It has long been found that the total (relative) variability of paralleled solar generators (or wind generators) is less than that produced by a single system (eg, Wiemken et al., 2001; Murala et al., 2009).

For example, the variability of 1 location in Figure 6.21 is compared to the variability of 25 locations within a 4km x 4km range. When the fluctuations produced by different locations are comparable but not correlated, the regional performance of the corresponding locations can be smoothly and precisely quantified (Mills and Wiser, 2010; Hoff and Perez, 2010). In this case, the variability of the individual systems that make up a system combination can be expressed as:

where is the variability produced by a single location and N is the number of locations. The results of variability are directly derived from the strong law of numbers. According to this law, as the number of observations approaches infinity, the mean of the sequence of independent and identically distributed random variables converges to the mean of the same distribution with probability 1 (Ross, 1988, 346).

For partial position correlation, we can intuitively understand that: ① If the positions of two systems are just adjacent, their fluctuations will occur almost simultaneously, and relatively speaking, the changes caused by the fluctuations are basically the same as the variability of each single position ; ② If the positions of the two systems are far apart, they will fluctuate independently, with a consequent smoothing effect [Equation (6.2)].

A smoothing effect occurs when between the two extremes, but to a lesser extent than the 1/√N trend:

Among them, ρ represents the pairwise correlation of positions, and its value is between 0 and 1. therefore,

It is critical that we determine how the correlation of pairwise positions varies with influencing factors. The influencing factors include: ① the distance between locations (D); ② the studied time interval (Δt); ③ the cloud velocity (CS) that produces fluctuations. Through the above discussion, we can understand the effect of distance as follows: the correlation of the same location is equal to 1, and as the distance between the locations increases, the correlation gradually decreases, and the correlation is 0 until the distance is enough to independently generate fluctuations.

• The time interval (Δt) that defines the fluctuation is relevant because it is related to the strength of the cloud disturbance causing the fluctuation. The fine structure of cloud fields (eg, small individual cloud layers) produces high-frequency fluctuations. The correlation of these fluctuations decreases sharply with distance. Low-frequency fluctuations are caused by large-scale structures such as entire cloud fields or weather peaks. On longer timescales, two unrelated locations at small structural levels may experience nearly the same variability in synchronization. Therefore, at this time scale, they are highly correlated.

Cloud velocity also affects pairwise position correlations, as it is the root cause of variability: simply, only moving clouds cause fluctuations. To demonstrate this, it is assumed that the moving cloud structure remains roughly constant over the time period we study. In this way, the faster the cloud structure moves, the ① the smaller the signal time shift between the two locations and the greater the correlation between the locations (when the cloud size is much larger than the sensor separation); and ② the moving along the cloud The greater the direction, the greater the distance between two locations that produce the same fluctuation in a given time interval, the longer they will exhibit a given correlation.

Note: The given cloud size and cloud velocity determine the time interval of the associated fluctuations.

We investigate the relationship between Δt, CS, and D with empirical evidence from multiple sources. For example, Mills and Wise (2010) analyzed data from the ARM network (Stokes and Schwartz, 1994), which also included GHI measurements at 32 locations at 20 s. They noted that om; exhibits an exponential decay as the distance between locations changes. And, they also observed that the rate of this exponential decay is a continuous function of the time interval Δt we are studying. However, since the shortest distance between any two locations in the ARM network is 20 km, they cannot observe the dynamics when Δt is less than 10 min. (See Figure 6.3.):

Hoff and Perez (2012) repeated this experiment using standard resolution (10km) hourly satellite irradiance. In their experiments, they observed a similar exponential decay and, at time intervals of 1h, 2h, and 3h, predicted the distance relationship from At. They also pointed out that there are different exponential decays in different study areas. They attribute these differences to the prevailing cloud velocity in the region. (See Figure 6.4.):

Perez et al. (2012) analyzed the 20s ARM data, and under the assumption that the cloud layer structure remained unchanged, the irradiance of each observation point was inferred through the cloud guided wind retrieved by the satellite, and each ARM observation was obtained. A single-dimensional grid of data around the station. Thus,

They were able to analyze the data at high frequencies (Δt=20s) and short distances. They quantified how the correlation decayed with distance and Δt, and defined the point at which fluctuations at two locations become uncorrelated as the uncorrelated tipping point. They also observed that distance is linearly related to Δt. It is important to note that their findings must be validated by analyzing real 2D, high-density network data, especially the negative correlation peaks evident in Figure 6.4. It is the result of a negative correlation between the unaltered cloud structure, passing from the real location to the downstream virtual location. In two-dimensional networks, only some of these negative peaks are apparent.

Hoff and Norris (2010) analyzed data from a modular network of 25 locations with coverage ranging from 400m × 400m to 4km × 4km (Figure 6.5):

They observed the same trends as in the one-dimensional virtual network, including a negative correlation in the direction of cloud velocity. Through qualitative observations, they found that cloud velocity, obtained from satellite cloud movement alone, affects the rate of decay.

Perez et al. (2011) systematically quantified distance trends due to changes in Δt and CS by using (true 2D) high-resolution (lkm, lmin) satellite observations in some regions of the United States ( Figure 6.6), and proposes the following experimental formula with variables , A, , CS, and D:

The linear relationship between the uncorrelated critical distance and the time interval studied, proposed by Perez et al. (2012), was validated, but adjusted to reflect the dependence of this relationship on cloud velocity.

Bing et al. (2012) selected a highly variable 30-day study from a network of 66 newly constructed measuring stations within the Sacramento Municipal Division (SMUD) area, which covers an area of approximately 200 km². The radiance at each location is measured with a time rate of change of lmin. The researchers obtained high-altitude cloud velocity from satellite images, and their results confirmed the preliminary empirical relationship between Δt, CS, and D.

Read more: Direct, scattered and total solar radiation and instrumental measurements