1. Introduction
There are various modern examples where unforeseen disruptive events deprive people of access to electrical power. These events can cause anything from minor inconveniences to situations (leading to mortal danger). Since the 2000s, there has been a drastic increase in weather-related power outages (affecting more than 50,000 customers [
1,
2]). One recent example is the winter storm that hit the lower 48 U.S. states in 2021, leaving 4 million people without electricity. A distributed generation microgrid (MG) can help make energy access more reliable and resilient to unforeseen disruptions while increasing energy independence. A properly configured MG can be run as the only energy source (island mode) or as extra energy to power all or some of an area’s critical energy demand nodes (DNs). DNs, such as hospitals, fire stations, and grocery stores, can be prioritized during disruptions to maintain safety and comfort. MGs, in these situations, must fully or partially power all necessary DNs with minimal investments, operations, and maintenance costs. MGs with PVDG sources are promising but require additional considerations and equipment to utilize their potential fully. In an island mode situation where the MG is expected to provide emergency power, which is the focus of the work, these issues are critical to optimal solutions.
According to a report by the National Renewable Energy Laboratory [
1], photovoltaic distributed generation (PVDG) systems could supply electricity during grid outages resulting from extreme weather or other emergencies. In order to take advantage of this capability, the systems must be designed with energy uncertainties in mind and combined with other technologies, such as energy storage systems (ESSs). DGs that utilize renewable energy sources (RESs) can help provide environmentally friendly energy sources but cause unique problems. First, PVDG energy output ties directly to environmental conditions, such as the amount of solar radiation an area receives and the area’s current sky conditions (cloudiness). Thus, energy output from the PVDG is intermittent, so there is a need to regulate the power received by DNs to see a consistent voltage profile even when solar resources are low. Second, high energy output from the PVDG can pose issues. Excess unused energy can cause reverse power flow (RPF) in systems with renewable energy sources. If the system generates more power than the DN requires, issues in the primary energy grid, such as loss of voltage control, increased dangers from short circuits, and degradation of the reliability of protective systems, can occur [
3,
4,
5]. Thus the appropriate sizes or capacities of PVDG systems are crucial to help minimize these issues.
ESSs can help control voltage fluctuations and the RPF that RESs can cause by storing and discharging energy as needed. The appropriately chosen ESS capacity is key to ensuring that the ESS is optimally utilized. Too large of a capacity leads to wasted monetary resources due to unused capacity; too small, and there may not be enough storage to help mitigate potential RPF or provide enough backup energy. However, adding the appropriately sized ESS will only improve conditions with appropriate charging strategies. A good charging strategy must respond with power when the system is at low output, be reactive to the intermittent production of RESs, and be able to store excess energy at the appropriate times [
6]. Charging unnecessarily when the system is at a low output or discharging when the system is at a high output will only amplify RES-related problems. As more renewable sources become a part of the energy grid, assignments, sizing, and storage management strategies become increasingly important.
Due to the varied capacity sizes, possible locations, and different charging strategies of DGPV/ESS systems finding optimal solutions can be problematic and computationally expensive. In addition, a feasible solution must consider the uncertain nature of the environmental conditions upon which the RESs depend on. Therefore, researchers are active in the optimal DG/ESS placement and sizing (ODGSP) problems, as well as ESS energy management. Many have sought to solve these problems using various techniques, including purely numeric, heuristic, machine learning [
7], and simulation [
8].
This work seeks to tackle the ODGSP and energy management using a genetic algorithm (GA) and simulation-based approach for a system in an island mode situation. The selected MG configuration must be the sole source of energy during the sunlight hours over the course of an extended disruption in service to act as emergency power. GA-based heuristics search for optimal solutions based on the results of simulations of each purposed configuration of the MG. The simulation provides insight into how a proposed solution will perform in a real-world environment based on historical data to gauge solution feasibility realistically.
In this work, we used a test set of twenty-five DNs and co-located PVDG/ESS units to test the optimization capabilities of the algorithm. The goal was to evaluate the algorithm’s ability to find solutions with minimal costs, maximal energy demand fulfillment, and minimized RPF MG configurations, to provide an appropriate energy management strategy that allows DN-type prioritization for load shedding. The results show that the method can achieve these desired goals. The following section will describe previous research on ODGSP, distributed energy storage sizing, placement, and management, and distributed generation in island mode microgrids.
The method’s objective is to optimize several factors (at a high-level model of a MG) from the perspective of operations research for critical strategic decision-making. First, the investment and operating costs for a set of PVDG/ESS units can be high, so this work seeks to minimize the sizes of these constraints below a predefined start-up budget while minimizing average operational costs. Second, meeting the demands of the critical DN is the emergency MG’s primary goal; thus, the solutions must fulfill maximal average hourly demands. Third, specific types of DNs may be deemed more critical than others in extreme situations. A heuristic was employed based on adjustable node priorities to aid load shedding decisions. Therefore, the described algorithm allowed the specified facility or DN types to prioritize their energy demands. Fourth, the PVDG requires an appropriate energy management strategy, and the method provides hourly ESS management strategies that exhibit peak shaving and voltage regulatory characteristics. Lastly, the potential RPF must be minimized for the MG to be suitable as an auxiliary power source in normal non-isolated conditions. In comparison to other works, this study sought to tackle these five problems simultaneously with one unified methodology. To make the model tractable, the proposed MG model does not include factors such as power quality, temperature-based performance considerations, equipment degradation, fluctuating energy costs, and low-level electronics of charging and discharging the ESS units.
3. Model Formulation
The problem to be modeled can be described using the sets, variables, and parameters found in
Table 1,
Table 2 and
Table 3 as follows. A utility seeks to purchase and install a PVDG/ESS MG capable of providing, at most, a month’s worth of emergency power. There is a set
of critical DNs that must have a maximal amount of their energy demands met using a set of PVDG/ESS units during the sunlight hours of the day
in an islanded MG. A monetary budget
is available for the purchase and installation of the units that must be honored. Energy system management seeks the optimal MG configuration of PVDG/ESS units and an appropriate hourly charging strategy feasible for the specific location and environmental conditions that maximize demands fulfilled while minimizing RPF, startup, and operational costs. In addition, the utility seeks solutions that prioritize specific DN energy demands over others.
Three tasks can describe the problem. The first task is the selection of the locations and sizes of a set of co-located AC-coupled utility-scale (MW) PVDG/ESS units or DGs
from the set of possible units
. The selected units will energize the DN set
. The second task involves the assignment of each of the selected PVDG/ESS units
j to an independent subset of the DN
from
. The assignment allows each DG unit
j to be assigned to multiple DNs, and each DN
i to only be assigned to one DG unit (see Equation (
11)). A given assignment
is represented as a binary value where if the unit
j is assigned to the DN
i the value is 1 and 0 otherwise. An example assignment can be seen in
Figure 1 below.
The third task is to provide an appropriate energy management/charging strategy for each operation hour. For each of the sunlight hours, a general directive is to be given for all ESS units currently assigned from the set of possible directives that will help dictate if the units will charge, discharge, or remain idle over the hour.
The objective is to minimize the initial startup cost
, average hourly operational costs
, average excess unused energy or potential RPF
, and the average amount of unmet energy demands
. The sum of these factors
(see Equation (
21)) represents the optimization objective. The available budget
constrains the total startup costs for the selected PVDG/ESS units
for any potential solution
g (see Equation (
1)). It is assumed that other measures will be taken outside sunlight to energize the DN.
A desire to prioritize different DN energy demands over others and minimize potential RPF is also desired. When energy resources are scarce, decisions may need to be made on which of the critical DNs are more important. For example, a grocery store may be spotlighted over a gas station by setting the priority of the grocery store higher than that of the gas station so that its energy demands are prioritized. The value of is adjustable, so different types of DNs can be prioritized. Minimizing potential RPF is also a goal, so the system may be suitable for connecting to the primary grid.
In the following paragraphs, the mathematical models and expressions representing costs of investment and operation and the energy output for the units are given. Next, the energy demands and transmission costs for the DN and the expressions used to model their energy supply and demand dynamics are described. Finally, the GA and other heuristics applied to find an optimal solution to the three tasks are detailed.
The set of selected units
are expected to power the DN
throughout the sunlight (6 a.m.–6 p.m.) hours
. Each unit
j in the selected set
is located at various positions around the energy demand nodes and comes in different capacity parameter combinations for the PVDG and ESS units. The parameters (
Table 2), time of day (
h), and current sky condition govern each unit
behavior. Each unit
j consists of a DG component modeled as a mono-crystal single-axis PV energy generation unit and an ESS modeled as BESS based on Li-ion batteries. The PV and ESS units have investment costs based on the product of the nameplate power and energy rating of the component (
and
respectively) and an investment rate, which comes in the form of dollars per kW rate for the PV unit (
) and dollars per kWh for the ESS component (
). The total investment cost for a given unit
j is then the summation of the investment cost for the PV and ESS units, labeled
Cj in Equation (
1).
For a selected subset
of DG units representing a potential solution, the total investment cost
is then the summation of all units currently selected for assignment (Equation (
2)) for that given solution
g. The investment cost represents purchasing the units, necessary equipment, and installation.
This investment cost for a given solution
g corresponding to selection
represents the monetary investment that must be less than the set investment budget
(Equation (
3)).
Along with individual investment rates, each PV and ESS component has individual operation and maintenance (O&M) rates,
and
, respectively. These rates represent the dollars per kW costs of maintaining and utilizing DG unit
j at a level of energy output over an hour. For the PV component, the energy output over hour
h under the sky condition
w is represented by
. The maximum expected energy output for a given PV component is twenty percent of its rated capacity
(Equation (
4)). This efficiency factor is based on the average efficiency of mono-crystal PV units according to the most conservative modeling of utility-scale PV systems (as per [
25]). The product of the energy output over the hour and its O&M rate
is the monetary cost of operating the PV unit for the hour.
The fluctuating energy represented by
values simulates the intermittent nature of PV energy production due to uncertain weather conditions and the dependence on the amount of sunlight. The magnitude of energy output
is lowest in the early morning (low sun) and overcast conditions, with the highest occurring when the sun is at its apex, and the sky condition is sunny.
The probability of sunny, cloudy, and overcast are 0.266, 0.293, and 0.441, respectively, and are based on historical sky condition data [
26]. Similarly, the ESS component has output over an hour
, and the product of this and the O&M rate
is its O&M cost in dollars over the hour
h. For the ESS of unit
j, the energy output is based on its rated energy capacity
, the level of charge
at the time
h, the current energy demand and supply conditions, and the current energy management directive
. The nature of the ESS output is discussed in more detail in the next paragraph. Thus, the overall O&M costs
for unit
j is the summation of the O&M costs for its PV and ESS components over the hour
h at the output capacity in kW (Equation (
5)). The expression indicates that if the ESS unit is charging, thus producing a negative energy value, or when it is idle, there is no operation and maintenance charge for that hour for the ESS unit. Otherwise, its O&M costs are based on the energy output over the hour h and its rated costs.
The charging commands
indicate the suggested charging/discharging behaviors for all ESS units currently in operation. A given charging strategy
provides hourly charging directives for maximizing the demands met and minimizing excess unused energy over the sunlight hours
for each hour
h. Each ESS will follow the command based on the charge left in the ESS
and the difference between the energy produced by its corresponding PV unit and their assigned DNs
represented by
(
Table 3). ESS system commands consist of discharge (0), meaning to supply stored energy if available and required, charge (1), meaning to store energy from the corresponding PV unit, and remain idle (2), meaning to neither discharge nor charge. The amount of charge left
represents the amount of discharging hours a given ESS has experienced without an hour of charge. When the unit experiences an hour in discharge mode, its
value is incremented, and each hour in the charge mode decrements the value to a minimum of 0, meaning the unit is fully charged. When the value equals the rated hourly energy rating for the unit represented by
, the ESS unit
j is considered fully discharged and can provide no power until it has received at least an hour of charge. Each hour a given ESS unit can output
kW until fully discharged, and it receives
% of its possible output from its given PV unit
j during charging, reducing the available output from the PV unit. When the system is set to idle mode, no energy is output from the ESS unit, nor does it draw any from the PV unit. This logic is represented by Equations (
6) and (
7). This behavior is intended to model the discharging and charging behaviors of an ESS without considering the more complex thermal behaviors of ESS systems or the static discharge they exhibit.
Equation (
8) represents the initial state of all ESS units at the start of a given simulation run, indicating that each is “fully” charged at the start. Each hour, the
value for a given ESS unit is updated using Equation (
9). These expressions ensure that the maximum discharge hours that a given ESS unit can provide are limited to its specified amount.
The overall energy output for a given unit
j for hour
h is expressed in Equation (
10), detailing it as the sum of the energy output of its PV and ESS components. These expressions are intended to model the energy production of PV units with varying sunlight and sky conditions, the energy output of an ESS unit based on its ratings, charge level, and simple heuristics based on the need for extra energy based on supply and demand. The following paragraphs will describe the expressions used to model the demand nodes
and the energy exchange dynamics between the DG and DN.
There are five types of possible demand nodes, including hospitals, grocery stores, fire stations, gas stations, and police stations, each having varying hourly demands to simulate time-varying loads. The different types can have their demands prioritized using the type priority variables represented by
. Each of the selected possible DG units
j from the selected set
are assigned a disjoint subset of the DN
from the set
. Each
i in the set
can only be assigned to one unit
j for a given solution (Equation (
11)). Each DN
i will require varying energy over a given hour
based on historical data for the type of building during the sunlight hours, and must be supplied solely from its assigned PVDG/ESS unit
j.
Due to Equation (
12), there may be some amount of unmet demands for the demand nodes, and this value is the difference between what is demanded over the hour
and what its assigned DG unit
j could supply
. The total unmet demand for a given hour
is then the summation of all unmet demand for all DNs
as seen in Equation (
13). Each different type of DN has its unmet demand weighted by the priority value
. There is also an adjustable overall unmet demand weight
to control the optimization’s sensitivity to unmet demands and control for the potential difference in magnitude in costs and unmet demands.
To energize each DN, a transmission cost is incurred. The transmission cost for a given DN is the product of the energy supplied to it at hour
h, and the distance between the energy source and the DN
. The total transmission cost over an hour
h is the summation of all transmission costs for the DN, as seen in Equation (
14).
For each GA-generated solution
, there are DG selection and hourly charging variables (for each chromosome). The selection gene represented by
determines the investment cost by selecting a specific subset of the potential PVDG/ESS units from the set
. The selection of
, as well as the charging strategy gene
, will determine the average transmission, operation, and maintenance costs, along with the energy dynamics of the MG. Each solution
is given
day of
hour runs. After the simulated time, the average transmission, operation, and maintenance costs are calculated with Equation (
15) representing the average hourly monetary cost of operating for a given solution.
Along with the costs, the potential RPF for the selected units
labeled
are calculated each hour with Equation (
16). There is an adjustable weight
to control how sensitive the “most fit” solution
G is to potential excess unused energy. For each DG, Equation (
17) demonstrates the calculation of a given DG’s total energy supplied to its assigned DN. The total potential RPF for an hour
h is then the summation of each unit in
’s potential RPF with Equation (
18).
The average potential RPF
for a solution
g (Equation (
19)) is used in the objective function to determine fitness. The solution with the lower
is more fit.
To quantify solution
’s ability to meet the energy demands of the demand node
, the average unmet demand over the run is calculated as seen in Equation (
20), which is another metric for solution
fitness.
For each possible solution, its fitness is calculated by summing the investment cost for the selected DG units
, the average monetary costs
, average potential RPF
, and the average unmet demand
with Equation (
21).
For each generation of solutions
their, fitness scores
are converted into what are termed here residual fitness scores
to perform the minimization task. All fitness scores are summed, and then the difference between each and the sum is used for the survival of the fittest pair selection process with Equation (
22). This step ensures that the solution
g with the highest residual
is the solution with the lowest fitness
. The fittest or optimal solution is the solution
g, such that it achieves the minimum
score, i.e., the maximum residual score
as indicated with Expression (
23). This way, the solution with the highest residual is the fittest of a given generation
for a given set of solutions. Equation (
23) is the metric used to compare the solution’s fitness. The following section will briefly describe the data used for the simulation, how the simulation uses heuristics to adjust the behavior of the ESS units, and the assignment to ensure feasible and accurate solutions. Following this, we present a description of the model and simulation logic along with the application of the GA.