The impact of project financing in optimizing microgrid design

Microgrids are a burgeoning energy and power technology, which offer unique economic and resilience opportunities by taking advantage of several value streams¹ (see Ref. 2 for multiple microgrid definitions). These technologies are a portfolio of interconnected distributed energy resources (DER) which, when orchestrated properly, provide synergistic value over any individual asset in the mix.³ In turn, microgrids have been considered to provide a myriad of societal and economic benefit including resilience against natural disaster, capacity deferment, lower emissions from integration of renewables and point of delivery generation, and operational energy cost reduction for participants.⁴ Despite this promise, to date the majority of operational microgrids are public projects funded through grants or pilot projects self-funded through an organization's R&D budget.^2,5

One of the driving reasons for the lack of privately developed microgrid projects is the need for significant upfront capital to cover all of the assets in the portfolio and the distribution network, which define a microgrid. Such large capital expenditures are beyond the balance sheet of most interested parties, meaning they cannot directly fund with available budgets, thus requiring an infusion of external investors for these projects to be implemented. This is common with most large infrastructure projects, and not unique to microgrids. Reference 6 outlines a similar problem in the Indian energy network, where without external sources of funding the growth of renewables (and by extension microgrids) is extremely limited.

Several works have discussed the benefits of microgrids, promising business models, and novel financing strategies providing many avenues for potential investors. Reference 3 introduces (generally) a number of financing approaches that could be used for microgrids at a high level. Reference 5 discusses ten sources of capital for microgrid projects which are broken down in the context of end-users who have access to these capital sources (originally presented in Ref. 7). Only four sources were described to be accessible across all end-users: (i) private debt/equity, (ii) Power Purchase Agreements (PPA), (iii) performance based contracts, (iv) project revenue sharing. It should be noted that the latter three are actually business models (i.e., how the project will pay itself back) and still rely on a project financier to fund the project directly through their balance sheet or seeking private debt. Reference 8 also discusses project financing in terms of microgrid business models, finding that third party financing (i.e., private debt financing) of a microgrid project which pays itself back with energy savings and resilience is the most straightforward approach. Note, none of these works discussed these strategies in terms of the system design; rather, they are high level overviews of possibilities.

Unfortunately, several factors still limit the willingness of institutional investors to enter the market: (i) the plethora of unknowns around the operation of so many assets (mainly due to a lack of data);⁹ (ii) the lack of standardization around microgrids;¹⁰ (iii) outdated modeling practices, especially concerning financing strategies;¹¹ and (iv) the difficulty of quantifying intrinsic benefits of microgrids, such as avoided costs due to increased resilience without making gross assumptions on lost revenue.⁹ Thus, many stakeholders are questioning if microgrids will ever realize their potential.¹² 

To quell these concerns, the development of a standardized design framework which provides confidence in microgrid returns, a repeatable design approach, and detailed financial assessment would be a boon to the industry. This is underscored in Ref. 9, which aims to define a bankable microgrid. Bankability is financial and commercial viability, including not only attractive financial returns but also the ability to address all commercial risks that could threaten those returns.⁹ Here, the authors state that to attract project financing a microgrid project “must be large enough to provide a worthwhile investment value, be proven from a technological and regulatory standpoint, and be standardized by a replicable framework to manage various risks and cover debt service associated with the diverse mix of generation assets.” Simply put, from an investment perspective, a microgrid project must be well understood, come from standard design practices, and deliver worthwhile financial returns.

The literature on microgrid design has recognized this. References 13 and 14 each introduce a staged design framework for the economic design of microgrids, where modeling and implementation are carried out in distinct steps aimed at different aspects of the microgrid design. These approaches typically include a single conceptual design stage centered around a techno-economic model (preferably optimization¹⁵) to determine the microgrid configuration. These holistic approaches ensure consistency along the entire design and implementation process by systematically adding complexities to the same underlying model. The key is that the framework provides the functionality in a single process, such that a single modeler or team of modelers can perform the analysis without needing to translate to another model or pass to another team. While both authors provide a novel repeatable approach to the design process, the lack of discussion on how project financing is secured in these approaches, and where in the design cycle it fits, renders it incomplete for some stakeholders. This conclusion is supported by Ref. 16 which showed that under different financing models (self-financing, PPA, and “Joint-venture”), the optimal design (i.e., selection of assets) and expected returns in the form of annual cost reductions will change drastically.

Therefore, while both investment firms and microgrid developers recognize the need for a standard design methodology, there is a form of causality dilemma¹⁷ when it comes to project finances. Investors need to see that a project provides sufficient return on investment (ROI) to offer funding. However, microgrid design models optimize around the cost of technology purchases to provide potential returns, where solutions are wholly dependent on input assumptions. It is not hard to imagine how an iterative relationship between financier and project developer to finalize a design prevents widespread adoption of this technology.

In this work, we introduce a standard design approach which is aimed to remove this complexity. The design approach builds on the methodology introduced in Ref. 14 but breaks the conceptual design into two steps: feasibility and detailed. This allows for a “phased” financing approach. First, a simplified debt financing model provides economic feasibility and well understood returns to secure financing terms. Second, a detailed design is finalized using those terms. It should be noted that not every project requires this structure, and one or the other method could be sufficient alone; however, this is targeted at the typical project which is carried out in phases with increasing information availability. Through a case study, we are able to show the outcomes of the different financing models and the sensitivity to financing terms.

Therefore, the contributions of this work to the literature of microgrid planning are as follows:

• the introduction of financing considerations to a standardized microgrid framework, which aims to alleviate the causality dilemma between investors and developers;

• derivation of a flexible optimization framework including multiple financing approaches which can optimize the amount a project should receive in financing;

• documented sensitivity of the financing approaches to terms such as project lifetime, loan term, interest and discount rates, and the amount financed on microgrid optimal design.

The paper is organized such that the design framework, optimization model, and sufficiency metrics are introduced in Sec. II. Then a case study is used to walk through the introduced logic and showcase how it would work on a real microgrid design in Sec. III. We then leverage the use-case findings to draw generalized conclusions applicable to which should be considered during microgrid design projects in Sec. IV.

A standard microgrid design framework is introduced in Fig. 1. The framework can be summarized as follows. First, a customer interested in a microgrid project is identified, who will provide partial (or full) information about their energy needs and available credit. Second, a feasibility study will be performed to determine the economic viability of the project, assuming a simplified financing model which requires little detail (see “equipment lifetime financing” introduced in Sec. II B). The results of the feasibility study are used to provide a go/no-go decision if the project is to be considered further. If a “go” decision is made, the feasibility step provides approximate economics which can be used to solicit a financial institution for available funding terms. Third, assuming funding terms have been contracted, the simplified model can be retrofitted with the available funding terms (as well as more detailed data where appropriate) and a detailed analysis can be performed. This detailed analysis uses a familiar “loan term financing” approach for the optimal design (see Sec. II B). A typical detailed design step is the application of a multiyear model to incorporate the impact of degradation and a changing price landscape on the optimal design.¹⁸ Next, following the economic design phases, a detailed power system design and analysis phases are carried out to ensure the microgrid power delivery network operates as expected.

Following the design phases, the fifth step of the process is the management of the implementation of the microgrid. This step is critical to include in the design framework as modifications to the finalized design are often requested as part of the interconnection process with the electric utility.⁹ Finally, once the microgrid is built and commissioned, it is crucial that the same dispatch logic that was used in the design process is used for system control (otherwise, the economic returns will be suboptimal).¹⁴ An important aspect of this step is the monitoring of the microgrid. This is to the benefit of both the microgrid owner/operator, who is alerted to needed upgrades or routine maintenance as well as the design firm which could leverage the operational data to further improve their algorithm.

From this figure, it is clear that financing of the microgrid is involved in nearly every stage of the design and implementation process. In the first stage, the customer's credit and access to capital is investigated to determine project suitability. Step two is where the business case is made to the financier, and they are first engaged in the design process. Step three uses the agreed upon financing terms to finalize the system design. In step four, the budget for the power system equipment is refined and finalized. Step five is where the funds are spent to purchase and install the system. Finally, step six requires close monitoring of the system to ensure optimal performance and savings, ensuring that the owner can meet their financial obligations of operation and maintenance (O&M) and debt payments. Previous studies have not discussed how financing is involved in each step of the design process.

The economic design model is a mixed integer linear programing optimization based on the XENDEE optimization engine.^14,18–20 The objective function minimizes the equivalent annual cost (EAC) of the net present cost required to operate the microgrid over a defined project length (PL) (in  Appendix A, we prove that for the simplified financing scheme it is equivalent to minimizing total annual costs (TACs) for an undefined project length) by optimally sizing multiple energy sources considering their optimal operation in each time step study duration. The model can consider variable time step data including the use of continuous time series or peak preserving representative days^19,20 in the optimization. The model considers power flow using a lossy DistFlow formulation²¹ on up to twenty nodes, which is sufficient to accurately represent any complex distribution system.^22,23 A number of value streams such as resilience, energy exports, renewable certificates, and tax incentives are considered directly in the optimization model, but a simplified objective function which focuses on minimizing cost (emissions are not optimized in this paper) is presented below. Note we use the equivalent annual costs [instead of the net present value (NPV) directly] to model both financing structures introduced as follows:

subject to (among others)

Note the energy balance and debt service coverage ratio (DSCR) are selected as a representation of many more constraints in the model as they are most relevant for this work. The DSCR is a measure of cash flow availability to meet current debt obligations. Simply put, it ensures that there is sufficient return to meet the loan payments, where cash flow is measured by the operational expense (OPEX) savings such as utility, fuel, and maintanence costs realized when the optimized Microgrid is installed. A typical level is 1.2, meaning the project returns at least 20% greater cash flow than required to meet the loans. Note, this is considered as a constraint in XENDEE, see Eq. (3).

In this model, all OPEX terms are considered constant and independent of project year (the ability to optimize around changes in operating costs over time exists in XENDEE,¹⁸ but is not discussed here for simplicity). This take the form of OPEX = $ctariff+cCO2+cfuel+cO&M+cresilience−rsales$⁠,

For a detailed description of the objective function, please see Refs. 19 and 24.

The CAPEX term can take two related forms, allowing for the modeling of two distinct financing model schemes, which are introduced in Fig. 2. The first approach is a familiar loan based approach to debt financing where all DER are financed together using a singular loan amortized over a defined loan period (LT). Replacement of equipment is carried out by reserving capital each year of the PL, such that the cost to replace the DER has been accrued by the end of equipment lifetime. Note this is not necessarily the same cost as the initial purchase.

Loan term financing

where

Note that the loan (c_loan) is a constant cost over the LT (which must be less than or equal to the length of PL). The non-financed cost (c_nf) is a one-time, upfront cost paid at the beginning of a project (or covered through other investment vehicles, such as a tax equity investor). The amount not financed is controlled by γ (which can be user selected or optimized, see  Appendix B). Finally, the replacement cost (c_replacement) is a fixed $/year cost that is accrued as long as the DER must be replaced at least one more time before the end of the project. Note, it is not necessary to treat the replacement this way; instead, it can be a single lump sum paid at the time of replacement. However, a single large sum can create challenges with meeting the DSCR constraint, and thus we do not discuss this approach further in the paper (this is an option available to the user).

Note this approach relies on five key financial inputs to the model: (i) the length of the project (PL), (ii) the loan repayment term (LT), (iii) the interest rate on financed capital (IntRate), (iv) the discount rate applied to future cash flow (DiscRate), and (v) the percentage of capital expenses financed (γ).

For most projects, these terms are not defined until financing has been negotiated with a lender, who usually wants to see proven returns before providing such terms. To establish these returns for lenders, we introduce a simplified financing scheme which only relies on an assumed interest to provide an optimal microgrid configuration. This approach, referred to as “equipment lifetime financing,” assumes that each DER is financed (amortized) over its own lifetime, where the DER are replaced continuously through new loans at the end of their life. Although lifetimes of technologies can be variable, we represent this as a fixed lifetime, which is bolstered through a fixed O&M cost. Practically, this is accomplished through financiers/investors using a consistent portfolio of proven vendors over multiple projects, typically tied to a performance guarantee contracts over the loan term.

The key to this approach is that it allows for a constant annual cash flow, allowing the model to be represented through a single year. See  Appendix A for a proof of the equivalence of the simplified approach.

Equipment lifetime financing

where

Note the following assumptions for this approach: (i) 100% financing of all technologies (γ = 1), (ii) each technology has its own amortization period, and thus annuity rate [ANN_t, Eq. (17)], and (iii) the amortization period is the lifetime of the equipment. These assumptions provide a constant cash flow over each year of the project, making the objective function of minimizing the equivalent annual cost of project net present cost exactly equivalent to minimizing the total annual costs of a microgrid assuming a single typical year (see  Appendix A for the proof of this concept). Thus, we recover the standard approach for most microgrid optimization approaches (such as DER-CAM,²⁵ RE-OPT,²⁶ and MDT²⁷).

For capital-intensive projects, investors rely on a bevy of financial indicators to determine project return. Often times, these indicators can point to conflicting conclusions, where it is up to a savvy financier to clarify the underlying expectations.²⁸ These metrics include the following:

• Net present value (NPV): the sum of the present values of all the cash flows associated with the project.²⁹ 

• Internal rate of return (IRR): the return that an investment is expected to yield, calculated as the discount rate which returns zero NPV. Useful for understanding and comparing potential returns of project alternatives.³⁰ 

• Return on investment (ROI): the annual return from the project as a percentage of the plant's capital cost.³¹ 

• Simple payback period (SPP): the time (number of years) required recovering the initial investment (first cost), considering only the net annual savings.³² 

• Levelized cost of energy (LCOE): the relative measure of economic cost to produce a unit of energy over the lifetime of DER including capital costs, O&M costs, and fuel cost.³³ 

• Total annual cost (TAC): the sum of all annual costs required to operate a generator to provide a desired amount of energy, useful for comparing alternatives.¹⁴ 

While these metrics have become well understood for traditional energy projects,³⁴ microgrids introduce a new level of complication and often require new metrics.¹⁴ First, since these projects rely on multiple large re-investments during a given project, metrics such as SPP and ROI cannot provide an accurate picture of the lifetime of the project. Second, LCOE is a very poorly defined metric due to the reliance of microgrids on multiple value and energy streams, breaking the classic definition which considers only costs and electricity.^24,35 Finally, IRR is an unreliable indicator since its non-linear nature makes solutions challenging, specifically for cases where 100% of the project is financed (in this case, there is not necessarily a negative cash flow). Thus, NPV and TAC are generally more desirable metrics for defining microgrid project success and will be the metrics discussed throughout the rest of this paper.

To demonstrate the proposed design framework, a case study is carried out for the economic design of a real microgrid. The case study will first examine the economic feasibility step, considering the equipment lifetime financing model to generate project returns. In this paper, we assume the project returns calculated above are sufficient; thus, a detailed design will be run, where a sensitivity to the financing terms is performed to examine how the optimal design changes and compares to the feasibility step.

The microgrid was first described in Ref. 20 and further discussed in Ref. 36 and is set to meet demand for a small cluster of buildings located on the University of California, San Diego (UCSD) campus. The reference case against which the economic returns will be measured is assumed to purchase all energy from the utility. To isolate the impact of the financing mechanisms, identical input data are used in both stages. In both design stages, we optimize the economics (not carbon footprint) of a microgrid considering solar photovoltaic (PV), electricity storage system (ESS), and natural gas generators (NG), where XENDEE determines the optimal size of the assets. Debt service coverage ratio (DSCR) of 1.2 is assumed in this study. Technology assumptions are shown in Table I.

The feasibility phase is carried out on the UCSD microgrid assuming two different interest rates of 4% and 8% (recall these are the only financial inputs needed for the equipment lifetime financing model). These represent the lower and upper bound of reasonable values observed in commercial projects, thus providing the upper and lower limit of returns for this project. The results of the analysis (shown in Table II) indicate that this project has good economic potential. For both interest rates, the TAC is reduced significantly over the reference case ($27 k/year for 8% and $53 k/year for 4%). Both rates also provide positive net present value by the 20th year of operation for the project, with increasing value by the 30th year. For the purposes of this study, we will assume that the projected economics are sufficient to propose to a project financier.

Following the demonstrated success of the feasibility phase, the process of negotiating financing terms and model refinement can begin. In this work, instead of focusing on any specific financing terms, we examine a range of the different terms to find the optimal solution according to the accepted financial indicators. More importantly, this exercise provides intuition into how the inputs of PL, LT, IntRate, γ, and DiscRate change the optimal solution and mix of DER. For simplicity, multiyear modeling (where prices and operation change each year) is ignored.

Here we examine PL from 10 to 30 years in increments of 1 year, LT from 10 to 20 years in increments of 1 year, DiscRate from 2% to 12% in increments of 2%, and IntRate of 4% and 8%. Discount rates are inherently variable as they depend on the goals of the project owner and the alternatives they have available, but this represents the typical range that a company would assume. Note that LT must be less than or equal to PL, thus limiting the total possible scenarios. For all of these scenarios, we assume 100% of the solution is financed. A separate sensitivity where we allow XENDEE to find the optimal γ is described in Fig. 8. The DSCR is not varied and is fixed at 1.2 for all analyses. The cost to replace the ESS, which is the only technology with a lifetime less than PL, is assumed to be the same as the purchase price.

Figures 3, 4, and 5 plot the equivalent annual cost, net present value at project end, and DER investment decisions for each optimization scenario examined, respectively. In terms of the studied microgrid's annual cost (Fig. 3), regardless of financing terms, the project is shown to produce cost savings over the reference case. However, only a limited number of scenarios provide projected savings greater than the equipment lifetime financing. This occurs only for discount rates less than the interest rate, and long project terms, which are generally scenarios in which debt financing does not make sense. It should be noted that the lifetime TAC is closest to the loan term financing cases for 30 year projects with IntRate = DiscRate (which makes sense since equipment lifetime financing assumes DiscRate = IntRate).

Similarly, all financing scenarios provide a positive net present value (see Fig. 4), indicating the project creates value for the owner. Comparing to net present value to the findings of the feasibility study, the results are seen to once again best align when IntRate = DiscRate. For both interest rates, only DiscRate < IntRate result in higher NPV's than the feasibility study, while DiscRate > IntRate generally results in lower NPV. The range of NPV is also observed to shrink with increasing DiscRate since high discounting renders later cash flows less impactful, instead the NPV is driven by only the first few cash flows.

DER investment is observed to have the most sensitivity to the financing approach, specifically project length. As observed in Fig. 5, the amount of PV and ESS increases with project length, up to 20 year projects, where the total installed capacity is approximately the same as what is optimized in the equipment lifetime financing feasibility step. For projects less than 20 years, the recommended NG investment is identical to the equipment lifetime financing cases regardless of project length. For projects longer than 20 years, there is a paradigm shift, where ESS and PV installation is dropped in favor of an extra NG generator.

This shift in trend at year 20 can be attributed to the 20 year lifetime of the ESS and shows the link between ESS and PV. For projects 20 years or less, replacement of the ESS is not needed; thus, there is no need to accrue any replacement funds. For projects longer than 20 years, replacement is needed at year 20 to finish out the project. Thus, for a project of length 23 years, savings are accrued to replace the battery for only additional 3 years of service, making it an unattractive asset compared to the NG which has a 30 year lifetime. It is noted that for 30 year projects (where now the NG needs a replacement), the amount of ESS and PV is seen to jump back up to the 20 year levels.

A common assumption in financing is that when DiscRate > IntRate the project owner should always maximize the amount of financing available, while when DiscRate < IntRate the amount financed should be minimized since differences point to a spread in the time value of money. As observed in Fig. 6, which shows the optimal γ determined as a function of the difference between IntRate and DiscRate, the latter is proved to be true, while the former is disproved. We observe, specifically for $IntRate≈DiscRate$⁠, that the highest NPV is produced when $γ<$ 100%. This behavior is driven by DSCR constrained projects, where the optimal investment is limited due to debt burden. In these cases, moving debt to upfront costs allows for greater investment and higher OPEX savings, which over the lifetime of the project overcomes the upfront costs to increase the NPV. A descriptive example showing the annual cash flows and cumulative NPV is given in Fig. 7. In this example, the project trades initial upfront costs and higher annual loan payments for the ability to purchase more DER and produce higher OPEX savings. In year 10, as the loan is paid off, the NPV accelerates eventually exceeding the fully financed case.

The case study showcases the design process in action on a realistic test case. Careful study of the presented plots reveals several trends in the data. Expanding some of this understanding from the specific test case, a few general principles can be observed in the data:

1. Lower interest rates result in better economics: Although it is an obvious outcome, it is supported by the analysis. Lower interest rates make purchasing DER effectively cheaper, thus improving economics through purchasing more DER [see Eqs. (11), (14), (16), and (17)].

2. Longer projects provide better economics: For any given DiscRate, IntRate, and LT, the lowest cost and highest NPV are observed for the longest project length (darkest marker in each sequence). This matches expectations since incremental savings accrue over time. Note, the same cannot always be said about LT length since the ratio of discount and interest rate can make a LT more or less relatively expensive [(see Eq. (5)].

3. Technology lifetime and project lifetime should be correlated: As observed with the ESS system, project lifetime can significantly impact the optimal solution. Thus, care should be taken to design the system and adequately select (or provide choices for) technologies with lifetimes that make sense for the project (see Fig. 5). It should be noted, however, that including an end-of-life value with the technologies could change this paradigm; however, this is not included as this value is rare in real projects.

4. Loans spanning the entire project $(LT=PL)$ result in identical solutions regardless of DiscRate considered: In the solutions in Figs. 3 and 4 where LT = PL (meaning the loan spans the entire project), it is shown that all solutions are the same regardless of discount rate chosen. This can be attributed to the projects having constant cash flows since the loan is present in each year of the project and no cash needs to be accrued since the longest loan matches the shortest tech life in the scenarios we ran. The authors imagine this behavior would not hold if there were reserves needed for replacement.

5. The ratio of IntRate and DiscRate drive project behavior: Several conclusions can be drawn here: (i) when DiscRate = IntRate, the length of the loan has little impact on project economics. This is noted through the flat EAC and NPV of the projects across all loan terms. (ii) When DiscRate < IntRate, shorter LT periods produce better economics. This can be attributed to the fact that the rate of interest outpaces the rate of discounting, meaning the more interest paid, the more money lost. (iii) When DiscRate > IntRate, longer LT periods produce better economics. This is the opposite of (ii), where discounting outpaces interest, meaning paying interest provides value over paying upfront money. Thus, a general conclusion of (i)–(iii) is that careful selection of DiscRate should be performed before agreeing to a given IntRate or LT term.

6. Financing percentage ($γ$) should always be minimized when IntRate > DiscRate to maximize NPV: The maximum NPV of a project which has DiscRate < intRate will always be γ = 0. See the portion to the right of zero in 6.

7. Financing should not always be maximized when IntRate < DiscRate, when the project is constrained by DSCR: When DSCR is constraining investment to suboptimal levels, shifting CAPEX from the loan to upfront can provide additional investment and greater savings, which over the course of the project outweighs the initial capital expense (see the descriptive example in Fig. 7). Note, this is most pronounced when DiscRate and IntRate are within a few percentage points.

It should also be noted that the outlined design approach which considers both financing schemes is not always a necessity and approach will vary with use-case. For example, the equipment lifetime financing approach may be the determined mode of purchase (i.e., individual loans paid off over equipment lifetime); thus, the second phase is not needed. Similarly, financing terms could already be known a priori; thus, the first phase is not needed. To outline the use cases for each approach, Table III is provided.

In this work, we demonstrate a novel approach to microgrid design which combines two different financing approaches into consecutive steps of the design process. This is demonstrated on a realistic and well published test case, which allows for a detailed comparison of the two approaches as well as the development of general considerations when using these financing approaches including the interplay of technical and financing parameters and how the amount financed should change in turn. The methodologies are independent of ownership model as the results from the optimization can be utilized to power on-site generation or drive a PPA agreement.

However, the introduced models still make certain simplifications, while many complexities can be introduced to better align the model with current financing practices. For example, the discussion only considers infusion of capital from a single financier. In reality, there may exist several sources of capital in a single project (which lessens the debt burden) such as crowd funding, grants, and tax-equity investors who provide project capital in turn for tax credits that the project will accrue, such as the federal investment tax credit. Similarly, the detailed design phase should include multiyear modeling, where changes in energy prices, inflation, and equipment degradation are considered as part of the model. Although all these considerations are built into the XENDEE commercial model, they complicate the comparison between the two approaches; thus, they are not documented in this work.

This paper does not discuss the relationship between valuing resilience and investment decisions. Conceptually this has been discussed in literature; however, it has not been captured in a meaningful (standard) way that is acceptable to institutional investors. The authors plan to discuss this in detail in a future work.

All authors contributed equally to this work.

The authors would like to thank Adib Nasle and Scott Mitchell from XENDEE and Paolo Natali from Rocky Mountain Institute for their support of our work.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Indices

d

Day-type, D = {Week, Weekend, Peak}

e

End-use type, E = {Electric, Thermal, Natural gas}

f

Fuel type, C = {Natural Gas, Diesel, Hydrogen}

g

Discrete generation technologies (only available in discrete sizes), G = {Internal Combustion Engine, Combined Heat and Power, Fuel cell}

h

Hour, H = {1,2,…,24}

k

Continuous generation technologies (available in any size), K = {Photovoltaic, Energy Storage, Absorption Chiller}

M

Month, M = {1,2,…,12}

p

Tariff demand period, P = {Non-coincident, Off-peak, Mid-peak, Peak}

pr

Priority Level of Demand, PR = {Low Priority, Mid Priority, High Priority, Extreme Priority}

t

Technologies, T = G $∪$ K

y

Year, Y = {1, 2, …, PL}

Variables

$cx$

A cost term to be minimized, where x represents a set of equations

$CapContk$

The capacity of continuous generation technology k purchased

$DCm,d,h,pr,e$

Demand curtailed of priority level pr and end use e during time m,d,h

$FPm,d,h,f$

Fuel purchased of type f at time-step m,d,h

$Lm,d,h,e$

Energy demand of type e at time m, d, h

$Pm,d,h,e,t$

Energy provided (or consumed) by technologies t for end use e at time-step m,d,h

$PurBk$

A binary indicating purchase of technology type k

$PurchNumg$

The number of units of discrete generators g purchased

$rx$

A revenue term to be maximized, where x represents a set of equations

$Sm,d,h,e,t$

Energy exports by technology t of type e at time m,d,h

$Um,d,h,e$

Utility energy purchased for end use e at time-step m,d,h

Parameters

$Am,d,h,e$

$∈$ [0,1], binary availability of external energy providers

$ANNt$

Investment annuity rate of technology type t

$CapGeng$

The capacity of generator g

$Cdeme,p$

Demand charge applied to period p for energy type e

$Ctax$

Carbon tax

$DiscRate$

The discount rate applied to the cash flow; a hurdle return to be met.

$EFt$

Emissions factor of technology t or the utility source

$FOMg$

Annual fixed O&M cost required for discrete generator g

$FOMk$

Annual fixed O&M cost required for continuous generator k

$FSm,f$

Fixed service charge for month m and fuel f

$Icapg$

Unit capital cost of discrete technology type g

$Ifixk$

Fixed investment cost required for continuous technology of type k

$IntRate$

Interest rate on investment

$Ivark$

Variable investment cost required for continuous technology of type k

$lifetimet$

Lifetime of technology type t

LT

The length of the loan term in years.

$NDm,d$

Number of days of type d in month m

PL

The length of the project horizon in years.

$PXm,d,h$

Exchange price for electricity at time-step m,d,h

$Rcapg$

Unit replacement cost of discrete technology type g

$SCm$

Utility standby charge for month m

$Rvark$

Variable replacement cost required for continuous technology of type k

$Ufixm$

Fixed utility charge for month m

$Vfuelm,d,h,f$

Volumetric fuel price at time-step m,d,h for fuel type f

$VOLLpr,e$

The value of lost load for demand in priority level pr of end-use type e

$VOMt$

Annual variable O&M cost required for technology of type t

$Vutilm,d,h,e$

Volumetric utility price at time-step m,d,h for energy type e

$γ$

The percentage of CAPEX which is financed through the loan

Theorem 1. For a project with constant cash flows over an infinite number of years, the annual equivalent of the net present cost is equal to that cash flow.

Proof. Recall the definition of equivalent annual cost of the net present cost for a microgrid introduced in Eq. (1),

For CAPEX and OPEX terms which do not change over time, they can be pulled out of the summation,

We can rewrite the term $∑y=0PL(1(1+DiscRate)y)$ as a geometric series as follows:

Simplifying this term in steps yields

□

To optimize the financing percentage of the project (see Figs. 6 and 7), which is helpful for DSCR constrained optimizations, we introduce a golden section search controlled optimization loop, as shown in Fig. 8. This loop runs the optimization iteratively, modifying only the percentage of the CAPEX which is financed. The solution stops when it reaches a solution which meets the DSCR constraint for all years of the project length. The algorithm is summarized in Fig. 8.