross] modules' surface 2976.7 m2 [3090.5 m2 ]

Table 1 – PV Module Specification Table 2 – PV Plant Specification

The PV Park located in Monteroni di Lecce (LE) –Italy – is diveded into 4 PV sub-plants:

• FV1: 960 kWp;• FV2.1: 990,72 kWp;• FV2.2: 979,20 kWp;• FV3: 84,43 kWp.

The plant under study is FV1, that is divided intotwo different module groups:

PV1Nominal Power:

353,3 kWpTilt: 3°

PV2Nominal Power:

606,7 kWpTilt: 15°

Figure 2 – PV Plant location

Fig. 1 – PV Park Shelves




DATA ACQUISITION SYSTEM

Fig. 5 – Acqisition and storagesystem flowchart

Fig. 4 – Software «Solar Data Extractor»

Fig. 3 – ESAPRO Web site

Fig. 7 – Java routine for MySQL conversion

Fig. 6 – Matlab routine for PV Powerproduction forecasting




ELECTRICAL TIME SERIES FORECASTING

ENERGY TIME SERIES FORECASTING STATE OF THE ARTINNOVATIVE FORECASTING

TECHNIQUESPROPOSED IN THIS THESIS

AIM OF THE

THESIS

Design innovative hybrid statistical modelshistorical data-based for PV Power forecasting

Evaluate the performance of these innovativemodels

Compare these performance with those obtainedwith already developed forecasting models

Fig. 8 – Power Forecasting Methods




TRAINING AND TEST DATASETS

Fig. 9 – Input dataset

Fig. 11 – Training and Test partition

Fig. 10 – Correlation between inputs and output power

Hourly PV Power; Solar Irradiation at 15°; Solar Irradiation at 3°; Module Temperature; Ambient Temperature.

The Acquired Dataset wasdivided into a Traning (65 %) and a Test (35 %) Dataset

Data correlation evaluated using the Pearson-Bravais (R2) coefficient:

Solar Irradiation 15° – PV Power: R2 = 0,9741 Solar Irradiation 3° – PV Power: R2 = 0,9726 Module Temeprature - PV Power: R2 = 0,3897 Ambient Temperature – PV Power: R2 = 0,1756




FORECASTING MODELS AND INPUT VECTORS

MODELS –INPUT VECTORS DESCRIPTION

MODEL I Elman Back-Propagation ANN

MODEL II Least Square – Support Vector Machine(LS-SVM)

MODEL III LS-SVM with Daubechies type 4 Wavelet Decomposition on 8 levels

INPUT VECTOR I PV Power

INPUT VECTOR IIPV Power, Solar Irradiation 3°, Solar

Irradiation 15°, Module Temperature, Ambient Temperature

Tab. 3 – Forecasting Models and Input Vectors

100)(

11 1

M

i iNi

ii

PMaxTP

NNMAPE

Performance evaluatuon are made by using Mean Absolute Error (MAE), Standard Deviation of the Error distribution (Std) and Normalized Mean Absolute PercentageError (NMAPE):

i: generic time instant;n: number of observations;Ti: Real PV Power at time instant i;Pi: forecasted PV Power at i;

Input Vector I Input Vector II

De Giorgi et. at. used the NMAPE as the best performance evaluation parameter.

Model I Model II Model IIIModel I Model II Model III




ARTIFICIAL NEURAL NETWORKS (ANNs)

Input vector I Input vector IITraining function TRAINGDX TRAINGDXAdapt learning function LEARNGD LEARNGDPerformance function MSE MSENumber layers 3 3Neurons (layer 1) l=1h

l=3hl=6hl=12hl=24h

213161121241

162651101201

Neurons (layer 2) l=1hl=3hl=6hl=12hl=24h

11163161121

8132651101

Neurons (layer 3) – output 1 1

Activation function hidden layer TANSIG TANSIG

Activation function output layer PURELIN PURELIN

Epochs 500 500

NotesTRAINGDX = Gradient descent with momentum and adaptive learning rate back-propagationLEARNGD = Gradient descent weight and bias learning function MSE = Mean Squared ErrorTANSIG = Hyperbolic tangent sigmoid transfer functionPURELIN = Linear transfer function

Fig. 13 - Elman Back-propagation ANN scheme

Tab. 4 - Elman ANN parameters

De Giorgi et. al. already applied an Elman ANN:

Feed-forward network Feedback from first layer output to first layer input Three layers of neurons Hyperbolic tangent sigmoid transfer function (TANSIG) applied for the first layer Linear transfer function (PURELIN) used for the second layer The “gradient descent weight and bias” was used as learning function (LEARNGD) to determine

how to adjust the neuron weights to maximize performance.

M.G. De Giorgi, P.M. Congedo, M. Malvoni, M. Tarantino, "`Short-term power forecasting by statistical methods for photovoltaic plants in south Italy"', 2013

Fig. 12 – ANN neuron




MODEL I FORECASTING RESULTS

Best performance reached using Input Vector IIfor all horizons

Better NMAPE reduction using Input Vector II forhigh forecasting horizons

Same NMAPE growing trend for both InputVectors except for Input Vector II – horizon +24hdue to low correlated data

Highest probability to have an NMAPE value inRanges ±1% ± 5% ± 10% using Input Vector II;

PredictionLength

Normalized AbsoluteAverage Error

Error Range Probability[-10%;+10%]


Model IInput Vector I

Model IInput Vector

II

Model IInput Vector

I

Model IInput Vector

II

Model IInput Vector

I

Model IInput Vector

II1 h 9.40% 6.49% 72% 78% 87% 91%3 h 15.11% 10.37% 56% 65% 76% 82%6 h 20.18% 13.46% 12% 53% 67% 78%12 h 21.12% 14.22% 17% 44% 37% 78%24 h 18.54% 19.60% 34% 31% 61% 57%

Tab. 5 – Model I performanceFig. 14 – Model I NMAPE comparison

Fig. 15 – Absolute error distribution comparison for Model I





MODEL I FORECASTING RESULTS

Fig. 17 – Error Distriution for Model I – Input Vector I - Forecasting Horizon +6h

Fig. 18 – Error Distriution for Model I – Input Vector II - Forecasting Horizon +6h

Underestimation of the Real PV Power usingInput Vector I;

Error Distribution Mean closed to zero usingInput Vector II;

Very similar standard deviation values for +24hhorizon;

Critical horizon +24h: very high standarddeviation values;

Bias error using Input Vector I when the Real PVPower has zero values;

No Bias error using Input Vector II when theReal PV Power has zero values;

Difficoult to follow abrupt changes of the Real PVPower (e.g. unespected passage of clouds);

For both of the Input Vector and forecastinghorizons, the forecasted PV power signalpresents a delay for the edge of the real powersignal and an advance for falling edge of the realpower signal.

Horizon +1h Horizon +3h Horizon +6h Horizon +12h Horizon +24h

Fig. 16 – Error distribution comparison between Model I – Input I and II




SUPPORT VECTOR MACHINES (SVMs)

Where k(x; z) is a kernel function. The formulated problem is a CQPand the solution allows to define the regression function in thefollowing form:

An optimal kernel function is the RBF:

Parametersto optimize

ε

σ2

CCROSS

VALIDATION

REGRESSION PROBLEMLEARNING A NON-LINEAR FUNCTION

The input data can be mapped from the input space to a higherdimensional feature space using a mapping function ϕ(x). Thelinear hyperplane estimator can be written as:

The Duality Theory and the use of kernel functions allows togeneralize the discussion to the non-linear regression models,similarly to what was done for the classification problems.The training problem is:

A classification problem involves the allocation of input vectors (xi) to a class membership, using the label value yi. If the membership classes are two, the classification problem is called binary, and the membership classes are identified with label values +1 and -1.The target is to define a machine capable of learning the relation xi → yi.

Vapnik, V.N. (1995), “The Nature of Statistical Learning Theory”. Springer-Verlag, New York, 1995Vapnik, V.N. (1998), “Statistical Learning Theory”, Wiley, New York




LEAST SQUARE SUPPORT VECTOR MACHINES (SVMs)

Note that in the case of RBF Kernels, one has only two additional tuning parameters (γ,σ2), which is less than for standard SVMs.

Parametersto optimize

γ

σ2

CROSS VALIDATION

A training set is given and the optimization problem in the primal weight space can be formulated as follows:

However, one should be aware that when w becomes infinitedimensional, one cannot solve this primal problem, so a dualLagrangian problem is constructed. The Lagrangian function is:

Imposing the conditions for optimality, after elimination of variables wand e and applying the kernel tick, the resulting LS-SVM model forfunction estimation becomes then:

Vapnik V.N. and Suykens et. al. proposed a modified form of SVM algorithm, called Least Square Support Vector Machines:

Vapnik, V.N. (1995), ``The Nature of Statistical Learning Theory``. Springer-Verlag, New York, 1995

Suykens J. A. K., Van Gestel T., Debrebanter J., 2002, ``Least Square Support vector Machines`` Singapore: World Scientific Publishing Co, 2002

The Training of the LS-SVM is more simple because it requires the solution of a set of linear equations (linear KKT systems).

LS-SVMs are closely related to regularization networks and Gaussian processes but additionally emphasize and exploit primal-dual interpretations.

For the productivity forecasting of this study, the Radial Basis Function (RBF) is used.

In literature many tests and comparisons showed great performances of LS-SVMs on several benchmark data set problems

Reduced computing time of the SVMs.

Where k are the Lagrangian multipliers.




MODEL II FORECASTING RESULTS

PredictionLength




Model IIInput Vector I

Model IIInput Vector II





1 h 7.53% 6.40% 70% 77% 88% 91%3 h 13.62% 10.18% 61% 65% 77% 82%6 h 18.22% 13.46% 12% 58% 70% 77%12 h 21.11% 14.53% 17% 44% 37% 75%24 h 18.52% 19.5% 34% 31% 61% 57%

Best performance reached using Input Vector IIfor all horizons

Better NMAPE reduction using Input Vector II forhigh forecasting horizons

Same NMAPE growing trend for both InputVectors except for Input Vector II – horizon +24hdue to low correlated data

Highest probability to have an NMAPE value inRanges ±1% ± 5% ± 10% with Input Vector II;

Critical horizon +24h: very high standarddeviation values

Tab. 6 – Model II performance

Fig. 19 – Model II NMAPE comparison

Fig. 20 – Absolute Error distribution comparison for Model II





MODEL II FORECASTING RESULTS

Fig. 22 – Error Distriution for Model I –Input Vector I

Forecasting Horizon +6h

Fig. 23 – Error Distriution for Model I –Input Vector II

Forecasting Horizon +6h



Very similar standard deviation values; Bias error using Input Vector I when the Real

PV Power has zero values; No Bias error using Input Vector II when the

Real PV Power has zero values; Difficoult to follow abrupt changes of the

Real PV Power (e.g. unespected passage ofclouds)

For both of the Input Vector and forecastinghorizons, the forecasted PV power signalpresents a delay for the edge of the realpower signal and an advance for falling edgeof the real power signal


Fig. 21 – Error distribution comparison between Model II – IV I and II




WAVELET TRANSFORM

Fourier Transform

Short Term Fourier Transform

Wavelet Transform

Time information are lost: is no longer possible todetermine when a particular event happened.

Even thought a signal is not stationary, but only stationary for short time intervals, thespectrum for this signal can be calculated by moving a stationary signal window onconsecutive signal segments, in order to realize a Short Term Fourier Transform (STFT). TheShort Term Fourier Transform is a compromise between time and frequency but its precisiondepends on the window amplitude and the amplitude can not be variate, but it is constant foreach frequency.

The Wavelet Transform uses adaptive windows in order to improve results obtainable usingSTFT. Adaptive windows encloses long time intervals to analyze low frequencies and shorttime intervals to analyze high frequencies. A signal is expressed as the combination ofchildren wavelets, results of the shifting and scaling from a mother wavelet:.

Fig. 24 – From Fourier Transform to Wavelt Transform

From a generic wavelet (a; b; t), where a and b are the shifting and scaling factors, the Continuous Wavelet Transform (CWT) is dened as the integral of the signal s(t) multiplied for the scaled wavelet:




WAVELET TRANSFORMIn continuous wavelet transform, the wavelet function is stretched and shifted along the signal in a continuous manner. This present an enormousamount of work. It turns out that if the scales and shifting are discretized based on powers of two so called dyadic scaled and positions thecomputing of the transform will be more efficient without any loss in accuracy. The Discrete Wavelet Transform decomposes the original signal inDETAILS (high frequencies components) and APPROXIMATIONS (low frequencies components).

Fig. 25 – Wavelet Decomposition procedure

Fig. 26 – 8 Level decomposition of the Input Signals using DB4 Wavelet Transform

Fig. 27 – Forecasting Model III Scheme 1

Fig. 28 – Forecasting Model III Scheme 2

MATLAB® WAVELET TOOLBOX® Used with Input Vector I Very simple implementation Normal CPU usage

Used with Input Vector II Sample forecasting

performance as Scheme 1 Lower computational

performance on normal CPU Very fast computing if

implemented in parallelcomputing algorithm




MODEL III FORECASTING RESULTS

Best performance reached using Input Vector II for +6h and+24h horizons;

Similar performance at +1h and +3h forecasting horizons; Better NMAPE reduction using Input Vector II for high

forecasting horizons; Same NMAPE growing trend for both Input Vectors except

for Input Vector II – horizon +24h due to low correlateddata;

Bigger difference between +24h NMAPE value with InputVector I and II than this detected with Model I;

Highest probability to have an NMAPE value in Ranges ±1% ±5% ± 10% with Input Vector I and II;

Absolute error distribution very similar for IV I and II;

PredictionLength




Model IIIInput Vector I

Model IIIInput Vector II


Model IIIInput Vector

II



1 h 6.57% 6.92% 74% 74% 81% 95%3 h 10.76% 10.35% 60% 56% 79% 84%6 h 13.52% 10.53% 52% 60% 74% 84%12 h 15.04% 12.09% 46% 54% 73% 79%24 h 12.91% 19.00% 47% 29% 77% 54%

Tab. 7 – Model III performanceFig. 29 – Model III NMAPE comparison

Fig. 30 – Absolute Error distribution comparison for Model III





MODEL III FORECASTING RESULTS

Fig. 32 – Error Distriution for Model III – Input Vector I - Forecasting Horizon +6h

Fig. 33 – Error Distriution for Model III – Input Vector II - Forecasting Horizon +6h



Very similar standard deviation values; Critical horizon +24h: very high standard

deviation values. Bias error using Input Vector I when the Real

PV Power has zero values; No Bias error using Input Vector II when the

Real PV Power has zero values; Difficoult to follow abrupt changes of the

Real PV Power (e.g. unespected passage ofclouds)

For both of the Input Vector and forecastinghorizons, the forecasted PV power signalpresents a delay for the edge of the realpower signal and an advance for falling edgeof the real power signal


Fig. 31 – Error distribution comparison between Model III – IV I and II




COMPARISONS BETWEEN MODEL I, II AND III

Normalized Absolute Average Error NMAE


Model IInput Vector II





1 h 9.40% 6.49% 7.53% 6.40% 6.57% 6.92%3 h 15.11% 10.37% 13.62% 10.18% 10.76% 10.35%6 h 20.18% 13.46% 18.22% 13.46% 13.52% 10.53%12 h 21.12% 14.22% 21.11% 14.53% 15.04% 12.09%24 h 18.54% 19.60% 18.52% 19.5% 12.91% 19.00%

Error Range Probability [-10%;+10%]







1 h 72% 78% 70% 77% 74% 74%3 h 56% 65% 61% 65% 60% 56%6 h 12% 53% 12% 58% 52% 60%12 h 17% 44% 17% 44% 46% 54%24 h 34% 31% 34% 31% 47% 29%

Error Range Probability [-20%;+20%]







1 h 87% 91% 88% 91% 81% 95%3 h 76% 82% 77% 82% 79% 84%6 h 67% 78% 70% 77% 74% 84%12 h 37% 78% 37% 75% 73% 79%24 h 61% 57% 61% 57% 77% 54%

Best global performance reached by Model III with Input Vector II For +1h and +3h horizons performance of Model III with IV II are very closed to those obtained with Model I and II with IV II +24h horizon with Input Vector I is always the critical one

Tab. 8 – Final performance comparisons between Models I, II and III

Fig. 34 – Final NMAPE comparison




COMPARISONS BETWEEN MODEL I, II AND III

Fig. 39 – Comparisons between error distribution obtainedwith all models and Input Vector I

Fig. 40 – Comparisons between error distribution obtainedwith all models and Input Vector II

Fig. 41 – Real and forecasted PV Power final comparison

Error distribution mean very closed to zero with Model III The performance improvements of Model III are clearly

perceptible using Input Vector I The performance improvements of Model III are barely

perceptible using Input Vector I Abrupt changes of PV power production always difficoult to

follow by the 3 Models Bias error of the forecasted PV Power not present using

Model III




CONCLUSIONS AND FUTURE WORKS RACCOMENDATIONS

No Numerical Weather Prediction are necessary Best global performance reached by Model III with Input Vector II Wavelet Transform is a good choice to treat non-stationary signals The forecasting Models developed can be used for industrial application due to the

high performance reached Input Vector I may be used only if a low computational time is necessary Abrupt changes of PV power production are not followed well. In this case a NWP

model is neccessary

Design a LS-SVM based model with NWP Design a LS-SVM hybrid based model with NWP Use a bigger input dataset with the collected data from the ESAPRO web-site Use the Multistep technique with every created models and for NWP-based

models

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