Expanding Existing Solar Irradiance Monitoring Network Using Entropy

1

Expanding Existing Solar Irradiance Monitoring Network Using Entropy Dazhi Yang, Nan Chen

Abstract—Many existing solar irradiance monitoring networks were built particularly for resource assessment purposes; they are often spatially sparse. In order for the networks to handle other increasingly important tasks, such as irradiance forecasting for grid integration, their spatial sparsity must be addressed by adding in new monitoring stations. Optimally expanding these networks using historical information thus becomes an important research topic for engineers. Variability of solar irradiance in space and time can be quantified using statistics such as entropy and covariance. The deployment of the additional monitoring stations should therefore utilize these statistics to reduce the variability. More specifically, we aim at maximizing the entropy of the network. A practical difficulty in statistical modeling of solar irradiance is that the data are not ideal. Properties such as stationarity and isotropy are not observed in irradiance random field. We therefore focus on hypothesis testing and transformation of the irradiance data, so that the design procedure is statistically justified. We propose the redesign framework in a solar engineering context, using data from 24 irradiance monitoring stations on a tropical island. In the case study, we demonstrate how to find 3 optimal stations from a pool of 100 potential future monitoring sites. Index Terms—solar irradiance, entropy, space deformation, network redesign.

I. I NTRODUCTION OLAR power plays an important role in meeting the future energy demands while decarbonizing the power sector. To increase the utilization of solar energy and reliably integrate it into the electricity grids, engineers and scientists need to perform tasks such as resource assessment and irradiance forecasting. Therefore, a knowledge of the spatio-temporal distribution of solar irradiance is crucial. The understanding of the irradiance distribution comes from two complementary sources, namely, remote sensing and ground-based monitoring network [1]. There is a rich literature on satellite-based irradiance models [2]. However, the estimates from these models are often found to include significant bias, and can be improved by considering the ground truth and post-processing corrections [3]. Besides complementing the satellite-based irradiance models, groundbased monitoring networks are used for spatio-temporal irradiance prediction [4]. It is shown that for a distance less than 35km, the ground-based network can achieve lower uncertainty in the interpolated data between sites as compared

S

D. Yang is with the Singapore Institute of Manufacturing Technology (SIMTech), Agency for Science, Technology and Research (A∗ STAR), Singapore (e-mail: [email protected], [email protected]), previously with the Solar Energy Research Institute of Singapore (SERIS). N. Chen is with the Department of Industrial and Systems Engineering, National University of Singapore, Singapore (e-mail: [email protected])

to the satellite-based estimation [5]. For these reasons, a welldesigned network of ground measurements maintained over long term is desired. Solar irradiance monitoring networks can be classified based on their spatial scales. Networks of continental or regional scales are built and used for resource assessment purposes [6]. Metropolitan scaled monitoring networks are used for hourly or intra-hour forecasts [7]. These forecasts are often employed when the spatio-temporal resolution of the satellite is coarse. The third type of irradiance monitoring networks has high spatial resolution, i.e., within a large solar farm [8], [9]. These networks are designed to capture the short-term movements of clouds using high frequency measurements. Common for all spatial scales is that the predictive performance of the network depends on the density of the monitoring stations. This fundamental consideration must be addressed when designing an irradiance network. The works on irradiance monitoring network design are surprisingly few. Ref. [10] uses principal component analysis to reduce the features of a satellite image sequence, the optimal station placement is then found through the k-means clustering algorithm. This method is later extended to a similar application with a smaller geographical scale [11]. Despite that the methodology adopted in these works is logical, this approach does not consider the predictive performance of the designed network. In this work, we consider the network design problem from another perspective. Instead of taking a data oriented approach, we consider the information embedded in the spatio-temporal irradiance random field. Statistical variability measures such as entropy and covariance are considered in the design framework. The network design approaches in solar engineering literature have two other key differences with the one herein described: (1) the approach in Refs. [10]–[12] uses satellite images as the input data to the design algorithms, (2) their approach designs the network from scratch, i.e., it does not consider the existing irradiance monitoring networks. Building new networks is costly, especially when the networks are expected to be maintained for long period of time. However, there are many available networks which have been operating for decades, e.g., the National Solar Radiation Data Base, the New Energy and Industrial Technology Development Organization. It is sometimes desired to revamp and expand these networks to better accomplish the needs for resources assessment and other emerging applications such as irradiance forecasting. Utilizing the existing monitoring networks is thus economically justified. An example project is described in a recent publication [13], where the irradiance monitoring

2

network maintained by the Australian Bureau of Meteorology is redesigned based on uncertainty modeling. Furthermore, the network construction projects are often multi-phase. After the first phase of the project, ground data become available, which in turn lead to better decisions and inference on stations’ locations in the following phases. Unlike Ref. [13] where satellite data and genetic algorithm are used for the redesign, we consider the ground data and entropy. Although there is no literature on statistical network redesign in solar engineering, rich literature is present in geosciences. Two most adopted approaches are the krigingbased redesign [14], [15] and the entropy-based redesign [16], [17]. The two approaches are being compared in Ref. [18]; we adopt the later in this paper. Before we reveal our redesign criteria, some practical issues associated with solar irradiance spatio-temporal data have to be considered first. In particular, two issues are constantly being overlooked by solar engineers during problem solving, namely, the statistical distribution of the (transformed) irradiance data and the isotropy of the spatial covariance structure. Many statistical procedures have assumptions on distribution of the inputs data. Violation of such assumptions of distribution may lead to unreliable and non-interpretable results. It is therefore mandatory to perform some preliminary analyses, such as the normality test, prior to modeling. The heterogeneous spatial covariance structure of an irradiance random field is another practical issue. In spatial statistics, isotropic variograms or isotropic covariance functions are often used to explain the spatial or spatio-temporal dependency in the data. However, isotropy is rarely observed in environmental processes; many works overlook this by fitting an isotropic model to the anisotropy data. This will result in suboptimal predictive performance at the unobserved locations and at future time. We will address both issues in sections II and III. A. Data The irradiance monitoring network with 24 stations in Singapore was completed in 2013 December. Fig. 1 shows the locations of these stations. The station locations are irregular due to the needs of our two-phase project. The first phase of the project was catered for PV systems performance monitoring and evaluation, therefore the locations of 11 stations (blue dots) follow the available PV systems. In the later phase, 13 additional stations (red triangles) were added to the station-sparse regions to collect data for irradiance forecasting task. Such two-phase or multi-phase projects are commonly encountered in solar engineering, thus we emphasize the importance of monitoring network redesign. The irradiance sensors are installed on the horizontal plane; they thus collect the global horizontal irradiance (GHI). The sensors are calibrated and certified by the Fraunhofer Institute for Solar Energy Systems at a regular time interval of 2 years. The data are sampled at 1s and logged at 1min interval; hourly averaged data are used here in correspondence of the spatial scale of the network. As Singapore is located near the equator, there is no significant seasonal variation in the clear sky irradiance [19]; there are also no significantly different

Fig. 1. Locations of 24 irradiance monitoring stations in Singapore. Blue dots are the stations built in phase one of the project; red triangles are the phase two add-ons. Source: Google Maps, retrieved 2014 December 12.

weather patterns in Singapore throughout the year. Data from 2014 July are therefore used in our analyses. As the focus of this paper is the methodology, the use of one month data is sufficient. II. P RELIMINARY ANALYSIS During a cloud free situation, ground-level irradiance is described by the clear sky models. In solar irradiance modeling, clear sky models are often used to detrend the irradiance time series. The detrended quantity is called the clear sky index which is obtained by dividing GHI by its corresponding clear sky irradiance. We detrend the hourly irradiance time series collected at the 24 stations using an empirical clear sky model [19]. A zenith angle filter of c, where the critical value c is found from the K– S table. After running the K–S tests for all 24 stations, only

3

2 tests reject H0 . Thus we have reasons to assume that our particular set of clear sky index data (all stations) is normally distributed. The station specific mean and variance will be dealt later. Although our data are approximately normal, such distribution of clear sky index data is rare. Asymmetrical distributions are more often observed for clear sky index data. If the observed distribution skews to either side, the Box– Cox transform can be used to make the data more normal distribution-like. Alternatively, ramp rate can be used instead of the clear sky index as the normalized measure for irradiance. B. Covariance function A spatio-temporal process {z(s; t) : s ∈ Ds ⊂ Rd , t ∈ Dt ⊂ R1 } defines a phenomenon such as solar irradiance which evolves through the spatio-temporal index set Ds × Dt . A covariance function C is spatially stationary if it satisfies: C(h; t, r) = cov{z(s; t), z(x; r)}

(1)

where h = s − x; s and x index the locations of the spatiotemporal process, and t and r index the time of the spatiotemporal process. Similarly, spatial isotropy is defined as: C(khk; t, r) = cov{z(s; t), z(x; r)}

(2)

Sometimes, it is preferable to use a normalized measure to model the proximity. A spatially isotropic correlation function is: ρ(khk; t, r) = corr{z(s; t), z(x; r)} (3)

Correlation [dimensionless]

In other words, a spatial correlation function is isotropic if it is only a function of the station separation distance. Figure 2 shows the empirical correlation versus the interstation distance plot using the described dataset. It is obvious that no isotropic correlation function can be fitted.

network for the entire region of interest. Ref. [22] proposed a so-called “spacebender” technique to iteratively divide the space into many strata with roughly equal variance within each stratum. The distances among the stratum centers are then made equal using the fat-ruler algorithm so that the resultant space is isotropic (same distance corresponds to same variance). However, this method requires a large number of observed spatial locations to equalize the variance. This is therefore not suitable for monitoring network applications, however, the method can be readily applied to satellite-derived irradiance data. Among several other alternatives, we consider the two-step method proposed in Ref. [23] most suitable for our application. We call this method the S–G method hereafter. III. T HE S–G METHOD The S–G method considers dispersion as the dissimilarity measure for the time series of spatial process. The algorithm takes two steps, namely, the multidimensional scaling (MDS) step and the thin plate spline (TPS) mapping step. The method considers two planes: the anisotropic geographical plane (G plane) and the approximately isotropic dispersion plane (D plane). Unlike the G plane, where the inter-station distance is calculated based on the geographical separation among the stations; the inter-station distance in the D plane is a monotone function of dispersion. In the MDS step, the dispersion matrix is used to retrieve the D plane distance. A one-to-one correspondence between the G and D planes is then constructed via TPS mapping. A. The D plane representation The goal of this section is to retrieve the D plane coordinates using the dispersion. The correlation dispersion between two locations si and sj is defined to be:
dij = 2 − 2corr z(si ; t), z(sj ; t) (4) where corr(·) denotes empirical correlation. The non-metric multidimensional scaling approach can be used to retrieve the stations’ location in the D plane [23]. The non-metric MDS aims at minimizing the stress:
X hij − ψ(dij ) 2 X (5) h2ij i