Proposal of an optimization tool for demand response in island electricity systems (IES) using the Simplex method and Generalized reduced gradient (GRG)

2.3.1 Development of a tool for GHG minimization

As already mentioned, the aim is to develop a tool to optimize the choice of power production equipment, with its application to those existing in Gran Canaria, based on their performance, fuel consumption, type of fuel, etc., which reduce GHG. In this case, a linear function to be optimized with linear restrictions is finally obtained, which is why the Simplex optimization method or Simplex algorithm has been used. As already indicated, the tool seeks to optimize fuel consumption and type and therefore minimize GHG emissions. Based on this, the decision variables, the objective functions, and the constraints will be defined.

DECISION VARIABLES: The instantaneous powers of the different production teams were taken as decision variables. This is:

TARGET FUNCTION: In terms of the target function, the aim is to minimize GHG emissions. This objective function defines the hourly emissions of energy production equipment as a function of the installed power or power produced in an hour. To achieve the objective function and determine the decision variables, we work with the emission factor equations of the producing equipment, transforming them until the appropriate objective function is reached. For each type of production technology, the following must be defined: $$F(tC{O}_{2eq}/MWh)=\frac{\mathit{GHG}(tC{O}_{2eq})}{\mathit{Energeticproduction}(MWh)}$$ $$F(tC{O}_{2eq}/MWh)=\frac{\mathit{GHG}/time(tC{O}_{2eq}/h)}{P(MW)}$$ $$\mathit{GHG}/time(tC{O}_{2eq}/h)=F(tC{O}_{2eq}/MWh)xP(MW)$$ $F\left(tC{O}_{2eq}/MWh\right)P\left(MW\right)GHG\left(tC{O}_{2eq}\right)$ is the emission factor of each technology, $P(MW)$ it's power by technology, and $GHG\left(tC{O}_{2eq}\right)$ are the greenhouse gases produced by each technology. Operating with this equation for each technology we obtain the following objective linear functions per technology. $${\mathit{GHG}}_{\mathit{TG}}/t({\mathit{tCO}}_{2eq}/h)=1.133·P_{\mathit{TG}}$$ $${\mathit{GHG}}_{\mathit{TV}}/t({\mathit{tCO}}_{2eq}/h)=0.812{·P}_{\mathit{TV}}$$ $${\mathit{GHG}}_{\mathit{CC}}/t({\mathit{tCO}}_{2eq}/h)=0.603·P_{\mathit{CC}}$$ $${\mathit{GHG}}_{D4T}/t({\mathit{tCO}}_{2eq}/h)=0.649{·P}_{D4T}$$ $${\mathit{GHG}}_{D2T}/t({\mathit{tCO}}_{2eq}/h)=0.649{·P}_{D2T}$$ The sum of all of them produces the overall objective function of Gran Canaria's energy system to minimise GHGs. $$\mathit{GHG}/t({\mathit{tCO}}_{2eq}/h)=1.133·P_{\mathit{TG}}+0.812{·P}_{\mathit{TV}}+0.603·P_{\mathit{CC}}+0.649{·P}_{D4T}+0.649{·P}_{D2T}$$ CONSTRAINTS: The constraints taken are the power intervals at which the equipment works, as well as the power needed to be reached at the time to cover the demand. These power ranges are those of the operation of the power-producing equipment, defined by the manufacturer itself.

2.3.2 Application of the Generalized reduced gradient method

As already indicated, the tool seeks to optimize fuel consumption and type and therefore minimize production costs. Based on this, the decision variables, the objective functions, and the constraints will be defined. Taking into account the behaviour of the production equipment, the current energy production of Gran Canaria is supported by 4 steam turbines, 5 diesel equipment, 5 gas turbines, 2 combined cycles with double gas turbine and steam turbine. Among the data studied is the variation of the efficiency (%) versus the instantaneous power (MW) produced, the fuel consumption as a function of its operating regime, type of fuel, etc. To achieve the objective functions, we work with the equations of the power-performance curves of the producing equipment, transforming them until the appropriate objective function is reached.

LOGARITHMIC HIGH CURVE: $$\begin{array}{c}{P}_{\mathit{TG}=}0.0003·{\mathrm{e}}^{33.318·\mathrm{\eta }}{\mathrm{\eta }}_{\mathit{TG}}=0.0288·\mathrm{l}\mathrm{n}{(P}_{\mathit{TG}})+0.2509R^2=0.9605\\ P_{\mathit{TV}=}0.0002·{\mathrm{e}}^{34.957·\mathrm{\eta }}{\mathrm{\eta }}_{\mathit{TG}}=0.0271·\mathrm{l}\mathrm{n}{(P}_{\mathit{TV}})+0.2447R^2=0.9461\\ \begin{array}{c}{P}_{\mathit{CC}=}0.0092·{\mathrm{e}}^{19.494·\mathrm{\eta }}{\mathrm{\eta }}_{\mathit{CC}}=0.0434·\mathrm{l}\mathrm{n}{(P}_{\mathit{CC}})+0.2738R^2=0.8451\\ P_{D4T=}2·{10}^{-5}·{\mathrm{e}}^{26.707·\mathrm{\eta }}{\mathrm{\eta }}_{D4T}=0.035·\mathrm{l}\mathrm{n}{(P}_{D4T})+0.4181R^2=0.9348\\ P_{D2T=}2·{10}^{-10}·{\mathrm{e}}^{48.417·\mathrm{\eta }}{\mathrm{\eta }}_{D2T}=0.0198·\mathrm{l}\mathrm{n}{(P}_{D2T})+0.4614R^2=0.9587\end{array}\end{array}$$

LOGARITHMIC LOW CURVE: $$\begin{array}{c}{P}_{\mathit{TG}=}0.0002·{\mathrm{e}}^{43.91·\mathrm{\eta }}{\mathrm{\eta }}_{\mathit{TG}}=0.021·\mathrm{l}\mathrm{n}{(P}_{\mathit{TG}})+0.2055R^2=0.9239\\ P_{\mathit{TV}=}1·{10}^{-8}·{\mathrm{e}}^{79.548·\mathrm{\eta }}{\mathrm{\eta }}_{\mathit{TV}}=0.0118·\mathrm{l}\mathrm{n}{(P}_{\mathit{TV}})+0.2319R^2=0.9378\\ \begin{array}{c}{P}_{\mathit{CC}=}2·{10}^{-8}·{\mathrm{e}}^{57.073·\mathrm{\eta }}{\mathrm{\eta }}_{\mathit{CC}}=0.0164·\mathrm{l}\mathrm{n}{(P}_{\mathit{CC}})+0.3203R^2=0.9334\\ P_{D4T=}0.0001·{\mathrm{e}}^{3.369·\mathrm{\eta }}{\mathrm{\eta }}_{D4T}=0.0415·\mathrm{l}\mathrm{n}{(P}_{D4T})+0.3927R^2=0.9708\\ P_{D2T=}2·{10}^{-17}·{\mathrm{e}}^{85.763·\mathrm{\eta }}{\mathrm{\eta }}_{D2T}=0.0101·\mathrm{l}\mathrm{n}{(P}_{D2T})+0.4588R^2=0.8625\end{array}\end{array}$$

HIGH POTENTIAL CURVE: $$\begin{array}{c}{P}_{\mathit{TG}=}6,087,683.73·{{\mathrm{\eta }}_{\mathit{TG}}}^{11.534}{\mathrm{\eta }}_{\mathit{TG}}=0.2581{{·P}_{\mathit{TG}}}^{0.0867}{R}^2=0.9432\\ P_{\mathit{TV}=}56,352,113.2·{{\mathrm{\eta }}_{\mathit{TV}}}^{13.245}{\mathrm{\eta }}_{\mathit{TV}}=0.2599{{·P}_{\mathit{TV}}}^{0.0755}{R}^2=0.9311\\ \begin{array}{c}{P}_{\mathit{CC}=}233,083.76·{{\mathrm{\eta }}_{\mathit{CC}}}^{10.288}{\mathrm{\eta }}_{\mathit{CC}}=0.3008{{·P}_{\mathit{CC}}}^{0.0972}{R}^2=0.8144\\ P_{D4T=}94,042.024·{{\mathrm{\eta }}_{D4T}}^{13.175}{\mathrm{\eta }}_{D4T}=0.4193{{·P}_{D4T}}^{0.0759}{R}^2=0.9201\\ P_{D2T=}254,075,634.4·{{\mathrm{\eta }}_{D2T}}^{25.062}{\mathrm{\eta }}_{D2T}=0.462{{·P}_{D2T}}^{0.0399}{R}^2=0.9531\end{array}\end{array}$$

LOW POTENTIAL CURVE: $$\begin{array}{c}{P}_{\mathit{TG}=}305,809,752.73·{{\mathrm{\eta }}_{\mathit{TG}}}^{12.52}{\mathrm{\eta }}_{\mathit{TG}}=0.2099{{·P}_{\mathit{TG}}}^{0.0799}{R}^2=0.9029\\ P_{\mathit{TV}=}1,480,368,843,832,330.0·{{\mathrm{\eta }}_{\mathit{TV}}}^{24.21}{\mathrm{\eta }}_{\mathit{TV}}=0.2363{{·P}_{\mathit{TV}}}^{0.0413}{R}^2=0.941\\ \begin{array}{c}{P}_{\mathit{CC}=}378,757,475,895.23·{{\mathrm{\eta }}_{\mathit{CC}}}^{23.75}{\mathrm{\eta }}_{\mathit{CC}}=0.3255{{·P}_{\mathit{CC}}}^{0.0421}{R}^2=0.9242\\ P_{D4T=}22,536.66·{{\mathrm{\eta }}_{D4T}}^{10.79}{\mathrm{\eta }}_{D4T}=0.3949{{·P}_{D4T}}^{0.0927}{R}^2=0.9623\\ P_{D4T=}12,710,359,419,211,600.0·{{\mathrm{\eta }}_{D4T}}^{47.62}{\mathrm{\eta }}_{D2T}=0.459{{·P}_{D2T}}^{0.021}{R}^2=0.8592\end{array}\end{array}$$ After this study, decision variables, target functions and constraints were defined.

DECISION VARIABLES: The instantaneous powers of the different production teams were taken as decision variables. This is:

OBJECTIVE FUNCTION: As for theobjective function, it seeks to minimize production costs, this function defines the economic cost-hour of the energy production equipment based on the installed power or power produced in an hour and its fuel consumption. To achieve the objective functions and determine the decision variables, we work with the equations that relate the performance of the equipment to its production, $\mathrm{\eta }=\mathrm{f}\left(\mathrm{P}\right),$ in potential and logarithmic form, transforming them until the appropriate objective function is reached. $$\mathit{Cost}(€/s)=m_{\mathit{combustible}}^{\prime }(kg/s)x\frac{{\mathit{Price}}_{\mathit{fuel}}}{m_{\mathit{fuel}}}(€/kg)$$ $$m_{\mathit{fuel}}^{\prime }(kg/s)=\frac{P\left(MW\right)}{\mathrm{\eta }(\%)·LCV(MJ/kg)}$$

$\mathrm{\eta }(\%)=a{·P}^b$ ; ecuación en forma potencial $\left(\mathrm{\eta }=f(P)\right)$ .

Replacing $$\mathit{Coste}(€/s)=\frac{P\left(MW\right)}{A{·P}^B·PCI(MJ/kg)}x\frac{{\mathit{Price}}_{\mathit{fuel}}}{m_{\mathit{fuel}}}(€/kg)$$ $$\mathit{Cost}(€/s)=C{·P}^{(1-B)}(€/s)$$ $m_{\mathit{fuel}}^{\prime }(kg/s)$ is the mass of fuel consumption of the equipment-technologies, $\mathit{LowCalorificValue}(LCV)(MJ/kg)$ is the lower calorific value of the type of fuel consumed, $\mathrm{\eta }\left(\%\right)$ is the performance of the equipment by technology, $P\left(MW\right)$ is the power by technology and A, B, C constant values resulting from the different operations. Working with this equation for each technology yields the objective nonlinear functions per technology in potential form. With the same procedure, the objective nonlinear functions would be obtained by technology in logarithmic form.

Logarithmic Target Function: $$\begin{array}{c}\mathrm{P}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{e}/\mathrm{t}\left(€/\mathrm{s}\right)=\frac{0.46504·{\mathrm{P}}_{\mathrm{T}\mathrm{G}}}{0.0288·\mathrm{l}\mathrm{n}\left({\mathrm{P}}_{\mathrm{T}\mathrm{G}}\right)+0.2509}·42.8885\\ +\frac{0.37998·{\mathrm{P}}_{\mathrm{T}\mathrm{V}}}{0.0271·\mathrm{l}\mathrm{n}\left({\mathrm{P}}_{\mathrm{T}\mathrm{V}}\right)+0.2447}·41.2124\\ \begin{array}{c}+\frac{0.46504·{\mathrm{P}}_{\mathrm{C}\mathrm{C}}}{0.0434·\mathrm{l}\mathrm{n}\left({\mathrm{P}}_{\mathrm{C}\mathrm{C}}\right)+0.2738}·42.8885\\ +\frac{0.37998·{\mathrm{P}}_{\mathrm{D}4\mathrm{T}}}{0.035·\mathrm{l}\mathrm{n}\left({\mathrm{P}}_{\mathrm{D}4\mathrm{T}}\right)+0.4181}·41.2124\\ +\frac{0.37998·{\mathrm{P}}_{\mathrm{D}2\mathrm{T}}}{0.0198·\mathrm{l}\mathrm{n}\left({\mathrm{P}}_{\mathrm{D}2\mathrm{T}}\right)+0.4614}·41.2124\end{array}\end{array}$$ Potential Target Function: $$\begin{array}{c}\mathit{Price}/t\left(€/s\right)=0.0420108480114703·{P_{\mathit{TG}}}^{0.9133}\\ +0.0354753388623078·{P_{\mathit{TV}}}^{0.9245}\\ \begin{array}{c}+0.0360472070204804·{P_{\mathit{CC}}}^{0.9028}\\ +0.019956797771242·{P_{D4T}}^{0.9601}\\ +0.0219891260918526·{P_{D2T}}^{0.9241}\end{array}\end{array}$$ CONSTRAINTS: The constraints taken are the power intervals at which the equipment works, as well as the power needed to be reached at the time to cover the demand. These power ranges are those of the operation of the power-producing equipment, defined by the manufacturer itself.

3.1.1 Application of the tool to minimise hourly emissions of energy production equipment

The developed tool is applied. It is analyzed 24 h a day on August 17, 2021, the day when the highest annual energy demand occurred at 2:53p.m., with 529.0 MW of instantaneous power. In this case, as already indicated, the tool seeks to minimize GHGs. As a result of this application, in the 24 h we have a combination of power-producing equipment that makes emissions as low as possible, meeting the energy demand. The following graph shows, among others, the “GHG” curve that indicates the actual emissions produced during the 24 h and the “GHG Minimized” curve, emissions produced by the best combination of equipment based on the restrictions and methodology established by the tool.

The algorithm established a new combination (it exclusively chose combined cycle technology for energy production), which decreases emissions in 24 h by 388.0 tCO_2eq, (8.83 %). In Fig. 2, in addition to representing the two curves: “GHG”, which are the actual emissions produced on August 17, 2021, and “GHG Minimized”, which represent the emissions that would have occurred on August 17, 2021 if the combination of equipment proposed by the algorithm to minimize GHGs had been used, The “GHG for Minimized cost” curve is also represented, which indicates the GHG emissions that would have occurred on August 17, 2021 if the combination of equipment proposed by the algorithm had been used to minimize costs. As will be seen in the corresponding section, cost minimization involves another combination of different equipment and therefore a different GHG production. This new combination of technologies results in a 24-hour decrease in GHGs by 338.3 tCO_2eq (7.61 %).Fig. 3..

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Fig. 2

GHG emissions.

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Fig. 3

Comparison of real GHG vs. estimated by the GHG Algorithm.

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3.1.2 Model validation to minimise emissions

When we have obtained the optimal solution to the system, we must validate the model, that is, check if the result obtained makes sense and the decisions can be executed ((Mathur et al., 1996)). Therefore, we will determine whether to accept the model and then apply it later or reject it in such a way that we will have to rework the modelling process. To determine the validity of the system, we will analyze the agreement between the observed data of the real model and those provided by the model (Ríos-Insua et al., 2006) by choosing the combination of equipment that was used in the real model and running the algorithm with this combination, and then comparing results. To validate the model determined to search for the minimum emissions, different simulations have been made with the simplex method of different combinations of operation of the production equipment, for which real emissions data are available and the differences between the algorithm and reality have been verified. The percentage of deviation has been estimated, all of which are less than 1.00 %. An example is the validation study that was carried out where the emissions produced on August 17, 2021 were studied.

It has been verified that the real GHG on that day was 4,781.0 tCO_2eq and those estimated by the algorithm 4,828.3 tCO_2eq, which is a 0.979 % deviation. Therefore, the tool for GHG estimation is considered validated.

3.2.1 Application of the tool to minimize energy production costs

The developed tool is applied. In the case of the logarithmic objective function, coherent results are not obtained and it is not applicable to the system followed, so we will work with the potential objective function that did give good results.

It is analyzed 24 h a day on August 17, 2021, the day when the highest annual energy demand occurred at 2:53p.m., with 529.0 MW of instantaneous power. In this case, as already indicated, the tool seeks to minimise the economic cost of energy production equipment depending on the installed power and/or production demanded. As a result of this 24-hour application, a combination of power-producing equipment is produced that makes the economic cost as low as possible while meeting the energy demand. The following graph shows, among others, two curves that indicate the real costs during the 24 h and the costs established by the tool, algorithm, with its new combination of producing equipment.

The algorithm established a new combination (it chose the combined cycle and two-stroke diesel engine technologies for energy production), which decreases the cost in 24 h by €56,015.3, (10.48 %). Fig. 4, in addition to representing the “Cost” curves, which are the actual costs incurred on August 17, 2021, and the “Cost Minimized” curves, which represent the costs that would have occurred on August 17, 2021 if the combination of equipment proposed by the algorithm had been used to minimize them, the “Cost to Minimized GHGs” curve is also represented“, which indicates the economic cost that would have occurred on 17 August 2021 if the combination of equipment proposed by the algorithm had been used to minimise GHGs. As seen in previous sections, GHG minimization involves another combination of different equipment and therefore another production of different costs. This new combination of technologies means a reduction in costs in 24 h by €49,255.3, (9.10 %).Fig. 5..

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Fig. 4

Economic costs.

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Fig. 5

Comparison of actual fuel consumed costs versus those estimated by the algorithm.

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3.2.2 Model validation to minimize economic cost

To validate the model determined to find the minimum fuel consumption, the most economical and most profitable, simulations have been made with the method of the generalized reduced gradient of operating data of the production equipment and the differences between the algorithm and reality have been verified, estimating the percentage of deviation, all of which are less than 1.00 %. An example is the validation study that was carried out where the costs produced on August 17, 2021 were studied.

It has been verified that the actual costs on that day were €590,416.9 and those estimated by the algorithm were €592,532.9, which represents a 0.357 % deviation. Therefore, the tool for estimating the cost of fuel consumption is considered validated.

3.2.3 Combined solution. Application to minimise energy production costs and GHGs

Fig. 4, as already indicated, also shows a third curve: “Cost to Minimized GHGs”, with which the costs produced by the combination of equipment that produce lower GHGs have been represented. As you can see, it improves the costs of the real case, and equalizes the costs of the best existing combination to minimize them. On the other hand, in Fig. 2, a third curve is also shown: “GHG for Minimized cost”, with this curve the GHGs produced by the combination of equipment that produces lower costs has been represented. As can be seen, it improves the GHGs of the real case, and equalizes the minimized GHGs.