3.1 Reliability of simulator responses

The experimental data are obtained with the aforementioned simulator; in order to demonstrate its reliability, two types of indexes were developed: approach to the measurement of solar radiation (Iam) and approach to the operation of the turbine-generator binomial (Iao), both represent the proximity of the information measured and that generated by the program. The simulator is considered reliable if the differences with the measured reality do not exceed 5% [¹⁰]. Iao was developed for the two possible input signal values. These indices were generated monthly, and 95% confidence intervals were obtained for all three. In addition, hypothesis tests were developed at 5% significance; the values tested in the hypotheses were 0.98 for the Iam and Iao signal 60, and 0.99 for the Iao signal 45; in all three analyzes, the value 1 represents the exact match to the measurements. Table 1 shows the monthly values of the indices. In the table, the 95% confidence intervals are very close to 1, and the averages obtained do not exceed 3% differences.

In this way, the simulator operates every 30 minutes for 17 intervals in the working hours of CEDER staff. It reports what is purchased or delivered in power units (kW) each day because it makes the sum of the differences between what is consumed and what is generated. The main reason for carrying out the monthly analysis is the billing; in this way, what is consumed, what is produced, and what is purchased from the external network are compared. Furthermore, in the best months of radiation, it is prioritized, from the simulator, not to deliver power to the supplier.

Table 1 Verification and approach to the real behavior of the data. 

3.2 Dual response statistical models in RSM applying CCD and Regression.

The analyzed sample space is shaped as follows: two factors that are the increase of the nominal power capacity of the 7 AFVs (X ₁) and the increase in the geometry of the turbine-generator backup system (X ₂); two experimental responses, the monthly average daily purchased power (Y ₁) and the standard deviation (SD) of . Y ₁ (Y ₂ ).By studying the simultaneous behavior of the two responses, the concept of dual response is formed.

MSR allows the researcher to inspect one or more responses that can be displayed as a surface when experiments look for the effect of varying quantitative factors on the possible values of a dependent variable or response; in other words, it tries to find the optimal values of the independent variables by maximizing, minimizing, or fulfilling certain response restrictions[¹¹].

The most efficient experimental designs are those that allow the study of curvatures. The CCD design, originally called Box-Wilson [¹²], has been shown to require fewer trials to obtain reliable information. It is made up of an augmented full factorial design, with central and axial experimental points. The inclusion of central replicas favors the estimation of the general variance of the process of changes of values in the factors. Adding axial points, whose axial distance “a” can be defined by the researcher, gives qualities to the general variance profile, namely: (1) axial distance in phase with α = 1 generates a variance profile similar to the full factorial design with an additional level of study; (2) rotatable axial distance with value α > 1 creates a variance profile that tends to be constant as a function of the radius in the plane of a pair of studied factors; (3) orthogonal axial distance with value α ≥ 1 produces a variance profile that tends to be uncorrelated with the quadratic effects under study; (4) spherical axial distance with α > 1 value generates a variance profile that tends to be constant as a function of the radius in a triple of studied factors. Obviously, the number of replicas considered by the researcher will modify the values of these distances, except for the axial distance in phase, which is always α = 1 [¹¹].

To exemplify the above, Table 2 shows some values of the factors under study (f), factorial replicas (n), axial replicas (a), central replicas (c). It can be seen in the table that the allowed values for the axial replicas must be less than or equal to the factorial replicas, the central replicas can be greater than or equal to the axial replicas and can exceed the replicas at each factorial point long as the statistical test of lack of fit (LOF) is not significant in all CCD design.

Table 2 Values of a for factor and replica conditions. 

In the case study, the axial distance α = √2 was selected, with a rotatable feature to achieve the tendency that the SD is a function of the radius from the center to the limit of the sample space whose domain is the plane X₁ − X₂. The CCD used contains two factors, two replicates at each factorial point, two replicates at each axial point and four replicates at the central point, the reason for having selected this combination is to take care of a homogeneous weighting at all the noncentral points of the domain. On the other hand, increasing central replicates improves the estimation of SD, but increases the probability of making significant the LOF.

According to [¹⁴] the MSR using CCD formulates that this type of experimental design is based on approaches (1) and (2). be the response surfaces for the mean Y₁ and standard deviation Y₂ of the combined study process.

Where B and C are matrices of size k × k containing the second-order terms of the estimated coefficients of each response model; b and c are vectors of size k × 1 with the first order terms of the estimated coefficients of each model; b₀ and c₀ are scalar additive constants and x is a k × 1 vector of control or design variables. [¹⁵] details the general considerations for estimating models (1) and (2). On the other hand, the regression technique allows handling polynomials of any degree such that their effects are significant; this facilitates the study of curvature changes in fitted models. Regression uses least squares methods leaving the degree of the polynomial free. The theoretical support for this technique is based on the linear statistical model shown in Equation (3).

The previous model is expressed in a matrix in Equation (4):

The matrix X is of size nx(p + 1) where n ≥ p + 1, vector y is nx1, vector b is (p + 1)x1 and vector ε is nx1. Where n represents the total number of experimental points, p represents the number of parameters β _j of the regressor variables xi1, xi2, . . . , xip, β₀ represents an additive constant that, since there are no effects caused by the regressor variables, usually represents the mean of the response variable y. The estimator of vector b in expression (4) where X′ X is the variance-covariance matrix is defined, according to [¹⁶] by Equation (6).

For the regression analysis, as in the CCD technique, the same responses were studied (Y₁, Y₂), in this case they are represented by Equations (7) and (8).

On the other hand, the increases allowed in the AFVs and the power of the backup system, as well as the values studied are found in Table 3.

Table 3 Nominal values, allowable increments and coded values. 

The encoding of x₁ and x₂ shown in the table, is in the function of adjusting the domain of the sample space; these values are used in MSR and in regression. In particular, for the regression, the domain point (x₁, x₂) = (−α,−α) corresponds to the set of values introduced to the simulator of the current situation of CEDER.

The array of values used for Y₁ and Y₂ are found in Figure 2. The factorial points of the CCD are inscribed in the circle allowing for curvature analysis in four directions, and the rotatable axial distance is at the ends of the X₁ and X₂ axes. On the other hand, the regression technique leaves the SD profile free according to its possible polynomial adjustment.

Figure 2 Configuration of the sample space for the 2 techniques. 

The sample space for CCD is formed exclusively within the circle; on the contrary, the regression technique includes the outer points (Pcurrent, PaA, Pmax and PAa). Where X₁ and X₂ are the factors under study given in percentage increments. On the other hand, the geometry of the experimental points is shown in Table 4.

Table 4 Factor positions and replicas of the CCD and regression. 

The responses Y₁, Y₂ were obtained by creating the experimental simulation of 20 and 28 years of operation for CCD and regression, respectively. When obtaining the data for each year, the number of days of power delivery (kW) per month to the supplier's electricity grid was counted. The categorical analyzes direct both studies to the months where greater participation of the supplier is required. The minimization in the regressions was carried out with the Solver complement of the Excel, under the condition of the Nonlinear Generalized Reduced Gradient (GRG) and the statistical analyses were done with [¹⁷].

4.1 CCD analysis

November has a significant behavior of the variable Y₁ in the effects: X₁, X₂, X₁ X₂, and , on the other hand, the variable Y₂ is significant in the effects X₁, X₂ and . The adjusted model can be observed in the prediction profile section of Figure 3. It should be noted that all statistical analyses meet significant relationships with α ≤ 0.05.

Figure 4 represents the three-dimensional behavior of Y₁ (brown) and Y₂ (blue). Point A (1,1) is the minimum obtained using the CCD technique and is valid for both responses.

The contour plot, Figure 5, details the demand savings regions. The blue area represents the lowest demand section required. It is contained between the following points (x₁, x₂) : (−0.5, 1.2); (0, α); (1, 1); (α, 0); (1.3,−0.2); (0.7, 0); (0.5, 0.5); (0, 0.8) and (−0.5, 1.2) the most recommended area is the first quadrant of (x₁, x₂) ∈ E².

Figure 3 Statistical analysis of the CCD with a dual response for November. 

Figure 4 Three-dimensional schematic of the fitted models of 𝑌 1 and 𝑌 2 . 

Figure 5 Categories of possible savings in November. 

In November, the region with the least required demand is intermediate in size, December covers a larger area, and January presents a smaller savings section. If greater savigns are needed, it is recommended to locate the increase decisions in the smallest region, so the effect in the months of December and November will be greater, and in January, an acceptable one will be reached, on average. Figure 6 splices the 3 previous zones. The most recommended area in all three cases is the proximity to point (1,1), that is, the first quadrant. Table 7 indicates the requested demand of the three months with the modification of X₁ and X₂ at that point.

Figure 6 Regions with the highest savings in the three months studied. 

If CEDER had the possibility of increasing the two factors to the point (1.1), the estimated average daily savings with reference to the current situation are: November (87.87%), December (83.53%), and January (74.04%). Additionally, a significant decrease in SD is achieved. On the other hand, the relative increase in CV is due to the greater decrease in Y₁ compared to the decrease in Y₂.

4.2 Regression analysis

Applying the regression technique for the month of November, the approximations of the two responses were obtained. Figure 7 shows their behaviors.

Figure 7 Dual Response Regression Statistical Analysis for November. 

Unlike CCD, when doing regression, the cubic effect of X₁ in both responses was significant, allowing to identify the change in curvature in this variable. Furthermore, it confirms that the FV contribution is dominant. By having a larger sample space, the minimum values of the responses are found at (alpha, 0.681) and (alpha, 0.969), respectively. For this reason, regression does not recommend increasing X ₂ to alpha, since the value of both responses would be higher.

Since the primary response is more important than the secondary one, the results obtained in it are prioritized; in this way, regression suggests locating the minimum in (alpha, 0.68087). Figure 8 shows three-dimensional curvature changes of the cubic models of Y₁ and Y₂, as described in the prediction profile in Figure 7. The minimum point A (alpha, 0.681) is shown in Figure 8.

The boundary graph, Figure 9, details the demand-saving regions. The blue area represents the lowest demand section required. It is contained between the following points (−0.7, α); (α, α); (α,−0.25); (1.3,−0.25); (0.7, 0); (0.5, 0.5); (0, 0.8);(−0.6, 1.2) y (−0.7, α). Again the most recommended area is the first quadrant of (x1, x2) ∈ E2;this region contains what is proposed by the CCD technique.

Table 7 Increased state at the optimal point of the CCD Naa (1,1) 

Figure 8 Three-dimensional schematic of the fitted models of 𝑌 1 and 𝑌 2 . 

Figure 10 shows the regions where the best minimizations of the three months studied by the regression technique are achieved. The change in the order of the sizes of the areas lies in the fact of having increased the sample space allowing polynomial adjustments of a higher degree.

It can be seen that for the months of January and December, the regression favors the backup system, while for the month of November both techniques are more stable in their minimum region. Table 8 indicates the requested demand for the three months if X₁ and X₂ were modified at point A.

Table 8 Increased state at the optimal point A of the regression fit. 

If CEDER had the possibility of increasing the two factors to point A, the estimated average daily savings with reference to the current situation are: November (92.77%), December (85.09%), and January (70.08%). In the comparative study of the three months, January is the month with the lowest savings and the highest SD.

Figure 9 Categories of possible savings in November. 

Figure 10 Regions with the highest savings in the three months studied.