A Generalized Method for Rightsizing the Design of a Hybrid Microgrid

Abstract

As the demand for sustainable and resilient energy systems grows, providing planners with effective tools for microgrid design becomes increasingly important. This research addresses the need for such tools by introducing a new method for distributed energy resource sizing in microgrid capacity planning. The planning process begins with a comprehensive assessment of the required capacity based on a given set of power load requirements. Rather than providing a single solution, as is common in related works, the sizing method introduced in this paper efficiently identifies a wide range of microgrid design options that satisfy the stated power needs. The benefit of this multi-solution approach is that it allows decision makers to consider vastly different possibilities, such as varying levels of renewables and battery storage, and weigh trade-offs between these potential designs before selecting one or more solutions for further detailed design planning. The proposed method is constructed as a three-step heuristic search procedure: (1) an exhaustive search identifies an initial set of candidate solutions; (2) a global binary search builds a diverse set of microgrid design options; and (3) a local linear search refines those options. A computational experiment is presented to demonstrate the method’s effectiveness at identifying diverse solutions sets and its computational tractability. The results show that increasing the number of capacity levels considered per distributed energy resource from 11 to 41 increases the size and diversity of the microgrid design set; however, further increasing the number of capacity levels beyond that point is not beneficial. The method presented is implemented and released in Microgrid Planner, an open source software platform.

1. Introduction

1.1. General Context

Microgrids are controllable systems for distributing power within a small geographic area, which could be a community, a company base or a military installation. A hybrid microgrid is composed of multiple types of distributed energy resources (DERs), examples of which are diesel generators, photovoltaic (PV) systems, wind turbines and battery energy storage systems (BESSs). A microgrid may be the sole energy source for an off-grid location, it may supplement the electrical grid, or it may be a backup in the event of a grid outage. Microgrids may provide decentralized renewable energy and help with gaining access to affordable and clean energy for unserved populations, as noted in United Nations Sustainable Development Goal 7 [1]. This paper presents a new DER sizing method that is implemented and released in Microgrid Planner [2,3], an open source software platform designed to deploy analytical methods for microgrid planning.

1.2. Motivation

In the context of microgrids, objectives may include several factors, such as cost, sustainability and reliability. Combining these into a singular weight that may then be optimized inherently results in a loss of information. Decision makers may prefer to review a range of options, weigh trade-offs between factors they understand well, and then further investigate a small set of alternatives they deem worthy of consideration.

A decision maker intuitively may know that a range of potential solutions exists. For example, a decision maker may understand that, at one end of the spectrum, it is possible to produce all power required using diesel generators and that, at the other end of the spectrum, it is possible to produce the same power using wind turbines and batteries. Producing a wide range of microgrid designs with varying, appropriately sized combinations of DERs empowers decision makers to play an active role in the decision process.

1.3. Literature Review

The design of general-purpose microgrids has been an active area of research in recent years [4,5]. A central element of designing a microgrid is selecting appropriate DER capacities. Particular emphasis has been placed on optimization-focused methods that provide a decision maker with a single best solution for selecting the sizes of DERs. Many efforts have concentrated on optimizing a single DER type, notably battery storage capacity. Alsaidan et al. [6] and Chen et al. [7] apply mixed-integer linear programming (MILP) to optimize battery size. Liu et al. [8] develop a search heuristic for battery sizing and compare its performance against an MILP formulation. Fossati et al. [9] also focus on optimizing battery size but apply a genetic algorithm. Hussain et al. [10] also use genetic algorithms but for optimizing the size of diesel generators. Bahmani-Firouzi and Azizipanah-Abarghooee [11] optimize battery size using an evolutionary algorithm. Chatterji and Bazilian [12] focus on homes and optimize battery storage using stochastic programming.

Other efforts tackle higher-dimensional problems. Dong et al. [13] use MILP to optimize battery and generator sizing, concentrating on extreme events. Rodríguez-Gallegos et al. [14] and Lan et al. [15] formulate non-convex optimization problems to size up to three DER types and use metaheuristic approaches to identify solutions. Bukar et al. [16] use a grasshopper optimization metaheuristic to size up to four DER types. Al-Shamma’a and Addoweesh [17] and Zhao et al. [18] apply genetic algorithms, and Cao et al. [19] use evolutionary algorithms to also optimize microgrid capacity with up to four DER types. The Distributed Energy Resources Customer Adoption Model (DER-CAM) is a notable MILP model for more general DER investment optimization, and many efforts have built upon it [20,21,22].

The “rightsizing” method of Reich and Oriti [23] is a fundamentally different approach to DER sizing. It does not seek to find the single best solution for optimizing microgrid performance on a specific measure of interest. It is instead focused on the problem of identifying many potential solutions that satisfy a given set of power load demands, a problem that has received little attention in existing literature. Their simulation-based search approach is sufficiently abstract to allow for the customization of DER models and energy management system logic. They provide nested binary searches for a three-dimensional DER sizing problem with diesel generation, photovoltaic and battery energy storage systems. Via construction, they are able to provide all potential microgrid designs that could meet an input power load profile requirement, within a specified search-grid spacing, with sufficient computational tractability to be used in practice. However, their search algorithm is limited to a fixed set of three DER dimensions, so it cannot be applied to higher-dimensional problems where, for example, wind turbines are also considered as a potential DER. The latter limitation is a major obstacle for practical use and prevents the method of [23], for example, from being implemented in Microgrid Planner.

1.4. Contributions and Scope

The method presented in this paper can be used to “rightsize” any stand-alone microgrid, whether it is always islanded or it operates in islanding mode when the utility grid is down. The aim is to provide a more general extension of the “rightsizing” method [23] by enabling decision makers to consider both more than three DER types and different sets of DERs, not being restricted to diesel generators, photovoltaic systems and battery energy storage systems. Moreover, the goal is not only to create a method that can solve the problem in computational experiments but one that is made readily available to practitioners. The contribution of this paper is to develop a new DER sizing method and implement that method in the open source Microgrid Planner platform, making it available for use.

Delivering this contribution requires developing a method that is sufficiently abstract that it can (1) accommodate any set of DER technologies; (2) allow those technologies to be flexibly modeled, including the energy management system logic that governs their operation; and (3) solve microgrid capacity planning problems with more than three DER dimensions. Unlike the work of [23], the approach in this paper does not identify every potentially “rightsized” microgrid design because this would quickly become computationally intractable in higher-dimensional problems. Instead, the proposed method provides a sufficiently varied subset of potential microgrid designs.

1.5. Document Structure

The remainder of this paper is organized as follows: Section 2 formally defines the problem, introduces performance metrics for candidate microgrid designs, presents a novel DER sizing method, and defines computational experiments. Section 3 provides computational results, discusses their significance, summarizes findings, and provides concluding remarks.

2. Materials and Methods

2.1. Problem Statement and Metrics

The aim is to identify a diverse set of microgrid design options composed of DERs with sufficient capacities to meet the power demand over a given time horizon. To state this formally, consider a function $f(g,p)=y\in {\{0,1\}}^{\left|T\right|}$ that maps an input microgrid design g and power load p to binary outcomes y, where $y_t=1$ if and only if the power load $p_t$ is not fully satisfied over time interval $t\in T$ with duration $d_t$. Then, $\overline{y}={\sum }_{t\in T}{y}_t{d}_t/{\sum }_{t\in T}{d}_t$ provides a performance measure representing the proportion of time a power load is not fully satisfied. The function f can take the form of an optimization or simulation model that incorporates the control logic for an energy management system.

The notion of dominance between two microgrids $g^1$ and $g^2$ can be defined with respect to a given f and p. Let $c_i^1$ and $c_i^2$ be the capacities of DER i for microgrids $g^1$ and $g^2$, respectively. Then, $g^1$ dominates $g^2$, denoted $g^1\succ g^2$, if ${\overline{y}}^1\le {\overline{y}}^2$ and $c_i^1\le c_i^2$ for every $i\in I$. In other words, if one microgrid performs at least as well as another without having a higher DER capacity, then it is dominant. The set of all “rightsized” microgrid designs can be stated as $S=\{g^1,\cdots g^m\}$ such that $g^j\nsucc g^k$ for any $(g^j,g^k)\in S\times S$, $j\ne k$. S is a finite set because capacities defining DERs are discrete. The goal is to find a diverse subset of S, where diversity can be measured by the difference between capacities of any two microgrids $g^j$ and $g^k$.

Table 1. An example of non-dominated microgrid designs generated by the DER sizing method. Ratio values equal to −1 correspond to DER capacities of 0.

	BESS					Diesel
Sizing	Charge &		BESS		Diesel	Gen.	Diesel		PV
Grid	Discharge	BESS	Time	BESS	Gen.	Time	Gen.	PV	Time	PV
Deficit	Power	Energy	Steps	Unused	Power	Steps	Unused	Power	Steps	Unused
Ratio	(kW)	(kWh)	Ratio	Ratio	(kW)	Ratio	Ratio	(kW)	Ratio	Ratio
0.004	65	130	0.24	0.93	70	0.55	0.46	350	0.5	0.7
0.0077	130	260	0.26	0.77	45	0.55	0.27	350	0.5	0.63
0.005	257.5	515	0.28	0.81	25	0.55	0.15	350	0.5	0.55
0.0002	325	650	0.31	0.85	25	0.59	0.14	210	0.5	0.39
0.0004	192.5	385	0.3	0.85	45	0.66	0.26	140	0.5	0.4
0	0	0	0	−1	120	1	0.67	0	0	−1
0	65	130	0.031	0.69	95	1	0.58	0	0	−1
0.0006	130	260	0.5	0.96	70	1	0.43	0	0	−1
0	325	650	0.51	0.9	45	1	0.09	0	0	−1
0.0034	257.5	515	0.51	0.88	45	1	0.095	0	0	−1
0.00079	290	580	0.28	0.83	25	0.55	0.15	350	0.5	0.55
0	325	650	0.3	0.85	25	0.58	0.14	245	0.5	0.44
0.0056	192.5	385	0.27	0.79	35	0.55	0.2	350	0.5	0.6
0.0004	225	450	0.27	0.82	35	0.55	0.2	350	0.5	0.6
0.0063	257.5	515	0.4	0.87	35	0.78	0.086	70	0.5	0.12
0	257.5	515	0.29	0.85	35	0.59	0.19	210	0.5	0.46
0.0012	257.5	515	0.32	0.85	35	0.66	0.19	140	0.5	0.33

shows an example set of results generated by Microgrid Planner using the DER sizing method proposed in this paper. The column “Sizing Grid Deficit Ratio” provides the performance measure $\overline{y}$. Practical candidate solutions have a value of 0 or very close to 0, so solutions over a selected value are filtered out, which in the example shown is set to 0.01. Although this value may be too high for critical applications, it may be suitable in some contexts, so it is used for illustrative purposes. Microgrid Planner allows users to adjust their $\overline{y}$ upper bound to ease or restrict the level of performance required. This is a post-processing step and does not impact the solutions obtained, only the ones presented. Only non-dominated solutions are shown to the user.

Detecting whether excess capacity exists for a potential microgrid design is not as straightforward as detecting deficits, because excess power is to be expected at many points in time in order to have sufficient capacity at the times with the highest power demand or lowest power availability. An individual performance metric is computed for each component in any candidate microgrid design identified, as shown in Table 1 in the column “{DER NAME} Unused Ratio”. This ratio measures the proportion of time steps where power from a given DER was available but not used. The metric is nuanced because not all DER types may supply power at all times; for example, photovoltaic systems are useful only during daylight hours. Therefore, to measure utilization, the domain of consideration is limited to those daylight hours, and the unused proportion must be calculated accordingly.

2.2. Sizing Method

The microgrid sizing method takes a function f (as input) that operates the microgrid, a power load profile, DER capacity bounds, and both a small and larger number of discrete capacity levels. No limiting assumptions on f are introduced, so the function can incorporate any desired control logic, a requirement for its integration into the open source Microgrid Planner platform.

The sizing method is illustrated in Figure 1 and can be broken down into a three-step process:

• Generate an initial set of microgrids via an exhaustive search using a small number of capacity levels.
• Generate a more diverse set of microgrids by executing a binary search algorithm using a larger number of capacity levels, as well as by using the initial set of microgrids identified in the prior step.
• Generate a refined set of microgrids by executing a local search algorithm for each non-dominated microgrid identified in the prior step, skipping ones with power deficits.

2.2.1. Exhaustive Search

The exhaustive search method is presented in Algorithm 1. All potential solutions are explored, but the search order is controlled to track when power deficits arise. A deficit indicates that at least one DER’s capacity may be insufficient, and the computationally more expensive step of simulating the microgrid can be skipped for solutions with lesser or equal capacity in all DER dimensions. In other words, potential microgrid designs are not simulated when ones with equal or greater capacities for all DERs have already been found insufficient to meet power demands at all points in time. Counterintuitively, this could result in pruning a solution that actually meets the power demand, due to subtleties in the battery charge states resulting from the energy management system logic. However, such cases are rare.

The first loop in Algorithm 1 iterates over each DER and (i) initializes the capacity levels to explore in decreasing order; (ii) initializes an ordered set $S^0$ of all possible combinations of DER capacities, from highest to lowest capacity levels by DER, using the Cartesian product; (iii) creates an empty set $S^1$ for storing potential solutions; and (iv) creates an empty set $S^2$ for storing pruned solutions. The next loop iterates over the set of all possible solutions $S^0$ and its inner loop iterates over all DERs, increasing the capacity temporarily to detect if a higher capacity solution has already been pruned. If no pruned solution is found, then the current solution is simulated and added to the potential solution set $S^1$. If the current solution has a power deficit, it is also added to the pruned solution set $S^2$.

Algorithm 1 Microgrid sizing exhaustive search algorithm

Input: f (function for operating microgrid) Input: p (power load, indexed by time) Input: $l_i,i\in I$ (capacity lower bounds by DER) Input: $u_i,i\in I$ (capacity upper bounds by DER) Input: $n_i,i\in I$ (number of capacity levels by DER) Output: $S^1=\{g^1,\cdots ,g^m\}$ (set of microgrids)    for $i\leftarrow 1$ to $\left|I\right|$ do       $c_i={\displaystyle \frac{u_i-l_i}{n_i}}$       $s^i=\{l_i+n_i{c}_i,\cdots ,l_i+2c_i,l_i+c_i,l_i\}$    end for    $S^0=s^1\times s^2\times \cdots \times s^{\left|I\right|}$    $S^1=\left\{\right\}$    $S^2=\left\{\right\}$    for  $g\in S^0$do       for  $i\leftarrow 1$ to $\left|I\right|$ do         $g^{\prime }=g$         set $c_i^{\prime }$ to capacity of DER i in $g^{\prime }$         $c_i^{″}=min\{c_i^{\prime }+c_i,u_i\}$         set capacity of DER i in $g^{\prime }$ to $c_i^{″}$ to obtain $g^{″}$         if $g^{″}\in S^2$ then            $S^2=S^2\cup \left\{g\right\}$            break         end if       end for       if $g\in S^2$ then         continue       end if       $y^g=f(g,p)$       $S^1=S^1\cup \left\{g\right\}$       if ${\overline{y}}^g>0$ then         $S^2=S^2\cup \left\{g\right\}$       end if    end for    return  $S^1$

Note: If the number of capacity levels considered is too small, the diversity of discovered solutions may be lacking; if it is too high, it will negatively impact runtime. Tuning may be required to balance these effects.

2.2.2. Binary Search

The binary search procedure is presented in Algorithm 2. If a power deficit is detected, then capacity is increased until no deficit exists, at which point the search direction is reversed; otherwise, capacity is decreased. A capacity adjustment is made on only one DER component per iteration. The step size is initialized to the power of 2 less than the number of capacity levels; it is halved whenever a deficit is encountered until the step size reaches 1—the stopping condition. The second and third level ‘for loops’ allow the binary searches to be run for DERs in differing orders to explore multiple branches of the exhaustive search tree. These loops are responsible for producing a wide range of microgrid designs by randomizing search directions.

Algorithm 2 Microgrid sizing binary search algorithm

Input: f (function for operating microgrid) Input: p (power load, indexed by time) Input: $l_i,i\in I$ (capacity lower bounds by DER) Input: $u_i,i\in I$ (capacity upper bounds by DER) Input: $n_i,i\in I$ (number of capacity levels by DER) Input: $S^0=\{g^1,\cdots ,g^m\}$ (set of microgrids) Output: $S^1=\{g^1,\cdots ,g^{m^{\prime }}\}$ (set of microgrids)    $S^1=S^0$    for  $g\in S^0$do       for  $i\leftarrow 1$ to $\left|I\right|$ do         $c_i={\displaystyle \frac{u_i-l_i}{n_i}}$         $s^i=\{l_i,l_i+c_i,l_i+2c_i,\cdots ,l_i+n_i{c}_i\}$         update capacity of DER i in g to closest capacity in $s^i$       end for       $y^g=f(g,p)$       for  $l\leftarrow 1$ to $\left|I\right|$ do         if ${\overline{y}}^g==0$ then            decrease_capacity = true         else            decrease_capacity = false         end if         $g^{\prime }=g$         for  $i\leftarrow 1$ to $\left|I\right|$ in a random order do            $h=2^{\lfloor log\left(n_i\right)\rfloor }$            while $h>=1$ do              $g^{″}=g^{\prime }$              while True do                 $g^{\prime }=g^{″}$                 set $c_i^{\prime }$ to capacity of DER i in $g^{\prime }$                 if decrease_capacity then                   $c_i^{″}=max\{c_i^{\prime }-h{c}_i,l_i\}$                 else                   $c_i^{″}=min\{c_i^{\prime }+hc,u_i\}$                 end if                 set capacity of DER i in $g^{\prime }$ to $c_i^{″}$ to obtain $g^{″}$                 if $c_i^{″}$$!=$$c_i^{\prime }$ then                   $y^{g^{″}}=f(g^{″},p)$                   $S^1=S^1\cup g^{″}$                   if ${\overline{y}}^{g^{″}}>{\overline{y}}^{g^{\prime }}$ then                      break                   end if                 else                   if not decrease_capacity ${\overline{y}}^{g^{\prime }}==0$ then                      decrease_capacity = true                   end if                   break                 end if              end while              $h=h/2$            end while         end for       end for    end for    return $S^1$

Algorithm 2 starts by initializing a set $S^1$ of potential solutions to the set $S^0$ of solutions identified by the exhaustive search in Algorithm 1. The outermost ‘for loop’ iterates over each potential solution. The first nested ‘for loop’ iterates over each DER and (i) initializes its capacity levels and (ii) sets the starting point for the potential solution to the closest levels compared to the original values from the coarser capacity levels used during the exhaustive search phase. The potential solution is then simulated. The second nested ‘for loop’ iterates several times over the same potential solution and initializes the search direction based on whether a power deficit arose, increasing capacity if a deficit was detected and decreasing it if not. This ‘for loop’ combined with the next inner ‘for loop’ allows for randomizing the order in which DERs are explored, so that the capacity of one DER can be adjusted sooner than other DERs in some iterations and later in other iterations. The ‘while loops’ execute the binary search on the current DER, proceeding until the current solution (i) has a power deficit more than the prior solution or (ii) has a power deficit of zero with a search direction of increasing capacity. In the latter case, the search direction is then reversed and the search continues to decrease capacity until a power deficit arises. Every solution simulated is stored in $S^1$. After the algorithm is terminated, all dominated potential solutions are pruned from $S^1$.

2.2.3. Linear Search

In the final phase of the heuristic, all the non-dominated solutions generated by Algorithm 2, aside from those with power deficits, are finessed through the local search procedure presented in Algorithm 3. Capacity is decreased until either no further decrease is possible or until a power deficit arises.

Algorithm 3 Microgrid sizing local search algorithm

Input: f (function for operating microgrid) Input: p (power load, indexed by time) Input: $l_i,i\in I$ (capacity lower bounds by DER) Input: $u_i,i\in I$ (capacity upper bounds by DER) Input: $n_i,i\in I$ (number of capacity levels by DER) Input: $S^0=\{g^1,\cdots ,g^m\}$ (set of non-dominated microgrids) Output: $S^1=\{g^1,\cdots ,g^{m^{\prime }}\}$ (set of microgrids)    $S^1=S^0$    for  $g\in S^0$do       $y^g=f(g,p)$       if ${\overline{y}}^g>0$ then         continue       end if       $g^{\prime }=g$       for  $i\leftarrow 1$ to $\left|I\right|$ do         $c_i={\displaystyle \frac{u_i-l_i}{n_i}}$         while True do            set $c_i^{\prime }$ to capacity of DER i in $g^{\prime }$            $c_i^{″}=max\{c_i^{\prime }-c_i,l_i\}$            if $c_i^{″}==c_i^{\prime }$ then              break            end if            set capacity of DER i in $g^{\prime }$ to $c_i^{″}$ to obtain $g^{″}$            $y^{g^{″}}=f(g^{″},p)$            $S^1=S^1\cup g^{″}$            if ${\overline{y}}^{g^{″}}>0$ then              break            end if            $g^{\prime }=g^{″}$         end while       end for    end for    return $S^1$

Algorithm 3 starts by initializing a set $S^1$ of potential solutions to the set $S^0$ of non-dominated solutions identified by the binary search in Algorithm 2. The outer ‘for loop’ iterates over each potential solution, first simulating it and skipping it if a power deficit exits. The inner ‘for loop’ iterates over each DER and the nested ‘while loop’ executes the linear search on the current DER’s capacity, proceeding until (i) it reaches the minimum allowable level or (ii) a power deficit arises. Every solution simulated is stored in $S^1$. After the algorithm is terminated, all dominated potential solutions are pruned from $S^1$, ensuring that no rightsized designs are unintentionally pruned during the search.

2.3. Computational Experiment

Microgrid Planner is intentionally designed for researchers to update DER models with ease. Therefore, the specific assumptions regarding the operation of DERs that are employed in the computational experiment are not central to the main focus of this paper. The reader can find details on the simulation method and DER models employed from Microgrid Planner in [23] (note: we used version 1.1, prior to the integration of historical weather data in version 2.0).

The computational example is motivated by a military deployment with a planned duration of two weeks, using the baseline power load shown in Figure 2, which is the same as in [23,24]. The time horizon is discretized into 4-min intervals, which translates to 5040 time steps at which the microgrid’s operation is simulated. If substituting hourly data that encompass typical discretization in commercial software products, the same problem size would accommodate a 7-month planning horizon.

Lower bounds of 0 are set for all DER capacities. The upper bounds are scaled by the peak power demand in the power load profile with adjustments to account for losses. For diesel generators and wind turbines, a multiplier of 1 is applied. For photovoltaic systems, a multiplier of 3 is applied to compensate for non-peak and night hours with less or no generation. For battery energy storage systems, a multiplier of 5 is applied to allow for high levels of energy reserves, and energy is treated as the primary capacity with Microgrid Planner maintaining energy-to-charge-power and energy-to-discharge-power ratios specified in an input microgrid template. Parameter settings of 11, 21, 41, 81 and 161 levels are tested with respective capacity spacing of 10%, 5%, 2.5%, 1.25% and 0.625% of the allowable maximum for each DER type. For generating the initial solution set from the exhaustive search algorithm, 6 levels are used with respective capacity spacing of 20%. The exhaustive search algorithm, shown for comparison purposes, becomes computationally intractable with more levels.

3. Results and Discussion

The heuristic sizing method with 11 levels (10% spacing) simulates 359 candidate microgrid designs when considering three DER types: diesel generators, photovoltaic and battery energy storage systems. Of these, the 17 solutions shown in

Table 2. An example of non-dominated microgrid designs with no power deficit generated by the DER sizing method. Ratio values equal to −1 correspond to DER capacities of 0.

BESS					Diesel
Charge &		BESS		Diesel	Gen.	Diesel		PV
Discharge	BESS	Time	BESS	Gen.	Time	Gen.	PV	Time	PV
Power	Energy	Steps	Unused	Power	Steps	Unused	Power	Steps	Unused
(kW)	(kWh)	Ratio	Ratio	(kW)	Ratio	Ratio	(kW)	Ratio	Ratio
0	0	0	−1	120	1	0.67	0	0	−1
65	130	0.031	0.69	95	1	0.58	0	0	−1
325	650	0.51	0.9	45	1	0.09	0	0	−1
325	650	0.3	0.85	25	0.58	0.14	245	0.5	0.44
257.5	515	0.29	0.85	35	0.59	0.19	210	0.5	0.46
192.5	385	0.29	0.85	45	0.62	0.26	175	0.5	0.45
130	260	0.27	0.88	60	0.66	0.39	140	0.5	0.47
65	130	0.025	0.59	80	0.62	0.53	175	0.5	0.54
160	320	0.5	0.96	70	1	0.43	0	0	−1
97.5	195	0.25	0.94	70	0.66	0.47	140	0.5	0.5
160	320	0.27	0.81	45	0.55	0.27	350	0.5	0.63
130	260	0.28	0.95	70	0.78	0.36	35	0.5	0.4
225	450	0.36	0.88	45	0.78	0.21	70	0.5	0.22
192.5	385	0.51	0.92	60	1	0.33	0	0	−1
290	580	0.35	0.88	35	0.71	0.15	105	0.5	0.23
160	320	0.3	0.91	60	0.78	0.36	70	0.5	0.38
130	260	0.061	0.83	80	1	0.5	0	0	−1

encompass all the non-dominated solutions with no power deficits at any point in time. The diversity of these candidate solutions is evident in the number of discrete capacity levels they cover. Specifically, 8 out of 11 diesel generation capacities, 9 out of 11 photovoltaic system capacities, and 10 out of 11 battery energy storage system capacities are present in these 17 candidate microgrid designs. The results from the sizing method can be compared with those obtained from an exhaustive search because the instance is sufficiently small. An exhaustive search produces 18 solutions with no power deficits, with none being identified as dominated, but 16 of which are identified by the heuristic. The heuristic misses two solutions that the exhaustive search identifies, one of which exhaustive search prunes, because of a subtle detail regarding energy management system logic and resulting battery state of charge.

The heuristic sizing method results are compared with those of the nested binary search “rightsizing” method presented in [23]. The approach in [23], with step sizes of 5 for all DER type capacities, produces 816 “rightsized” designs, 326 of which are within the capacity limits set in our heuristic sizing method. The heuristic sizing method with 41 capacity levels identifies 42 non-dominated microgrid designs with no power deficits. Although the heuristic only identifies a fraction of the number of solutions of [23], that fraction is diverse in the mixes of DER capacity levels, as shown in Figure 3. Moreover, the hundreds of options produced by [23], many of which are quite similar, are neither needed nor necessarily desirable. The heuristic algorithm produces smaller numbers of potential solutions with smaller numbers of capacity levels, increasing from 17 non-dominated microgrid designs with no power deficits with 11 levels, to 32 such designs with 21 capacity levels, to 42 designs with 41 levels, to 44 designs with 161 levels, as shown in Figure 3c and Figure 4. Increasing the number of capacity levels past 41 does not have a significant impact on the number of microgrid designs identified.

The comparison with [23] is explored further in

Table 3. Comparison of candidate microgrid designs produced by the sizing method versus their closest counterparts using the “rightsizing” method of [23].

Sizing Method			Reich & Oriti [23]			Difference
Diesel	Photovoltaic	Battery	Diesel	Photovoltaic	Battery	Diesel	Photovoltaic	Battery
25	245	650	25	240	635	0	5	15
35	105	580	35	100	570	0	5	10
35	210	515	35	210	500	0	0	15
45	0	650	45	0	610	0	0	40
45	70	450	45	65	455	0	5	−5
45	175	385	45	170	370	0	5	15
45	350	320	45	335	320	0	15	0
60	0	385	55	0	360	5	0	25
60	70	320	55	70	315	5	0	5
60	140	260	60	140	240	0	0	20
70	0	320	70	0	275	0	0	45
70	35	260	70	35	250	0	0	10
70	140	195	70	140	185	0	0	10
80	0	260	75	0	245	5	0	15
80	175	130	80	160	130	0	15	0
95	0	130	95	0	125	0	0	5
120	0	0	120	0	0	0	0	0

. The difference between solutions generated with the two approaches is less than the 10% spacing set by the input number of levels (11). For example, the largest difference encountered is in the microgrid with 70 kW diesel generation and no photovoltaic system, where the sizing method sized the battery energy storage system at 320 kWh, which is 45 kWh more than the method of [23]. This difference may seem large at first glance. However, when considered in context of the discretization of battery capacity levels in the sizing method, it actually represents the lowest possible level corresponding to the “rightsized” value, as decreasing another level to 260 kWh results in a non-zero power deficit. The same is true for all differences reported. In other words, given the discretization of capacity levels used in the computations, the 17 non-dominated microgrids are all “rightsized”.

To further reduce the differences between the solutions generated by the two methods, capacity level spacing must be reduced by increasing the number of levels per DER. Reducing capacity level spacing also increases the number of potential “rightsized” solutions. A summary of results for the various capacity level spacings tested is provided in

Table 4. Summary of the sizing method versus exhaustive search for varying numbers of DER types and levels per DER type. “N/A” denotes six levels per DER-type instance for which the sizing method is not applicable, as it would return the exhaustive search results. “-” denotes instances for which an exhaustive search was not solvable within the 48-h time limit.

			Sizing Method			Exhaustive Search
# DER	# Levels	# Potential	# Candidate	# Sim	Run	# Candidate	# Sim	Run
Types	per DER	Microgrid	Microgrid		Time	Microgrid		Time
		Designs	Designs		(min)	Designs		(mins)
3	6	216	N/A	N/A	N/A	7	114	2
3	11	1331	17	359	8	18	665	14
3	21	9261	33	615	15	62	4562	101
3	41	68,921	42	816	20	132	21,324	770
3	81	531,441	42	920	23	-	-	-
3	161	4,173,281	44	1160	28	-	-	-
4	6	1296	N/A	N/A	N/A	29	1104	33
4	11	14,641	90	2196	64	132	12,691	295
4	21	194,481	176	3711	109	-	-	-
4	41	2,825,761	223	5652	168	-	-	-
4	81	43,046,721	236	6480	188	-	-	-
4	161	671,898,241	254	9287	216	-	-	-

. In total, 42 out of the 44 solutions obtained by the sizing method when using 161 levels per DER exactly match the solutions generated by the method of [23]. The other two differ by only 5 kWh of battery capacity, which is the rounding precision set for generating the discrete capacity levels to be considered. A similar comparison is not possible in the four DER-type instances, as the method of [23] is limited to three DER dimensions.

The exhaustive search exceeded the 48 h time limit with 81 levels for three DER types, and with only 21 levels with four DER types. The run time of the sizing method increases at a decreasing rate as the number of levels are increased at an exponential rate, so it scales well to achieve capacity precision. However, it does not scale as well when increasing the number of DER types. Microgrid Planner does not yet include a fifth type of DER, so further research may be required to improve the efficiency when more DER types are supported. Avenues exist for doing so, such as decreasing the number of levels in the exhaustive search used to provide the initial set of microgrid designs, reducing the number of iterations in the ‘for loops’ currently used to expand the search and pruning more solutions.

The sizing method introduced in this paper was developed to meet the planning needs of partners at military installations. The search heuristics presented are efficient for the size of the current DER technology portfolios being actively planned for current microgrid investments. These heuristics are customizable and may be tuned as additional technologies are considered; however, the DER technology set will likely remain relatively small in the near future.

The method has notable limitations. It is a heuristic. Moreover, is not guaranteed to find rightsized solutions and may discover local optima as it branches. To mitigate this, the binary search method explores the search space in different orders of DER capacities, for example, adjusting the photovoltaic capacity before diesel generation capacity on one iteration of third-level and fourth-level for loops. Then, it may explore those DER capacities in the opposite order on another iteration. The computational results suggest that this approach is effective in practice. Moreover, Microgrid Planner allows the user to easily simulate the performance of any microgrid design being considered, review the results, and interactively adjust capacities, so the user can be confident in any solution being considered for adoption.

The method is deterministic. Another limitation is that it does not account for the inherent stochasticity in weather conditions, upon which renewable energy sources, such as photovoltaic systems and wind turbines, rely. Sensitivity analysis [24] can be applied to subject microgrid designs of interest to a range of generated scenarios and assess their performance. Future work may explore methods that integrate stochastic analysis directly in the DER sizing method.

Future work may also include experimentally validating the performance of microgrid designs produced by the search heuristics. The construction of the function for operating the microgrid has an effect on when power deficits may arise and thereby impacts the results produced. Therefore, a future study may compare performance of different control strategies on planning outcomes.

The DER sizing method has been released in Microgrid Planner [2,3]. By contributing to open source software, the sizing method proposed in this paper provides a low-cost planning capability that may be configured and extended, which is a significant advantage compared to proprietary microgrid planning software.