A Stochastic Model for Short-Term ProbabilisticForecast of Solar Photo-Voltaic Power
Raksha Ramakrishna, Student Member, IEEE, Anna Scaglione, Fellow, IEEE, Vijay Vittal, Fellow, IEEE
AbstractIn this paper, a stochastic model with regime switch-ing is developed for solar photo-voltaic (PV) power in order toprovide short-term probabilistic forecasts. The proposed modelfor solar PV power is physics inspired and explicitly incorpo-rates the stochasticity due to clouds using different parametersaddressing the attenuation in power. Based on the statistical be-havior of parameters, a simple regime-switching process betweenthe three classes of sunny, overcast and partly cloudy is proposed.Then, probabilistic forecasts of solar PV power are obtained byidentifying the present regime using PV power measurements andassuming persistence in this regime. To illustrate the techniquedeveloped, a set of solar PV power data from a single rooftopinstallation in California is analyzed and the effectiveness of themodel in fitting the data and in providing short-term point andprobabilistic forecasts is verified. The proposed forecast methodoutperforms a variety of reference models that produce pointand probabilistic forecasts and therefore portrays the merits ofemploying the proposed approach.
Index TermsSolar PV power modeling, Short-term solarpower prediction,probabilistic forecast, Roof-top solar panels,Dictionary learning, Hidden Markov Models
Solar power generation, both from PV farms and roof-top solarpanel installations is on the rise leading to their increasingpenetration into traditional energy markets. Hence, considera-tion of solar PV power resource while analyzing electric gridoperations is gaining great significance. Accurate models thatcan not only provide solar power forecasts, but also capturethe uncertainty in the random process are necessary to addressdecision problems such as stochastic optimal power flow(SOPF) , probabilistic power flow studies , designingmicrogrids , solar power shaping  and reserve planning.
A vast array of literature exists in the area of short-termpoint forecasts for solar power. A majority of the approachestaken can be broadly classified as being physical, statistical ora hybrid of the two methods (see e.g.  for a review). Phys-ical methods employ astronomical relationships , meteoro-logical conditions and numerical weather predictions (NWPs)for an improved forecast , . Such studies are based onmodeling the clear sky radiation using earth-sun geometry,panel tilt and orientation, temperature and wind speed ,. Some also use irradiance information available fromdatabases to determine the value of power for a geographicallocation considered. Other papers use static images of cloudsin the sky recorded by a total sky imager (TSI)  or utilize
The authors are with the School of Electrical, Energy and ComputerEngineering (ECEE), Arizona State University, Tempe, AZ 85281, USA. Thiswork was funded in part by the Advanced Research Projects Agency- Energy(ARPA-E), U.S. Department of Energy, under Award Number de-ar0000696and by National Science Foundation under grant number CPS-1549923 . Theviews and opinions of authors expressed herein do not necessarily state orreflect those of the United States Government or any agency thereof.
a network of sensors recording cloud motion  to predictsolar power. These models rely on a deterministic mappinggiven additional information to produce an estimate of thepower generated by the panel.
The most prevalent statistical methods for solar powerforecasting include time-series modeling such as using au-toregressive (AR) models . One of the advantagesof these methods is that they are power data-driven and donot depend on having additional information like the previousliterature cited, and are adaptive. However, these methods aredesigned to model stationary normal processes and assume thatsome transformation such as dividing by the clear sky powertime series makes the series stationary. Such assumptions maynot be enough to fully capture the non-stationarity of solarpower production. Alternatively, there are other approachessuch as autoregressive integrated moving average (ARIMA)and autoregressive with exogenous input (ARX)  based non-stationary methods for solar power prediction.
In the same class of statistical methods there exist otherworks that capture variability in solar PV power and black box models like using artificial neural networks(ANN)  and support vector machines (SVM) based pattern matching techniques to predict solar power whenclass labels are known. Additionally, there are also methodsbased on the Markovianity assumption of solar power such as in order to forecast solar PV power.
Contrary to solar forecasting methods that provide pointforecasts, there exist only a few works in the field ofprobabilistic forecasting of solar PV power. In , a non-parametric kernel density estimation method is used to fita probability distribution to PV power. In , a higherorder Markov chain is used to characterize solar PV powerand operating points based on temperature are also usedto classify different PV systems and then Gaussian mixturemodels (GMM) are used for probabilistic forecasts. One canalso consider all of the AR based time series methods sincethey can essentially be used to obtain probabilistic forecasts.
In the proposed methodology the advantages offered by bothphysical and statistical approaches are exploited. The proposedmodel provides a statistical description of stochasticity of theelectric power signal that is inspired by the physical behaviorof solar PV power, while being completely adaptive. Theadvantage of going by the modeling approach motivated bythe physics of the problem is that it helps in understanding theunderlying phenomenon and provides an easier interpretationof the results obtained. Also, solar PV power is non-stationaryby nature and this needs to be captured by the modelingtechnique.
The model at a macro-level defines a regime switchingprocess  which says that solar irradiance emanates from
one of the three classes: sunny, partly cloudy, overcast. Thestochastic models for sunny and overcast are simple Gaussiandistributions whereas for the partly cloudy regime, a hiddenMarkov model (HMM) is proposed.
Such an approach simplifies the understanding of temporalvariations in solar PV power by examining each regimeseparately and associating a physical meaning to the hiddenstates. It is also important to note that no assumption ofstationarity is made while describing the regime switchingprocess and no attempt is made to estimate this time-varyingtransition probability. This is usually not the case in most ofthe other works. In this manner, the proposed method uniquelycaptures the non-stationarity in solar power which is not justdue to its diurnal structure.
Prior work in  by the authors briefly described in SectionII involved the development of a parametric model that wasproven to efficiently capture the effect of clouds on solar PVpower while providing a compact representation. In this paper,the prior modeling technique is utilized and extended to fita switching process to solar PV power, using which a solarpower prediction algorithm to provide short-term probabilisticforecasts is designed. The resulting low order model ensuresreduced computational complexity for the proposed algorithm.
The key contributions of this paper are:
The proposition of a regime-switching process for solarPV power that consists of periods that can be classifiedas sunny, overcast and partly cloudy and development ofstochastic models for the three regimes. This is detailedin section III.
A hidden Markov model (HMM) for the partly cloudyregime whose latent states are the support of sparseparameters pertaining to attenuation of power. This isdescribed in subsection III-3.
A change detection algorithm to identify the presentregime using solar power data. No other auxiliary infor-mation such as temperature or wind speed is used.
The design and analysis of a computationally efficientonline algorithm for short term solar power predictionby employing the switching process and the relevantstochastic models for sunny, overcast and partly cloudyas outlined in section IV.
The prediction results using the proposed method, as seen insubsection V-B, indicate the validity of the approach. In fact,the proposed prediction also outperforms multiple referencemodels including smart persistence , diurnal persistence,ANN based prediction method, AR model for the stochasticcomponent of solar PV power and multiple AR models,one each for sunny, partly cloudy and overcast with regimeswitching (c.f subsection V-B2).
Also, one could complement the proposed method by usingweather prediction and cloud imagery as additional informa-tion in order to improve the performance of the proposed fore-cast method. Since stochastic models are available, the methodprovides probabilistic forecasts which are very useful whilemaking decisions under uncertainty. Section VII includes theconclusions and future work.
day of the year
time of day
movement of sun
Bn[k i, jn j] dn[k i]
direct component diffuse component
Fig. 1. Figure representing the suns path across the sky over the days on aplane. The orange dot marks the position of the sun at time k on day n. Thewind trajectory is described by blue lines.
II. DISCRETE TIME MODEL