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Demand response (DR) can be very useful for an industrial facility, since it allows noticeable reductions in the electricity bill due to the significant value of energy demand. Although most industrial processes have stringent constraints in terms of hourly active power, DR only becomes attractive when performed with the contemporaneous use of battery energy storage systems (BESSs). When this option is used, an optimal sizing of BESSs is desirable, because the investment costs can be significant. This paper deals with the optimal sizing of a BESS installed in an industrial facility to reduce electricity costs. A four-step procedure, based on Decision Theory, was used to obtain a good solution for the sizing problem, even when facing uncertainties; in fact, we think that the sizing procedure must properly take into account the unavoidable uncertainties introduced by the cost of electricity and the load demands of industrial facilities. Three approaches provided by Decision Theory were applied, and they were based on: (1) the minimization of expected cost; (2) the regret felt by the sizing engineer; and (3) a mix of (1) and (2). The numerical applications performed on an actual industrial facility provided evidence of the effectiveness of the proposed procedure.

It is well known that battery energy storage systems (BESSs), due to the number and variety of services they can provide, are powerful tools for the solution of some challenges that future micro grids will face [

Possible applications of BESSs that seem particularly useful are load leveling, reducing end-users' electricity bills, improving end-users' power quality and reliability, and spinning reserve [

In the frame of the above applications, we focused on the optimal sizing of a BESS installed in an industrial facility to reduce the facility's electricity bill. In the most general case, reducing the electricity bill can involve both energy [energy charge (EC)] and peak power (demand charge) [

When sizing a BESS, a cost analysis should be conducted that takes into account investment costs, maintenance costs, and benefits associated with the installation of the BESS. These savings and benefits depend on the control strategy performed during the operation of the BESS. The optimal size of a BESS should be the size that can meet the anticipated needs at the minimum total cost.

However, to apply the sizing procedure for reducing the electricity bill, we must have some input data (“sizing framework”). In particular, the load demand of the industrial facility and the relationships that quantify the electricity bill,

In defining the sizing framework, the engineer who is sizing the BESS (hereafter referred to as the “Decision Maker” or DM) can operate under the hypothesis of either a deterministic framework or under the uncertainty of the data associated with the problem. In the first case, which has been the more popular approach in the past, certain conditions are assumed and used as input data. In the second case, uncertainties are introduced and modeled probabilistically.

We contend that the problem of sizing storage systems to be installed to reduce the electricity bills of an industrial facility must be solved with uncertain data related to the problem. Our position is based on the fact that this is the only way the DM can properly include both short-term and long-term factors in the sizing procedure. Of course, future systems will be subject to random perturbations that unavoidably result in uncertainties in the sizing calculations. Thus, traditional, deterministic paradigms can lead to uneconomic or unreliable solutions.

Although sizing a BESS for an industrial facility is characterized by unavoidable uncertainties related to energy costs and loads, to the best knowledge of the authors, no papers have been published in the relevant literature dealing with the probabilistic sizing of battery storage systems to reduce facilities' electricity bills. However, some papers have addressed the probabilistic sizing of storage systems, but the systems were installed to reduce the uncertainties associated with wind power and photovoltaic power.

For example, with reference to wind power, in [

With reference to photovoltaic power, in [

In this paper, the problem of sizing a BESS for reducing the end-user's electricity bill was solved by using a probabilistic approach based on a stepwise procedure,

Then, the original objectives of this paper were: (i) to propose a new method for sizing a BESS when uncertainties exist and (ii) to apply a decision theory-based process to obtain the best sizing alternative considering the various uncertainties involved in the sizing framework. The new method involves the solution of a constrained optimization model for the daily optimal operation of the battery with the aim of minimizing the total cost incurred for energy.

In this paper, we focused mainly on sizing BESSs for industrial applications. However, the proposed procedure can be extended easily to other types of end users, e.g., domestic and commercial loads.

The remainder of this paper is organized as follows: Section 2 formulates the BESS sizing problem and shows the procedure proposed for solving the problem; Section 3 presents the practical application of the proposed procedure to an actual industrial facility; our conclusions are presented in Section 4.

Let us consider an industrial facility's electrical distribution system in which one or more transformers connect the distribution grid to the lines of the users' power system. A BESS is connected at the secondary side of the transformers with the aim of reducing the electricity bill. We propose to solve the problem of BESS sizing under uncertainty with a four-step procedure,

A set of possible futures is specified, and each future is characterized by a probability assigned by the DM. In this paper, each future is associated with a different industrial facility's load demand and the way in which the electricity bill is calculated, depending on electricity use at each time of the day.

Several possible BESS design alternatives are specified. Each design alternative is based on the BESS energy ratings, with its associated installation and maintenance costs.

The total BESS costs are calculated for each future specified in the first step and for each sizing alternative specified in the second step. The total costs take into account the installation cost, maintenance cost, and the benefits derived from the operation of the BESS. The benefits are obtained by solving an optimization problem in which the objective function to be minimized is the electricity bill and the constraints of which include the need to maximize the BESS's lifetime.

Decision theory is applied to choose, among the alternatives of the second step, the best BESS sizing solution by considering the futures with their probabilities, as specified in Step 1. The applied decision theory approaches used the future probabilities assigned in Step 1 and the total cost of the BESS calculated in Step 3; they are the minimization of the expected cost, the min-max regret, and the stability areas' criteria. These approaches have been used extensively and successfully for the solution of several important power system planning problems [

We note that the DM, based on her or his understanding of the nature of the uncertainties relevant to the BESS sizing problem, selects possible alternatives and futures of Steps 1 and 2 and assigns the future probabilities [

The first approach is based completely on the observed information.

The second approach is based completely on the subjective judgment of the DM.

The third approach is a mix of the above two approaches, and it combines the DM's judgmental information with the observed information.

In this paper, we used the second approach (subjective judgment of the DM). In fact, even if it may seem unsound to assign values of probabilities with little or no empirical information, surprisingly positive results can be obtained when the DM has a good understanding of the nature of the uncertainties relevant to the problem and uses this understanding to assign probabilities in a subjective manner [

In the next subsections, we show the details of the optimization problem to be solved in Step 3 and the decision theory criteria of Step 4.

As shown in

Then, the aim of this subsection is to show how to calculate the total BESS cost. In the most general case, the calculation should be effected taking into account the investment costs, maintenance costs, and benefits derived from the installation of the BESS, that is:
_{T1} is the BESS total cost;

While installation and maintenance costs depend on the BESS size, and as shown in [_{NOB} is the electricity bill without BESS; and _{withB} is the electricity bill with the BESS. In _{NOB} is clearly independent of the size of the BESS. We are searching for the best alternative for the size of the BESS, and then the following total cost can be considered instead of

The _{withB} in

However, the evaluation of the daily cost is not an easy task, since, while the BESS operation is aimed at reduction of the electricity bill, at the same time technical constraints able to maximize the battery lifetime has to be met. In particular, constraints on the depth of discharge and the number of charging/discharging cycles per day have to be satisfied [_{obj} is an objective function to be minimized; _{eq} and _{iq} are the number of equality and inequality constraints to be met, respectively.

Before specifying the objective function and constraints in _{T} time intervals of length Δ_{t} and that, in order to limit one charging/discharging cycle, the day is separated into three intervals, as shown in

The time steps in which the discharging mode starts and ends (
_{T}). Based on the daytime steps of

We outline also that we limited the BESS to only one charging/discharging cycle per day since its use can be profitable only if there is a large enough number of charging/discharging cycles to obtain significant economic benefits during the lifetime of the BESS. While a greater cost benefit could be obtained for a given day by multiple charging/discharging cycles, we must take into consideration the fact that operating in this manner ultimately decreases the lifetime of the BESS, producing an adverse effect on total cost. In our experience, the overall total cost benefits (

The objective function of the _{N}_{t}_{t}_{t}

The first equality constraint in _{N}_{t}_{B}_{t}_{L}_{t}

Further equality constraints require that the daily balance of charging and discharging energy is satisfied:
_{ch} and η_{dch} are the BESS efficiency in charging and discharging mode, respectively; _{T} is the number of day time intervals; and

Moreover, the inequality constraints require that the BESS can only charge during the charging period and only discharge otherwise:
_{max} is the maximum power that the BESS can supply or absorb. The state of charge during the discharging stage cannot be less than a minimum value (which depends on the maximum depth of discharge of the BESS):
_{0} is the energy stored in the battery at the beginning of the day; and

It has to be highlighted that, during the lifetime of the battery, its features (e.g., efficiency, maximum storage capacity, and minimum storage capacity) vary with time based on the battery's aging characteristics [

The above optimization model was solved with a hybrid approach based on a genetic algorithm (GA) and a linear optimization that operated inside the GA as an inner loop. The GA was used to obtain only the time intervals in which the BESS operates in the charging and discharging modes, while the linear optimization determined the state of charge of the BESS at the beginning of the day and the optimal charging/discharging powers of the BESS inside the above intervals to minimize the electricity bill cost function in

In more detail, the GA created populations in which the individuals referred to the times that the discharging mode started and ended (

When the inner linear optimization problem converges, the value assumed by the objective function in

As previously shown in Steps (1) and (2) of the proposed BESS sizing procedure, several futures are specified, with each future characterized by an assigned probability, and several design alternatives for the BESS are specified in terms of the energy to be produced by the BESS. In addition, in Step (3), for each future specified in the first step and for each alternative specified in the second step, the total cost of the BESS was calculated by optimizing the operation of the BESS.

Decision theory was used in Step 4 to choose, among the alternatives of Step 2, the best solution with respect to the size of the BESS by considering the futures with their probabilities as specified in step 1 and considering the total costs calculated in Step 3.

To choose the best solution, let the uncertainties in the sizing of the BESS be represented by a set of _{F}_{k}_{F}_{k}_{t}_{L,t}_{a}_{i}_{a}

minimizing the expected cost;

the regret felt by the DM;

a combination of (i) and (ii).

It should be noted that the application of decision theory requires knowledge of the total cost incurred in each future for each alternative. This has positively influenced the choice of the problem formulation in terms of a single-objective function in which the only objective is the total cost.

Approach (i) may be applied as follows. The expected value of the cost associated with all the _{F}_{ik}_{k}_{i}_{i}_{F}

Among all the possible alternatives, _{i}_{a}

The solution of the optimization problem in _{opt}

In more detail, Approach (ii) indicates the best solution as the one that minimizes the regret felt by the DM after verifying that the decision he or she made was not optimal with respect to the future that actually occurred. The criterion is based on the calculation of the regret felt for having chosen a certain alternative _{i}_{ik}_{i}_{k}_{ik}_{i}

Finally, the sizing alternative, _{opt}_{a}

It should be noted that a critical aspect of both the above criteria (based on the expected cost and the regret) is the assignment of the probabilities _{k}_{F}

In order to overcome the above problems, it may be convenient to refer to the “stability areas” concept proposed in [

When the results of (i) and (ii) criteria are superimposed, the “stability area” of each sizing alternative is the area that corresponds to the probability for which both the Approaches (i) and (ii) give the same recommended solution for sizing the BESS. Based on the knowledge of all of the sizing alternatives characterized by a stability area different from zero (and of the corresponding area value), the DM can determine the sizing solution he or she considers to be the best. For example, the DM's final choice (

It should be also noted that assigning alternatives and futures is a further important aspect in the proposed approach. In the decision-making context, the DM identifies alternatives and futures on the basis of her/his understanding of the nature of the planning problem to be solved. In case of BESS sizing, these choices should be affected also considering: (i) that a maximum amount of investment cost can exist, imposed by the owner of the industrial facility; (ii) that there is a range of sizes within which the DM can forecast that the optimal solution will occur more frequently; and (iii) that the use of a very small number of futures can generate final decisions that will lead to bad performance in the future.

The optimal procedure proposed in Section 2 was used to size a BESS to be connected to the secondary side of the transformer that connects an actual industrial facility to the medium voltage distribution grid. Among the batteries that are commercially available, either Li-ion or redox batteries can be used, both characterized by a long useful lifetime even with a significant depth of discharge [

In order to better show the proposed sizing procedure, two different case studies are presented:

Case 1: only three futures are considered (_{F}

Case 2: nine futures are considered (_{F}

With reference to the sizing alternative, 16 sizes for the BESS were considered (A1 = 0, A2 = 100, A3 = 200, A4 = 300, A5 = 400, A6 = 500, A7 = 600, A8 = 625, A9 = 650, A10 = 675, A11 = 700, A12 = 725, A13 = 750, A14 = 775, A15 = 800, A16 = 900 kW h). The sizing alternatives are chosen considering that: (i) a maximum amount of investment cost exists, imposed by the owner of the industrial facility, constraining the maximum value of the size to 900 kW h; and (ii) there is a range of sizes (between 600 kW h and 800 kW h) within which the DM forecasts that the optimal solution will occur more frequently, as will be shown later. Alternative A1 = 0 means that no BESS is installed.

The following three futures were considered:

Future 1: the hourly EC profile reported in [

Future 2: same as Future 1 except that the profile of the industrial facility's load demand was taken from

Future 3: same as Future 1, except that the profile of the industrial facility's load demand was obtained by multiplying the profile in

We considered three possible demand profiles, as suggested in [

Then, the three decision theory approaches were taken into account [approaches (i), (ii) and (iii) of Section 2.2]. For the application of the first two criteria, initially the following probabilities were assigned to each future, _{1} = 0.2, _{2} = 0.3, and _{3} = 0.5.

From the analysis of the results in

The stability areas for Approach (iii) (_{1}, _{2} and _{3} while meeting _{3} = 1 − _{1} − _{2} and an _{1}, _{2}, the color of which distinguishes the optimal size obtained, is reported in _{1}, _{2}, only the optimal solutions that contemporaneously satisfy both Approaches (i) and (ii) are shown with a marker, the color of which distinguishes the optimal size obtained. The white area corresponds to couples of probabilities that furnish different solutions when Approaches (i) and (ii) are applied.

The analysis of the stability area in

It also is interesting to observe that the solutions in _{1} = 1 and that _{2} = _{3} = 0. Then, from

In order to verify the effectiveness of the constraint of one cycle per day, some further simulations were performed by allowing more than one cycle. However, the results were that one cycle per day is always the optimal solution.

The peak price and the gap between the minimum and maximum prices can have a strong influence on the benefits derived from the use of the BESS and, therefore, on the sizing of the BESS. Motivated by the above consideration, two price profiles were considered in addition to the profile in

As an example, the following probabilities _{i}_{1} = 0.1, _{2} = 0.1, _{3} = 0.1, _{4} = 0.1, _{5} = 0.1, _{6} = 0.1, _{7} = 0.2, _{8} = 0.1 and _{9} = 0.1 for the application of the first two criteria.

From the analysis of the results in

The effect of unequal probabilities is evident in _{7} = 0.2 but all other probabilities are 0.1.

From the results in

As a final consideration on the sizing procedure, it should be noted that, even if the DM chooses a very high number of futures (much greater than nine) and if each optimization problem shown in the previous section is solved using GA and linear optimization, this does not result in excessive computational effort because the computations occur in the planning stage and new computers and configurations (parallel distributed processing and environment) can easily handle massive computational requirements.

This paper addressed the problem of determining the optimal size of a battery storage system to be installed in an industrial facility to reduce the facility's electricity bill. The main original contribution of the paper is that the sizing was conducted by using a probabilistic approach that took into account the unavoidable uncertainties involved with the electricity bill cost coefficients and the profile of the industrial facility's load demand. The choice of the optimal size for the BESS was made by using a stepwise procedure based on the application of decision theory. Different decision theory-based approaches were used, and the results were compared.

The main observations and outcomes of our analyses are that:

The probabilities of the futures can significantly influence the optimal BESS sizing.

The BESS optimal sizes obtained using the decision theory approaches involved various optimal sizing solutions with different stability areas, thus furnishing extensive and useful information for the DM's use in identifying the best solution.

Decision theory appears to be a powerful tool in that it was able to solve the BESS sizing problem for industrial applications even when there were significant uncertainties, just as it has been for several other important problems associated with planning power systems.

As a final consideration, we stress that the slight differences in terms of cost and regret values for the assigned futures were not surprising. They were due to the high investment costs associated with an actual BESS that tend to mask their economic advantages. The future, worldwide-forecasted reduction in the investment costs associated with the installation of BESSs makes us confident that, in the near future, the economic advantages of BESS installations will be recognized and, consequently, there will be a pressing need for an optimal sizing procedure.

Future research will be devoted to the BESS sizing problem when the input data are treated as random variables characterized by their probability density functions; the results of that approach will be compared with the results obtained by assigning subjective probabilities and using decision theory, as we did in this paper. Future research also will consider other tariff schemes that involve both energy consumption (energy charge) and peak power (demand charge).

This paper is funded in the framework of the GREAT (GREAT: Research for Energy and Technology) Project supported by the GETRA Distribution Group (Italy).

The authors declare no conflict of interest.

_{2}

Flowchart of the proposed procedure. DM: decision maker; BESS: battery energy storage system.

Daytime steps.

Hourly energy price.

Load demand daily profile.

Stability areas: (

Decision matrix: total cost (k$)—Case 1.

| |||
---|---|---|---|

A1 = 0 | 3,247.81 | 3,820.96 | 4,394.10 |

A2 = 100 | 3,239.27 | 3,812.41 | 4,385.55 |

A3 = 200 | 3,230.72 | 3,803.86 | 4,377.01 |

A4 = 300 | 3,222.17 | 3,795.32 | 4,368.46 |

A5 = 400 | 3,213.63 | 3,786.77 | 4,359.91 |

A6 = 500 | 3,205.08 | 3,778.22 | 4,351.37 |

A7 = 600 | 3,769.68 | 4,342.82 | |

A8 = 625 | 3,203.66 | 3,767.93 | 4,340.69 |

A9 = 650 | 3,204.90 | 4,338.55 | |

A10 = 675 | 3,206.63 | 3,767.43 | 4,336.41 |

A11 = 700 | 3,208.87 | 3,767.91 | 4,334.38 |

A12 = 725 | 3,211.12 | 3,768.69 | 4,332.86 |

A13 = 750 | 3,213.36 | 3,769.67 | |

A14 = 775 | 3,215.61 | 3,771.10 | 4,332.53 |

A15 = 800 | 3,217.85 | 3,773.04 | 4,332.95 |

A16 = 900 | 3,226.82 | 3,782.01 | 4,337.29 |

Decision matrix of weighted regrets ($)—Case 1.

| |||
---|---|---|---|

A1 = 0 | 8,986.37 | 16,136.42 | |

A2 = 100 | 7,277.12 | 13,572.58 | |

A3 = 200 | 5,567.90 | 11,008.74 | |

A4 = 300 | 3,858.67 | 8,444.89 | |

A5 = 400 | 2,149.46 | 5,881.05 | |

A6 = 500 | 440.23 | 3,317.21 | |

A7 = 600 | 0.00 | 753.37 | |

A8 = 625 | 156.69 | 229.75 | |

A9 = 650 | 404.63 | 0.00 | |

A10 = 675 | 750.22 | 79.57 | |

A11 = 700 | 222.81 | 1,064.98 | |

A12 = 725 | 457.85 | 307.48 | |

A13 = 750 | 751.09 | 0.00 | |

A14 = 775 | 1,179.10 | 139.41 | |

A15 = 800 | 1,760.41 | 348.65 | |

A16 = 900 | 4,452.57 | 2,522.22 |

Expected value of the costs (k$) and maximum weighted regret ($) of each alternative—Case 1.

_{i} |
| |
---|---|---|

A1 = 0 | 3,992.90 | 30,925.44 |

A2 = 100 | 3,984.35 | 26,652.42 |

A3 = 200 | 3,975.81 | 22,379.36 |

A4 = 300 | 3,967.26 | 18,106.28 |

A5 = 400 | 3,958.71 | 13,833.20 |

A6 = 500 | 3,950.17 | 9,560.14 |

A7 = 600 | 3,942.89 | 5,287.08 |

A8 = 625 | 3,941.46 | 4,218.80 |

A9 = 650 | 3,940.41 | 3,150.52 |

A10 = 675 | 3,939.76 | 2,082.28 |

A11 = 700 | 3,939.34 | |

A12 = 725 | 1,647.61 | |

A13 = 750 | 3,939.70 | 2,096.30 |

A14 = 775 | 3,940.71 | 2,545.00 |

A15 = 800 | 3,941.95 | 2,993.69 |

A16 = 900 | 3,948.61 | 4,788.46 |

Decision matrix: total cost (k$)—Case 2.

| |||||||||
---|---|---|---|---|---|---|---|---|---|

A1 = 0 | 3,247.81 | 3,767.98 | 3,820.96 | 4,394.10 | 4,394.10 | 5,053.21 | |||

A2 = 100 | 2,768.37 | 3,239.27 | 3,710.16 | 3,255.55 | 3,812.41 | 4,369.27 | 3,742.72 | 4,385.55 | 5,028.39 |

A3 = 200 | 2,776.11 | 3,230.72 | 3,685.33 | 3,263.28 | 3,803.86 | 4,344.44 | 3,750.46 | 4,377.01 | 5,003.56 |

A4 = 300 | 2,783.84 | 3,222.17 | 3,660.50 | 3,271.02 | 3,795.32 | 4,319.62 | 3,758.19 | 4,368.46 | 4,978.73 |

A5 = 400 | 2,791.58 | 3,213.63 | 3,635.67 | 3,278.76 | 3,786.77 | 4,294.79 | 3,765.93 | 4,359.91 | 4,953.90 |

A6 = 500 | 2,799.31 | 3,205.08 | 3,610.84 | 3,286.49 | 3,778.23 | 4,269.96 | 3,773.66 | 4,351.37 | 4,929.07 |

A7 = 600 | 2,812.44 | 3,593.31 | 3,294.23 | 3,769.68 | 4,245.13 | 3,781.40 | 4,342.82 | 4,904.25 | |

A8 = 625 | 2,816.86 | 3,203.66 | 3,590.46 | 3,296.49 | 3,767.93 | 4,239.37 | 3,783.33 | 4,340.69 | 4,898.04 |

A9 = 650 | 2,821.66 | 3,204.90 | 3,588.14 | 3,299.59 | 4,234.74 | 3,785.27 | 4,338.55 | 4,891.83 | |

A10 = 675 | 2,826.88 | 3,206.63 | 3,586.14 | 3,303.57 | 3,767.43 | 4,231.30 | 3,787.20 | 4,336.41 | 4,885.62 |

A11 = 700 | 2,832.54 | 3,208.88 | 3,585.21 | 3,307.72 | 3,767.91 | 4,228.10 | 3,789.22 | 4,334.38 | 4,879.54 |

A12 = 725 | 2,838.20 | 3,211.12 | 3,584.04 | 3,312.14 | 3,768.69 | 4,225.25 | 3,791.68 | 4,332.86 | 4,874.04 |

A13 = 750 | 2,843.86 | 3,213.36 | 3,582.87 | 3,316.72 | 3,769.67 | 4,222.62 | 3,794.91 | 4,869.59 | |

A14 = 775 | 2,849.52 | 3,215.61 | 3,581.70 | 3,321.68 | 3,771.10 | 4,220.51 | 3,798.90 | 4,332.53 | 4,866.16 |

A15 = 800 | 2,855.17 | 3,217.85 | 3,580.53 | 3,327.08 | 3,773.04 | 4,218.99 | 3,803.00 | 4,332.95 | 4,862.89 |

A16 = 900 | 2,877.80 | 3,226.82 | 3,349.71 | 3,782.01 | 3,821.70 | 4,337.29 |

Decision matrix of weighted regrets ($)—Case 2.

| |||||||||
---|---|---|---|---|---|---|---|---|---|

A1 = 0 | 0.00 | 4,493.19 | 19,213.80 | 0.00 | 5,378.81 | 17,978.80 | 0.00 | 6,185.09 | |

A2 = 100 | 773.58 | 3,638.56 | 13,431.00 | 773.58 | 4,524.19 | 15,495.92 | 1,547.14 | 5,330.48 | |

A3 = 200 | 1,547.16 | 2,783.95 | 10,948.19 | 1,547.17 | 3,669.58 | 13,013.20 | 3,094.31 | 4,475.87 | |

A4 = 300 | 2,320.73 | 1,929.34 | 8,465.32 | 2,320.74 | 2,814.97 | 10,530.39 | 4,641.48 | 3,621.26 | |

A5 = 400 | 3,094.31 | 1,074.73 | 5,982.58 | 3,094.32 | 1,960.35 | 8,047.52 | 6,188.63 | 2,766.64 | |

A6 = 500 | 3,867.89 | 220.12 | 3,499.77 | 3,867.90 | 1,105.74 | 5,564.78 | 1,912.03 | 7,618.72 | |

A7 = 600 | 5,180.79 | 0.00 | 1,746.64 | 4,641.48 | 251.13 | 3,081.98 | 1,057.42 | 5,135.92 | |

A8 = 625 | 5,622.38 | 78.35 | 1,461.74 | 4,868.12 | 76.58 | 2,506.24 | 843.76 | 4,515.22 | |

A9 = 650 | 6,102.76 | 202.32 | 1,229.30 | 5,178.03 | 0.00 | 2,043.18 | 630.10 | 3,894.52 | |

A10 = 675 | 6,624.64 | 375.11 | 1,053.02 | 5,575.57 | 26.53 | 1,698.69 | 416.46 | 3,273.81 | |

A11 = 700 | 7,190.33 | 599.46 | 935.96 | 5,991.15 | 74.27 | 1,378.59 | 213.00 | 2,664.84 | |

A12 = 725 | 7,756.02 | 823.81 | 819.01 | 6,432.75 | 152.62 | 1,093.69 | 61.50 | 2,115.62 | |

A13 = 750 | 8,321.72 | 1,048.15 | 702.02 | 6,890.75 | 250.36 | 831.10 | 0.00 | 1,669.90 | |

A14 = 775 | 8,887.42 | 1,272.50 | 585.01 | 7,387.11 | 393.03 | 620.18 | 27.88 | 1,326.96 | |

A15 = 800 | 9,453.11 | 1,496.85 | 468.01 | 7,926.81 | 586.80 | 468.00 | 69.73 | 1,000.08 | |

A16 = 900 | 11,715.9 | 2,394.23 | 0.00 | 10,189.6 | 1,484.19 | 0.00 | 504.44 | 0.00 |

Expected value of the costs (k$) and maximum weighted regret ($) of each alternative—Case 2.

_{i}) |
| |
---|---|---|

A1 = 0 | 3,815.65 | 20,032.75 |

A2 = 100 | 3,805.44 | 17,549.94 |

A3 = 200 | 3,798.52 | 15,067.13 |

A4 = 300 | 3,791.60 | 12,584.34 |

A5 = 400 | 3,784.68 | 10,101.42 |

A6 = 500 | 3,777.76 | |

A7 = 600 | 3,772.75 | 9,282.94 |

A8 = 625 | 3,772.01 | 9,669.74 |

A9 = 650 | 10,056.53 | |

A10 = 675 | 3,771.86 | 10,443.31 |

A11 = 700 | 3,772.27 | 10,847.43 |

A12 = 725 | 3,772.97 | 11,339.90 |

A13 = 750 | 3,774.07 | 11,985.35 |

A14 = 775 | 3,775.65 | 12,782.72 |

A15 = 800 | 3,777.44 | 13,603.88 |

A16 = 900 | 3,786.00 | 17,342.89 |