Target-oriented robust optimization of a microgrid system investment model

Abstract

An emerging alternative solution to address energy shortage is the construction of a microgrid system. This paper develops a mixed-integer linear programming microgrid investment model considering multi-period and multi-objective investment setups. It further investigates the effects of uncertain demand by using a target-oriented robust optimization (TORO) approach. The model was validated and analyzed by subjecting it in different scenarios. As a result, it is seen that there are four factors that affect the decision of the model: cost, budget, carbon emissions, and useful life. Since the objective of the model is to maximize the net present value (NPV) of the system, the model would choose to prioritize the least cost among the different distribution energy resources (DER). The effects of load uncertainty was observed through the use of Monte Carlo simulation. As a result, the deterministic model shows a solution that might be too optimistic and might not be achievable in real life situations. Through the application of TORO, a profile of solutions is generated to serve as a guide to the investors in their decisions considering uncertain demand. The results show that pessimistic investors would have lower NPV targets since they would invest more in storage facilities, incurring more electricity stock out costs. On the contrary, an optimistic investor would tend to be aggressive in buying electricity generating equipment to meet most of the demand, however risking more storage stock out costs.

Keywords

microgrid / renewable resources / robust optimization / target-oriented robust optimization

Introduction

Considering the rapid increase of population, the issue of energy crisis is one of the most urgent topics nowadays. Currently, it is projected that around 1.3 billion people globally still have limited or totally no access to electricity. By the year 2040, the world’s population is projected to increase from 7 billion to 9 billion. According to the World Energy Council, the global economy is expected to double by the year 2050, and the demand for global energy (110-trillion kilowatt hours annually) would also increase by at least a hundred percent. Furthermore, the said crisis not only concerns energy shortages but also compounds with environmental issues such as pollution and climate change . In a rapidly developing world, escalating energy demand is crucial as the focus shifts to improve access and usage of energy resources, thus, making it a top priority agenda for political leaders and economic giants worldwide [¹].

According to Woite [²], the most important step in power system planning is to properly estimate the capital investment costs for the power plant. However, large scale power plant infrastructure usually has low returns on investment. This leads to an imbalance between the supply and demand, which threatens the energy security and results in market malfunction. On the other hand, given that the investment costs for the renewable sources have been falling over the past decade, investing in low-voltage renewable energy sources such as the photovoltaic panel is becoming more attractive, especially for small communities and commercial districts.

The rapid improvement of technology has changed electricity generation and transmission for the past two decades. Since then, distributed energy resources, (DER) such as gas turbines, micro turbines, photovoltaic, fuel cells, and wind-power, have emerged within the distribution system. However, application of individual distributed generators can cause as many problems as it may solve [³]. A better way to realize the emerging potential is to take a more systematized approach which views the generation and associated loads as a subsystem or a “microgrid” [⁴].

Microgrids (MG) are small-scale, low voltage power supply networks designed to supply electrical load for a small community such as a university campus, a commercial area, a trading state and others [⁵]. Furthermore, MG is an innovative future electric power system that will improve the conventional electrical grid network to be more reliable, secure, cooperative, efficient, and cleaner. MG systems would improve efficiency by lowering distribution system loss since increasing the amount of onsite generation minimizes transmission and distribution line losses by up to 7% of the electricity generated [⁶].

Aside from ensuring electricity generation supply, MG systems could play a significant role in climate change mitigation as they have the capability to address greenhouse gas emissions (GHG) reducation targets. Soroudi and Ehsan [⁷] have presented in their paper that MG systems are able to have a significant impact fulfilling 20/20/20 by the year 2020 – goals of reducing the greenhouse gas emissions by 20%, increasing energy efficiency by 20%, and increasing the share of renewables to 20%. This is primarily due to how microgrids integrate the use of renewable sources of energy to traditional generation sources. Furthermore, the use of microgrids allows communities and developing economies to explore the viability of being less dependent on nonrenewable sources of energy while still maintaining reliability, quality and energy efficiency.

Most of the existing MG models such as the ones proposed in Refs. [⁷,⁸] have been formulated with a known demand as they have used historic market data. Furthermore, they have assumed that the supply is fixed for the entire period. However, Ref. [⁹] has mentioned that uncertain demand may occur due to load forecast error and fluctuating market prices in an open market setup. Therefore, considering the uncertainty in the load demand and market price would make the stochastic structure of the study more evident. In Ref. [¹⁰], the authors have addressed this concern by building a stochastic framework based on the scenario production technique that includes uncertainty associated with load forecast error, wind power, solar photovoltaic power, and market price. However, the use of a stochastic method has been criticized in Ref. [¹¹] which has stated that all stochastic optimization approaches assume the existence of an accurate probability distribution function (PDF) of the uncertain parameters, which is normally hard to obtain in practice. In addition, stochastic approaches usually lead to computationally intensive models that may even be intractable. For instance, Ref. [¹²] has required the computation of the convolution of PDFs which are approximated using the discrete methods. Otherwise, the exact computation of the convolution would have called for the use of a small step size, which in itself requires a significantly large number of Monte Carlo simulations.

The concept of classical robust optimization is being used by various studies as an alternative approach to integrate uncertainties [¹³,¹⁴]. It brings with it several advantages. First, it only requires the knowledge of the range of variation of the uncertain parameters as opposed to an accurate specification of the load and renewable energy source output PDF as in stochastic programming. Second, the robust optimization approach immunizes the design solution against all realizations of uncertainty that are governed by the uncertainty set of the problem parameters. This is also in contrast with stochastic programming that provides probabilistic guarantees for constraint satisfaction.

However, in most engineering optimization projects, rather than optimizing a single criterion, the objective is to attain a specific set of performance goals. For instance, minimizing carbon dioxide (CO₂) emissions might be another performance indicator of a microgrid system. In Ref. [⁷], the CO₂ emission constraints have been transformed into a cost function so that the main objective function is to minimize cost. On the other hand, multiple objective functions have been used in Ref. [⁸] to consider CO₂ emissions. However, the models of these studies only minimize either the cost or the CO2 emissions, but not both.

Alternatively, Ref. [¹⁵] has developed the target-oriented robust optimization approach (TORO), which provides a new framework for robust optimization. Instead of the classical way of either maximizing a profit function or minimizing a cost function, TORO maximizes the degree of robustness that guarantees the achievement of system targets. Specifically, this degree of robustness equates to the maximum level of uncertainty that the system could handle such that system targets will always be achieved. This, then, allows decision makers to consider multiple objectives simultaneously. Furthermore, the framework preserves computational tractability by preserving the properties of a linear programming model. Recently, Refs. [¹⁵,¹⁶] have showed the applicability of the approach in investment problems in the energy sector such as in electricity, polygeneration systems, and offshore gas field development that have multiple targets such as cost and emission budgets, quality and profit targets, among others.

Microgrid system

A microgrid, as shown in Fig. 1, is a small-scale centralized energy system that has generators, controllable loads, and storage batteries, which could generate, transmit, and store energy within a small geographic area. According to Ref. [¹⁷], microgrids usually use renewable sources of energy such as photovoltaic (PV) panels, wind turbines, bio-diesel generators, and the like. Moreover, it is an important feature that a microgrid remains connected to the main grid to ensure continuous supply of power. It can also be classified as connected or disconnected from the main grid for it to operate in grid connected mode or islanded mode, respectively.

There are three basic functions of a microgrid: first, it must be able to meet the customer’s electricity demand. Second, it must have the capability to manage the supply and demand of the electricity and take into account the power balance, voltage quality, flexibility, and electrical safety of the system [¹⁷]. Lastly, it must have the “plug and play” function on two levels: (1) flexible system for smooth implementation of new devices, and (2) disconnect from the main grid when enough power is produced and resynchronize a connection when needed [¹⁸].

An investor’s point of view will be considered in this paper. The investor of the microgrid system is the one who purchases the facility and uses the electricity generated. The investor and the consumer are considered as one entity in this paper and it is the one directly interacting with the main grid. Although it is considered as one entity, it does not necessarily mean that it only supplies electricity to one household. For example, a townhouse could be considered as one entity; in this case, the owner of the townhouse could be the investor. The investor will be the one to distribute the generated electricity to individual houses. Moreover, when it comes to the point that the electricity generation of the microgrid exceeds the demand, the owner of the townhouse will also be the one to sell the excess electricity to the main grid. Similarly, this setup could be compared to a university setup with multiple buildings which consume different amounts of electricity.

Model development

The indices together with the major parameters and variables of the model development for the microgrid system are initially presented in Tables 1 and 2. This would then be followed by a thorough discussion on the different components of the system. Additional notations used in the model such as cost and profit parameters will be introduced as needed in the succeeding discussion.

Generation technologies

The first set of components in the microgrid system is represented by the generation technologies. In this paper, three generation sources have been considered: photovoltaic, wind power, and diesel generator. Equation (1) shows that the total amount of electricity produced by the photovoltaic panels is equal to the capacity purchased, multiplied by the solar insolation. In addition to this, the amount of electricity is also subjected to change due to the efficiency delay per time period. This is computed by multiplying the amount of photovoltaic purchased per month to its corresponding efficiency given the time it has already stayed in the system. Equation (2) then shows the possible outputs of electricity from the photovoltaic panels. Similarly, the wind power in Eq. (3) shows that the total amount of electricity that can be produced by the wind power is equal to its overall capacity multiplied by a percentage efficiency. In Eq. (4), the supply of electricity has the option to be used by the consumer or to be sold to the grid. Lastly, Eq. (5) shows that the total amount electricity that can be generated in a month must be less than the capacity of the diesel generator purchased.

(1){{\displaystyle o}}_m={\displaystyle {\sum }_{m=\mathrm{1}}^m{\displaystyle {\sum }_{q=\mathrm{1}}^m{{\displaystyle c}}_{\text{'}pv\text{'},q}{{\displaystyle {}^{*}es}}_{m-q+\mathrm{1}}}}{}^{*}{{\displaystyle i}}_{m-q+\mathrm{1}}{{\displaystyle \forall }}_m,

(2)o_m=o{b}_m+o{c}_m^{}+o{s}_m\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall m,

(3)k_m={\displaystyle {\sum }_{m=\mathrm{1}}^m{\displaystyle {\sum }_{q=\mathrm{1}}^m{c}_{\text{'}\mathrm{wind}\text{'},m}}}*e{w}_{m-q+\mathrm{1}}*w_m\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall m,

(4)k_m=k{c}_m+k{s}_m\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall m,

(5){{\displaystyle j}}_{i,m}\le {\displaystyle {\sum }_{m=\mathrm{1}}^m{\displaystyle {\sum }_{q=\mathrm{1}}^m{{\displaystyle b}}_{\text{'}\mathrm{diesel\text{'},m}}}}{{\displaystyle {}^{*}eg}}_{m-q+\mathrm{1}}{}^{*}{{\displaystyle r}}_i{{\displaystyle {}^{*}s}}_m\forall m,

Equations (6) and (7) are the respective equations responsible for limiting the number of continuous generation technologies (kW) and the number of discrete generation technology (units) that can be purchased by the investor; as the capacity parameters signifies the allowable number of units that can fit in the project site.

(6){{\displaystyle cc}}_{i,m}={\displaystyle {\sum }_{m=\mathrm{0}}^m{{\displaystyle c}}_{l,m}}-{{\displaystyle bc}}_{l,m}\forall m,

(7){{\displaystyle cd}}_{i,m}={\displaystyle {\sum }_{m=\mathrm{0}}^m{{\displaystyle b}}_{i,m}}-{{\displaystyle bd}}_{i,m}\forall m,

Useful life

Equations (8) and (9) indicate that all the DER being purchased in month m would break down in month m plus its corresponding useful life.

(8)b{c}_{l,m+uc}=c_{l,m\hspace{0.17em}}\hspace{0.17em}\hspace{0.17em}\forall m,

(9)b{d}_{i,m+ud}=b_{i,m\hspace{0.17em}}\hspace{0.17em}\hspace{0.17em}\forall m.

Equations (10) and (11) compute for the net amount of DER in the system per month m. This is equal to the difference between the summation of all the DER purchased and DER that break down.

(10)n{c}_{l,m}={\displaystyle {\sum }_{m=\mathrm{0}}^m{c}_{l,m\hspace{0.17em}}}-b{c}_{l,m\hspace{0.17em}}\hspace{0.17em}\hspace{0.17em}\forall m,

(11)n{d}_{i,m}={\displaystyle {\sum }_{m=\mathrm{0}}^m{b}_{i,m\hspace{0.17em}}}-b{d}_{i,m\hspace{0.17em}}\hspace{0.17em}\hspace{0.17em}\forall m.

Equation (12) suggests that the cost in investment per month must be within the allowable budget. It is important to note that the budget is only considering the budget for the investment. The other expenses such as fixed costs, variable costs, and grid purchases are not part of it as these are already classified as operational costs.

(12)d{b}_m\ge c_{l,m\hspace{0.17em}}c{l}_{l,m}+b_{i,m}c{i}_{i,m}\hspace{0.17em}\hspace{0.17em}\forall m.

Energy storage

In this paper, battery is considered to be part of the system to ensure that when all sources are down, the community would still have the ability to operate. Equation (13) demonstrates that the capacity of the battery must always be greater than a certain limit relative to the load requirement per month. Equations (14) and (15) show the constraints on how the battery would work inside the system. A binary variable is attached to the right hand side of the equation to limit the model in such a way that if there is remaining space inside the battery, the electricity would flow into the battery. If the volume of electricity is greater than the remaining space, the amount that would flow inside the battery is simply equal to the difference between the capacity and the electricity previously stored. Equation (16) shows the equation in defining the total amount of electricity in the battery at any given time. There are certain loses from the electricity transferred and the electricity from the previous period because it is part of the nature of the battery to have certain losses in both transmission and storing processes. Equation (17) shows that at any point in time, the amount of electricity inside the battery must always be in a certain limit. This is to ensure that when an emergency occurs, the system would always have spare electricity for the community. Lastly, Eq. (18) shows that the amount of electricity stored for any time period must not exceed the overall capacity of the battery.

(13){\displaystyle {\sum }_{m=\mathrm{0}}^m{c}_{\text{'}\mathrm{batt}\text{'},m}}\ge {\displaystyle \frac{l_m}{s_m}}*d_{m\hspace{0.17em}}\hspace{0.17em}\hspace{0.17em}\forall m,

(14)o{b}_m\ge ({\displaystyle {\sum }_{m=\mathrm{0}}^m{C}_{\text{'}\mathrm{batt}\text{'},m}})\hspace{0.17em}-(f{o}_{m-\mathrm{1}}*fd)\hspace{0.17em}-(M*z_m)\hspace{0.17em}\hspace{0.17em}\forall m,

(15)o{b}_m\le ({\displaystyle {\sum }_{m=\mathrm{0}}^m{C}_{\text{'}\mathrm{batt}\text{'},m}})\hspace{0.17em}-(f{o}_{m-\mathrm{1}}*fd)\hspace{0.17em}+M*(\mathrm{1}-z_m)\hspace{0.17em}\hspace{0.17em}\forall m,

(16)f_m=(o{b}_m*f{c}_m)\hspace{0.17em}+(f_{m-\mathrm{1}}*f{d}_m)\hspace{0.17em}+c_{\text{'}\mathrm{batt}\text{'},m}-f{o}_m\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall m,

(17)f_m\le {\displaystyle {\sum }_{m=\mathrm{0}}^m{c}_{\text{'}\mathrm{batt}\text{'},m}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall m,

(18)f{o}_m\le (f{m}_{m-\mathrm{1}}*fe)-({\displaystyle \frac{l_m}{s_m}}*d_m)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall m.

Supply and demand

The following equations shows the relationship between the supply and demand of electricity. Equations (19) and (20) show that the electricity provided by the system must be equal to all the possible sources of electricity and must be greater than the demand of the consumer.

(19)h_m=o{c}_m+f{o}_m+k{c}_m+g_m+j_{i,m}\hspace{0.17em}\hspace{0.17em}\forall i,m,

(20)h_m\ge l_m.

Equation (21) shows that the amount of electricity sold back to the grid is equal to the excess amount generated. In short, it is equal to the amount of electricity provided minus the demand for that particular period.

(21)u_m=h_m+k{s}_m+o{s}_m-l_m.

System targets

There are two system targets for the microgrids. The first concerns the greenhouse gas emissions. The emission constraint in the model is a soft constraint wherein the total emissions in the system is limited by the carbon target ct. Any emissions being generated that is beyond the carbon target cm is subjected to a penalty cost. There are two sources of emissions in the model: diesel and the electricity purchased from the grid. The electricity obtained from the two sources is multiplied by a rate that converts the amount of electricity (kWh) to kilogram of carbon (kg-carbon).

(22)(j_{i,m}\cdot de)+(g_m\cdot me)\le c{t}_m+c{m}_m\hspace{0.17em}\hspace{0.17em}\forall i,l,m.

The second target is in the form of the NPV. First, the cost of the entire system is broken down into three categories: total profit, total investment cost, and total electricity purchased from the grid. The first part of the NPV shows the equation for the total profit. There are only two revenue streams in this system, which are the amount of money earned from selling electricity to the community and to the grid. Next to it are the different fixed and variable cost associated in operating both discrete and continuous generation technology. The output of continuous generation technology is equal to its capacity. Therefore, the operation cost is also equal to the capacity multiplied by the variable cost. On the other hand, the output of discrete technology can be the control of the system. Therefore, the total operating cost is determined by the total electricity generated by each technology multiplied by its corresponding operational cost. The last part of the profit equation is the penalty cost associated for each kg-carbon of emission being generated that exceeds the carbon target. The second part of the objective function shows the total investment cost in purchasing both kinds of technology. The total cost for continuous technology is equal to the total capacity purchased multiply to its cost per unit cl. However, the total cost of purchasing discrete generation technology is equal to the number of units purchased multiplied by its cost per unit ci. The last part of the objective function shows the cost of purchasing electricity from the grid. The overall objection function computes for the net present value of the entire system, which gives the investor an idea if it is worth it to invest in this kind of a system.

(23)\begin{array}{l}\mathrm{Max}\hspace{0.17em}\mathrm{NPV}=\{(l_m\cdot es{c}_m+u_m\cdot es{g}_m)-[{\displaystyle {\sum }_{l\in L}(n{c}_{l,m}\cdot x{c}_{l,m})+{\displaystyle {\sum }_{i\in I}n{d}_{i,m}}}\cdot x{d}_{i,m})]-\\ {\displaystyle {\sum }_{l\in L}(o_m\cdot v{c}_{\text{'}pv\text{'},m}+k_m\cdot v{c}_{\text{'}w\text{'},m}})-{\displaystyle {\sum }_{i\in I}(j_{i,m}\cdot v{c}_{\text{'}dg\text{'},m})-(c{m}_m\cdot cmc)\}-}{\displaystyle {\sum }_{l\in L}(c_{l,m}\cdot c{l}_{l,m}+c_{l,m}\cdot c{f}_{l,m})}.\\ \end{array}

Computation experiments

This section presents a hypothetical case study of a microgrid system. The discussion on how the parameter values is chosen alongside the actual values used can be found in Appendix. In the process of validating and solving the model, different extreme scenarios are likewise tested to determine if the model behaves as expected. The model is validated through the use of the CPLEX solver in Matlab.

Scenario A: Base scenario

The base case assumes that there are no uncertain parameters. Furthermore, the base model maximizes the NPV (Eq. (23)), with Eqs. (1) – (22) serving as the model constraints. The cost of the different generation technologies is set with reference to the current market prices. Figure 2 depicts the number of discrete and continuous generation technology purchased.

Initially, the model would select the cheapest source of electricity, non-renewable sources such as diesel generator “b_dg,” and purchase from the grid since they are the cheapest among all the electricity sources. However, due to the carbon emission constraint, the model limits the amount of electricity acquired from non-renewable sources and therefore, prioritizes the diesel generator which has a lower emission contribution. Next, it prioritizes wind power “c_w” over the other options for the reason that a lower cost is set to it. The battery “c_batt” capacity is set relative to the load. Therefore, it changes whenever the load changes. In this paper, part of the assumption is that the battery should have at least 3 days of backup electricity stored in it to be ready for emergency cases. Lastly, despite setting a higher cost for PV, the model still chooses it as part of the option mainly because it is the only source that can supply electricity to the battery. This goes to show that there would always be an option to purchase it. The oscillating behavior seen in the purchasing pattern is due to the efficiency decay and useful life of the equipment.

Figure 3 displays the amount of electricity being generated by each source together with the total amount of electricity sold to the grid “u.” As a result, despite having a higher carbon contribution, a minimal amount of electricity is still purchased from the grid “g.” This is because the generation capability at that time cannot meet the demand of the customer. In addition, the model cannot purchase any additional generating unit because the budget for that time period is already maxed out. The electricity generated by the diesel “j_dg” remains at a constant level because this is the level on which the carbon emission is maxed out. The electricity generated by wind power “k” shows an increasing trend because any budget left at the end of one time period is used to wind power. Since the customer’s demand or load “u” in the system is assumed to be constant throughout the entire time period. As the amount of electricity from the wind power increases, the excess electricity being sold to the grid “l” also increases.

Scenario B: Extreme cost scenarios

Table 3 lists the amount of electricity generated from the different sources for each scenario. Four scenarios are considered in this validation by setting the cost of the four electricity sources to 1 respectively. The first column shows the scenario and the other columns show the total amount of electricity generated for the entire time period. In the first scenario, the cost of PV is set to 1. As a result, the majority of the electricity needed is generated through the use of PV panels. The result in the second scenario is almost the same as that in the first one, except that this time, the cost of wind is set to 1. In the third scenario, the cost of diesel generator is set to 1; as a result, the model maximizes the allowable carbon target per month through the use of the diesel generator. However, since the electricity generated is limited by the emission constraint, additional electricity from the other sources are still needed to meet the demand. Lastly, the cost of purchasing electricity from the grid is set to 1. This time, the prioritization still has not changed between diesel and the grid. The model did not choose to maximize the carbon limit through the use of electricity from the grid; diesel still has a higher electricity generation as compared to the amount purchased. The reason for this is that the carbon emission per kWh of electricity from diesel is lower than the unit purchased from the grid. Therefore, given the limited amount of carbon limit, lesser kW of electricity can be produced if it would allocate all the allowable emission to the electricity from the grid. If that happens, the model would need to purchase additional generating units to compensate for the decrease in the electricity generated given the allowable carbon limit.

Scenario C: Reducing the target emission by half

This scenario examines the effect of reducing the allowable target emission by half. As a result, the overall capacity of electricity generators in the first period is not sufficient enough to satisfy the demand of the customer. The model would need to rely on the non-renewable resources which cause the model to exceed the carbon target as exhibited in Fig. 4. This scenario shows that by decreasing the carbon emission target, the investor would need to increase the initial budget so that the model would be able to purchase all the generating units needed to meet the demand without over relying on the diesel generator and the grid.

Scenario D: Limiting the capacity of wind

This scenario examines the effect of having a capacity limit for the wind power. This phenomenon may occur when there is a certain limitation that prohibits the expansion of a generation technology. As an effect, once the wind power reaches the set limit, the model switches to the next source which is PV. Referring to Fig. 5, it can be seen that for the first few months, the model purchased wind until it reaches the capacity limit. Afterwards, it started to purchase PV. It went back to purchase wind once the first batch of the wind has reached its useful life.

Monte Carlo analysis

The deterministic model is built upon an assumption that all the parameters are known, including the costs, efficiencies, budgets, and customer demands. However, in a real life situation, there are parameters that are uncertain, such as the customer demand. Therefore, this section evaluates the model if there is a need to consider uncertainty through the use Monte Carlo analysis. The objective of the Monte Carlo analysis is to see the performance of the model when it is subjected to parameters with variability.

Figure 6 plots the flowchart of the analysis. The process starts from fixing the optimal solution generated by the deterministic base run and testing it under uncertain demand parameters. The solutions are assumed to be the number of DER technologies needed in order to achieve the said optimal NPV. In this paper, the electricity load is the uncertain parameter because the electricity consumption of the community is said to be unpredictable. However, the random number only applies to the percentage change between periods to capture a realistic fluctuation, rather than subjecting the entire load to uncertainty.

Figure 7 shows the results of the Monte Carlo simulation. As can be seen, the mean of the Monte Carlo result is generally lower than the result of the deterministic base. With this, it implies that the deterministic base run solution might be too optimistic, and there is a high chance of not achieving the optimal NPV. Putting this in the context of the investor, the deterministic solution might show a promising result, but the investment might not be as profitable when it is subjected to uncertainty. Having said that, it is important to consider uncertainty in this type of investment model as the randomness of the demand could affect the profitability of the investor.

Target-oriented robust optimization

The Monte Carlo analysis shows that the solution of the deterministic base run is very sensitive to the uncertain demand. Thus, it could not give the investors a clear picture of the return they should be expecting. With that said, the TORO approach [¹⁵] will be used in order to address the uncertainty issue. The purpose of TORO is to optimize the model given that it has a target objective value. Since the objective function of this model is to maximize the NPV, the investor needs to set the NPV target. Then, the model will need to find a solution that could satisfy the target NPV, instead of finding an optimal NPV based on a set of solutions. With TORO, the investors would have a profile of solutions for different risk levels. Thus, it could give them the solution that fits their risk appetite.

Initially, a target (tau) is set using the following in Eq. (24). The alpha which also refers to the risk level is assigned a value from 1 to 0, with 1 being pessimistic and 0 referring to optimistic. Pessimistic in this case is being contented with setting a lower NPV target. On the other hand, optimistic risky investors in this case set a higher NPV target. This perspective also validates the high risk, high return concept in investing.

(24)\mathrm{Tau\; =}\alpha \mathrm{*\; (Minimum\; NPV)\; +\; (1\; -}\alpha \mathrm{)\; *\; (Maximum\; NPV)}.

The next step is to set up the tolerance limit and the range of the uncertainty of the model. The tolerance signifies the accuracy of the answer to the model. A lower tolerance would mean that the iterated value would be closer to the target; thus, the solution will be more accurate. In this setup, the tolerance level is set to 0.01%, which means that the iteration would only allow a deviation of 0.01% from the target. As for the range of uncertainty, this follows the range of random numbers as discussed in Sub-section 4.5.

Another component of the iteration process are the variables “a_hi” and “a_lo” which affects the increase or decrease of the uncertain parameter “d_a.” To initiate the iteration process, the initial values for “a_hi” and “a_lo” is set to 1 and 0 respectively. For each iteration, a new value for the variable “d_a” would be computed which serves as the percentage change between loads. In effect, Fig. 8 shows that when there is a higher value of “d_a” when the investor is more pessimistic and a lower “d_a” when the investor is optimistic. The reason for this is that an optimistic investor would assume that the demand would be low and therefore, use the majority of the budget to purchase electricity generating units. However, a pessimistic investor would assume that there would be an increase in the demand and therefore, spend more of the budget on the battery in compliance with the three day requirement. This, therefore, affects the NPV because less budget would be allocated for wind power.

(25)d_{-}a=d_{-}b-d_{-}c*a.

The value of “d_a” would then be used in the TORO model to compute for the electricity demand or “load.” The new load is computed by multiplying the previous load to the variable “d_a” as expressed in Eq. (26). To convert the variable “d_a” from a percentage to a decimal, it is multiplied by 0.01. The TORO model would optimize the model following the new load. Afterwards, the objective function would be compared to the value of “Tau.” If the value of the objective function is greater than “Tau,” the current “a_hi” would be the new “a”; else, the new “a” would be equal to “a_lo.” The iterations would end if the difference between the “a_hi” and “a_lo” is less than the tolerance. After satisfying the first “Tau,” the model would keep on running until all targets are satisfied.

(26)l(\mathrm{1},\mathrm{1},p)=I(\mathrm{1},\mathrm{1},p-\mathrm{1})*d_{-}a*\mathrm{0.01.}

Shown in Fig. 9 is the graph of the objective values at different alpha levels. It is observed that the more pessimistic solutions would have lower objective values and the more optimistic solutions would have higher objective values, which is in line with the definition of optimism and pessimism in the second paragraph of this subsection. The orange line in Fig. 9 represents the objective value in the base case scenario. With the deterministic base run, the intersection would be at around 0.15 alpha level (85% optimistic) which shows that the optimal solution obtained is more of an optimistic solution.

Since the model requires the demand to be fully satisfied and the pessimistic solution assumes that the DER purchased are not sufficient enough to satisfy the required demand, the model is forced to buy additional electricity from the grid in spite of high carbon emission stock outs. These purchases would tend to incur higher costs due to the high penalties for carbon emission stock outs, which is reflected in the total operational cost (gray line) as can be seen in Fig. 10. Furthermore, the behavior of the total investment cost (orange line) in Fig. 10 is just a straight line. Since the budget is the same for all alpha levels and the model decides to max it out, and because of the two costs, the total cost (blue line) is the net cost that affects the NPV.

The results of the model provide the investor with an optimal solution (mix of DER technologies) for a specific target that the investor will choose. To further understand the concept, Fig. 11 shows the change in the DER purchase profiles from one alpha to the other. Pessimistic solutions would tend to invest in more batteries to satisfy battery constraint when the demand increases. Then the excess budget is used in buying electricity generating DER technologies such as wind turbines. Batteries are prioritized in pessimistic scenarios as this constraint cannot be violated when the demand increases. In contrast with the electricity generating DER technologies, the investor will always have an option of buying electricity from the grid with higher costs in carbon emission stock outs. On the other hand, the model will tend to purchase less batteries in optimistic scenarios because it is expecting a lower demand that is needed to be satisfied, allowing more budget to be spent on wind turbines to satisfy the most of the demand.

Given the set of solutions, the final decision would still depend on the personal target of the investor. For example, if the goal of the investor is to break even in the investment, he/she can decide to have an alpha level of 0.4 or 0.5 and follow the decision making in these particular optimistic levels. Although all investors would always want to aim at the highest profitability, there would be trade-offs to be considered. As it can be seen in Fig. 11, a pessimistic solution is more protected in the case where the electricity demand is expected to increase, compared to the optimistic solution.

The objective of conducting a Monte Carlo analysis on the solutions generated by the TORO base run is to validate whether these solutions are more robust compared to the solution generated by the deterministic model. In addition, it also aims to determine which of the 11 solutions would generable have a more favorable result.

Similar to the process of the deterministic base run, the TORO approach also has an optimal solution for each alpha that should be tested under uncertainty. The 11 solutions from TORO would be sets of decision variables that satisfy each target objective value. The 11 sets of decision variables “cl” and “ci” from TORO would then be fixed and used in the TORO Monte Carlo simulation. In order for the validation to be consistent, the range of random numbers will also be set between 80% and 120%, similar to the deterministic validation.

The mean and standard deviation of the NPV from the 11 sets are plotted in Fig. 12. The x-axis corresponds to the NPV and the y-axis shows the value of the standard deviation. The standard deviation in this case pertains to the variation of the NPV given the uncertain demand. The label beside each point signifies the alpha, which is derived from the TORO model. Given the following figure, the recommended points are those levels that have a low standard deviation and a high NPV. In this case, they are alpha 0.1 and 0.0. An alpha of 0.1 has a higher NPV but a lower standard deviation as compared to an alpha of 0.0. Up to this point, it is now up to the investor if he/she would either prefer a higher NPV by following the decision at point 0.1 or be pessimistic by choosing the other points.

Conclusions

There has been a continuous stream of research that highlights and solves investment problems regarding microgrid systems. This research extends existing microgrid investment models with a target-oriented robust optimization approach on the uncertain demand parameter while considering multi-period and multi-objective investment setups to contribute to the growing literature in this field of study

In the initial setup, the microgrid model was able to provide investors with the mix of distribution energy sources that will maximize the NPV with consideration of the allowable budget and emission constraint in multiple periods. Through the NPV, the return of investment of the investor would be determined for the period that the model was run. Upon the analysis of the model, it was seen that the model chooses to prioritize the used of diesel generator until the emission constraint was maxed out, then moved to wind power then photovoltaic power. The use of electricity from the grid only happens when the demand for electricity is greater than the generating capacity of the system.

In conducting the validation and sensitivity analysis, the model was run indifferent scenarios to see which factors do affect the decision of the model in choosing one source over the other. As a result, it was seen that there were four factors that affected the decision of the model: cost, budget, carbon emissions, and useful life. Since the objective of the model is to maximize the NPV of the system, the model would choose to prioritize the source having the least cost among the different distribution energy resources (DER). However, when the emission and useful life enter the picture, the model would integrate the three factors and determine which DER has the least cost for the more output. Through the scenario analysis, it was validated that the model could generate the purchasing pattern given the different parameters. This gives the investors the set of tools that they need to invest in each period based on their limited resources.

After doing the analysis, the group further examined if there is a need to consider the uncertainty of load in the study through the use of Monte Carlo Simulation. The resulting DER solution of the deterministic model was used as an input parameter for the Monte Carlo Simulation; by running the solution of the model under uncertain demands. As a result, it was seen that upon consideration of the uncertainty parameter, the average NPV of the model is below the value from the deterministic model. This means that the results from the deterministic model might be too optimistic and it might not be achievable in a real life situation.

The issue with uncertainty was addressed through the application of the TORO approach. Upon the integration of TORO, the study was able to come up with a profile of solutions that would guide investors in their decisions while considering the uncertain demand. As a conclusion, the results show that pessimistic investors would tend to have lower NPV targets since they would only meet a portion of the demand, resulting in more stock outs. On the other hand, optimistic investors have the highest potential of getting larger profits as they tend to be aggressive in buying electricity generating equipment, to meet most of the demand.

With the profile of solutions generated by the TORO base run, these were further validated once again with the Monte Carlo simulation to determine if these solutions were more robust than the one obtained from the base run. The results of this validation show that there are solutions that are more favorable compared to the base run, being favorable in the sense that the solution must have a higher NPV and a lower standard deviation after the Monte Carlo simulation.

Recommendations for future works

The only parameter in the study that was considered uncertain was the change in the load or the customer’s demand from one month to the other. This, therefore, presents an opportunity to look into other uncertain parameters in order to see the behavior of the model in such cases. One parameter that can be subject to uncertainty for future studies would be the supply efficiency of the PV and wind. Since PV and wind depend on the solar insolation and wind efficiency, this would be highly variable depending on the country where it is applied.

This research considered four main distributed energy resources – PV, wind, battery, and diesel generator. In this regard, the inclusion of other distributed energy resources such as hydropower or waste to energy plant should be considered in the future as these are valuable in developing the perfect mix of energy supply to satisfy the demand. Furthermore, it is also recommended that further studies could identify a particular product of PV, wind, battery, and diesel generators in order to consider the variation in sizes, capacity, useful life, and such.

Finally, the model considered limited periods which could be further extended if technology allows in the future. All in all, future work could include the extension of the model with other parameters under uncertainty, and consider other energy sources, different design and perspective, and longer planning horizons.

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