Percentage Match, Inequality coefficient, Correlation Coefficient, Residual area, Shared area

Matching Statistical Indicators

To assess the temporal match between demand and supply profiles, a number of different statistical methods are employed to define the match or goodness of fit between two profiles. There are two elements that require to be considered: magnitude and phase. Ensuring the magnitude of supply corresponds to that of the demand is the basis of energy matching and has been applied in other work. However, focusing on match magnitude alone could lead to the energy being supplied at times when demand is low. To ensure generation is not wasted, matching must also encompass the relative phases of demand and supply.

Consider a small UK commercial demand supplied from a 3 kW wind turbine and a 3 kW PV installation over a one week period in June. Figures 1 and 2 illustrate the temporal match between the demand and the wind system and the demand and the PV system respectively. Visual examination of these figures highlights the difficulty in determining which, if any, of the RE systems is best matched to the demand. The wind system output is seen to increase throughout the week, with its most productive period occurring over the weekend where the demand is at its lowest. The PV system output is seen to exceed the demand over the midday hours on all but one day. Neither system meets the entire demand over the period and both graphs contain periods when the supply exceeds the demand, which would either require to be grid exported or wasted.

Figure 1: Matching with a wind system Figure 2: Matching with a PV System

An energy self-sustenance index can be defined for different energy measures based on temporal end use as described by Eqn (1). A building’s self-sustenance is the ratio of the demand displaced by on-site generation, to the demand without generation. The optimal value for self-sustenance is taken to be unity, whereby all the sites demand could be displaced by on-site generation. The difficulty with this index is that although it accounts for displaced energy, it neglects excess production.

(1)

where ESS_x is the site energy self sustenance for energy type x, disp_x,t the energy displaced by the generation system for energy type x at time t, dem_x,t the energy demand of a building without generation for energy type x at time t and n the total number of time steps.

Similarly, match evaluation could be based on the area common to both profiles. The Shared Area, SA, is described by Eqn (2) as the union, denoted by U, between the area under the demand and supply profiles. This value can be approximated by evaluating the area between the x-axis and the lowest value between supply and demand for every time step.

(2)

where D(t) is the demand profile, S(t) the supply profile and n is the time period.

When comparing scenarios where one of the profiles is common, i.e. a demand profile matched to a number of supplies or vice-versa, the shared area can be used to compare the individual matches. For the above example, the shared area in the wind scenario is 63.93 kWh, and in the case of PV is 44.84 kWh. This indicates that the wind system is better matched to the demand. However, the shared area term becomes less meaningful when comparing matches between different demands and supplies. For example, if demand was doubled and the number of PV system panels doubled, the shared area term would double and so too would the residual. Thus, shared area needs to be expressed as a percentage of the demand area, as described in Eqn (3), to be used as a valid means for comparing different sets of profiles.

(6.3)

Although this term describes the portion of demand satisfied by the RE system, it gives no indication of excess supply. The excess supply, ES, can be obtained by subtracting the shared area from the supply area and expressing the result as a percentage of the supply as described by Eqn (4). However, this now requires that both terms are included to make valid comparisons between different sets of profiles. Additionally, a question of priority is introduced as to whether the importance is in meeting the demand or in using the supply. To achieve a perfect match both are equally important and, in the case of a perfect match, %SA would equal 100 and %ES zero. The addition of these terms yields an optimum value of 100, however non-perfect match values could range above and below this figure, making judicious comparisons difficult.

(4)

The Residual, r(t), of two profiles can be used to represent the combined profile and can be obtained by subtracting the supply at each time step from the demand, as described by Eqn (5). This is the profile the supply grid would ‘see’. There are two possible ‘net-profiles’, one where the excess supply is exported to the grid and one where it is dumped to some external load. The residual profiles from the above example are illustrated in Figure 3, where the excess is exported, and in Figure 4, where the excess is dumped. It can be seen that the variability on the grid is increased in the case of exporting surplus energy production.

r(t) = D(t) – S(t) (5)

Figure 3: Demand and residual profiles associated with and PV systems, grid export.

Figure 4: Demand and residual profiles associated with wind and PV systems, no grid export.

However, dumping the surplus supply will rarely promote the use of RE technologies. The two ranking statistics, mean and variance, which represent the energy content and variability of a profile, have been calculated for both export and non-export scenarios and are illustrated in Figure 5. Figure 51 illustrates the reduction in energy requirement due to RE deployment. In the cases associated with grid export, this is particularly pronounced due to the negative values associated with export. The variability is increased significantly through grid export, which if this were to occur on a large scale, could result in grid instability. Without export, the variability is only reduced moderately, and the unfavourable aspects of wasting RE are not accounted for. The rank of the demand profile can be seen to increase with RE integration without grid export, although with grid export improvements in rank become unclear. The match between demand and supply is not accurately described using this method.

Figure 5: Statistics associated with various matching scenarios.

Meanwhile, the least-squares approach can be used to quantify the magnitude of deviation between two sets of variables and has been used in quantifying differences in simulated time series. This method can be applied to supply and demand profiles, where the least-squares value is obtained using Eqn (6). This metric will always yield a positive value, with the lower limit of zero indicating a perfect match (there is no upper limit). The least-squares method can be used to compare different matches, with the lowest value indicating the optimal match. In the current example, the least squares metric between the PV and demand profiles is 31.5 and, for the case of wind, 20.9. It is clear that the match with wind is best. However, because of the lack of an upper limit, it is difficult to establish the difference in the quality of match. Where numerous profile pairs are to be compared, bands defining the quality of match are useful in processing various possibilities, although establishing such bands is difficult where a worst case cannot be defined.

(6)

Spearman’s Rank Correlation Coefficient describes the correlation between any pair of variables by calculating the degree to which the variables fall on the same least-square line. Calculation of this coefficient will result in a value between -1 and +1. A result of 1 indicates perfect positive correlation and –1 perfect negative correlation, i.e. as one variable tends to increase the other will decrease at the same rate. A value of zero denotes no correlation between the variables. The correlation coefficient, CC, between a demand and supply profile is calculated by Eqn (7). For the previous example, the correlation between the PV profile and the demand is 0.115, while for wind it is 0.604. The figures again demonstrate the match with wind to be superior. The coefficient is used to describe the trend between two data sets and does not consider the relative magnitudes of the individual variables. Thus, if the wind system were doubled in size the correlation coefficient would remain the same even though the excess supply would be far greater. Additionally, two profiles perfectly in phase with one another, but of very different magnitudes, would result in a perfect correlation, but not a perfect match. Nevertheless, it provides a measure of the potential match that could exist given changes to the relative capacities, i.e. through energy efficiency or altering the size of the RE system.

(7)

where D_t is the demand at time t, S_t the supply at time t, d the mean demand over time period n and s the mean supply over time period n.

The Inequality Coefficient, IC, which was originally used to validate building thermal simulation models, describes the inequality in a time-series due to three sources: unequal tendency (mean), unequal variation (variance) and imperfect co-variation (co-variance) as described by Eqn (8). The resultant coefficient can range in value between 0 and 1, with 0 indicating a perfect match and 1 denoting no match. The inequalities in the above example are 0.375 and 0.299 for PV and wind respectively. This metric is ideally suited to establishing bands of match, where matches resulting in inequalities between 0 and 0.1 could be termed good matches, with bad matches indicated by values between 0.9 and 1.

(8)

In Merit, the Percentage match is defined using this Inequality coefficient as described by Eqn (9).

Percentage match (%) = (1 – Inequality Coefficient)*100 (9)

Based on the Percentage match values, 10 categories evaluating bands are defined as below:

Percentage match >= 99: Perfect Match

Percentage match > 90 : Excellent Match

Percentage match > 80 : Very Good Match

Percentage match > 70 : Good Match

Percentage match > 60 : Reasonable Match

Percentage match > 50 : Poor Match

Percentage match > 40 : Very Poor Match

Percentage match > 30 : Bad Match

Percentage match > 20 : Very Bad Match

Percentage match > 10 : Almost No Match

Percentage match > 0 : No Match

Example of best overall search with statistics

Consider the example shown in Figure 6, which illustrates a mixed set of supply and demand profiles. The demand profiles range in weekly consumption from 100 kWh for a residential profile, up to 700 kWh for an industrial profile. The supply profiles include a number of different PV and wind turbines configurations ranging in size up to a 10 kW. This example demonstrates the difficulty of supply and demand matching, as the optimum combination could consist of a mix of demand and supply profiles. Following the best overall search procedure for these 14 profiles generates 16,129 possible combinations.

Figure 6: Example of matching numerous profiles.

These combinations were filtered to find the best match using a single variable search based on the inequality metric to define the quality of match between profiles. The result is illustrated in Figure 7. The inequality metric for this combination is 0.27 and the correlation 0.42. The combined supplies incorporate 20 mono-crystalline PV panels with solar tracking devices, a hybrid PV façade consisting of 30 vertical south-facing mono-crystalline panels, 10 ducted wind turbines facing south-west. These supplies are matched to the combined profiles of a small commercial and agricultural demand. From this result, it can be seen that the PV systems meet the mid-day peaks while the wind systems help to meet some of the base load.

Figure 7: Best supply and demand match based on the inequality metric.

A different result is obtained if the correlation coefficient is used as the filtering variable, as illustrated in Figure 8 where the correlation is 0.62 and the inequality 0.43. The best correlation is achieved with a small commercial demand profile, combined with a PV hybrid façade and 10 ducted wind turbines. These results enable the scope of the search to be refined, eventually leading to system sizing. The optimisation process is only valid over the period for which it is performed and should therefore be performed on a range of periods encompassing typical annual seasons over a full year.

Figure 8: Best supply and demand match based on correlation.

Carrying out a demand led search on the profiles used in the previous example requires the evaluation of 127 possible combinations for every demand profile, giving 889 in total. This reduces the data processing required for the best overall search by 95%. The results obtained from a demand led search for this example are detailed in Table 1. As can be seen, the small commercial profile is best matched with a combination of 20 PV panels with tracking, and 10 ducted turbines. Examining regions of supply deficit enables certain observations to be made that could increase the match. For example, in the match illustrated for the small commercial demand, an afternoon deficit is observed that could possibly be met by a west-facing PV array. However, the addition of such a system will also contribute to the supply peaks seen around noon, thereby increasing energy wastage or requiring export.

Table 1: Statistics from best per demand led search.

Demand	Supply	Inequality	Correlation	Shared Area	RE	Export	Import
Residential	10 ducted wind turbines, 100W	0.35	0.29	53.49	60.9	7.27	46.4
Small commercial	20 PV with tracking, 10kW	0.29	0.48	96.61	118.9	22.18	53.2
Industrial	20 PV with tracking, PV roof, 10 ducted wind turbine,100 W, & 10 kW	0.31	0.02	322.54	395.4	72.08	176.7
Agricultural	Hybrid façade, 10 kW turbine	0.29	0.26	219.55	293.3	73.08	79.7
Street lights	10 ducted wind turbines, 10 kW	0.53	-0.22	91.39	257.1	164.9	107.8
Medium commercial	PV roof 10 kW turbine	0.3	0.2	244.96	323.1	77.38	104.3
Large commercial	20 PV with tracking, PV roof, 10 ducted wind turbines, 100 W & 10kW	0.3	0.15	325.23	395.4	69.39	174.0