We sized a solar panel and storage microgrid in MATLAB, then tested it against 300 scenarios of uncertain loan rates, grid prices and carbon fees.
At $0.40 per kWh of grid power, the cheapest plan is to build almost nothing and buy from the grid. At $0.55 the optimizer switches to about 31,000 m² of panels and 19,000 kWh of storage, and the grid fills in only around the summer peak.
See where the design flipsHot Springs, North Carolina runs a real solar microgrid. We used it as the template for a town of 2,000 people, and first needed two inputs: how much energy the town uses, and how much land there is for panels.
Each resident draws between 500 and 1,000 W on average. We took the midpoint and scaled it to a full day for the whole town.
The brief asked for one full day of demand in reserve, so 36,000 kWh is also the storage target.
We traced the outline of the existing Hot Springs array on satellite imagery. The dark rows inside it are the shadows the panels cast, not panels, so we measured each shadow row and subtracted it.
The shadows covered more than half of the outline. Keep the 14,800 m² in mind: several designs later want three to four times more land.
The model has two design variables and three uncertain prices. For any choice of panel area and storage size, it simulates every hour of a year and returns the average cost of each kWh the town consumes.
Here \(L\) is the yearly loan payment on the panels and storage (it grows with \(A\), \(C\) and \(r_\gamma\)), \(M\) the operating cost and \(E_{\text{grid}}\) the energy bought from the grid (it shrinks as \(A\) and \(C\) grow), so \(J\) is the cost of each kWh in cents. Every kWh bought from the grid pays both the utility price and the carbon fee, so those two prices push in the same direction. Building more panels and storage replaces those fees with loan payments, and the optimizer looks for the point where the two balance. We solved it with sequential quadratic programming (SQP), starting from 15,000 m² and 28,000 kWh.
In the first runs both variables ended on their lower bounds with positive multipliers, \(\lambda_A\approx 20.0\) and \(\lambda_C\approx 3.8\). A positive multiplier on a lower bound means the cost was still falling there: the optimizer would have built even less if the bound allowed it. The two inequality constraints had \(\lambda=0\) in every run. They never bound, so each design was set by the cost tradeoff and the variable bounds alone.
Inside the simulation, a fixed dispatch rule decides where each hour's energy comes from. Solar serves the town first, any surplus charges storage, storage covers a shortfall, and the grid supplies whatever is still missing.
Three panel types times four storage technologies gives 12 designs. We optimized each one at baseline rates: grid power at 12¢ per kWh, a carbon fee of 40¢ per kWh and a 6.5% loan. Choose a measure to shade the grid, and select a design for its details.
Li-ion = lithium-ion battery · PSH = pumped-storage hydro · CAES = compressed-air energy storage. “Over site” marks designs that need more than the 14,800 m² we measured.
The cheapest designs don't fit. At these rates the lowest costs come from low-efficiency panels with pumped hydro (34.2¢) or compressed air (33.6¢). Both fill the 60,000 m² upper bound, about four times the site.
High-efficiency panels aren't built at all. Area and storage stay at their floors, and the cost is just the grid price plus the carbon fee: 12¢ + 40¢ ≈ 52¢. At baseline rates the expensive panels never pay for themselves.
Bulk storage wins every row. Pumped hydro and compressed air were the two cheapest storage options for every panel type.
The grid shows one set of prices. To find the tipping point, we held the grid price at 12¢ and the loan at 6%, and raised the carbon fee. Each plot shows a full year of hourly energy flows into the town, with summer in the middle.
Storage, in cyan, now carries the night, and the grid only fills in around the summer peak.
Grid price 12¢, loan 6%.
Between $0.55 and $0.75 the fee rises by 20¢, but the cost of energy rises by only 1.5¢. Once the panels and storage are built, the town buys so little grid power that the fee hardly touches it.
We also halved and doubled all three rates (clipping each to its range) and tried a cheaper loan. The two marked rows differ only in the loan rate.
| Scenario | Loan | Grid price | Carbon fee | Panel area, m² | Storage, kWh | Cost, ¢/kWh |
|---|---|---|---|---|---|---|
| All rates halved | 4.0% | $0.09 | $0.10 | 19.00 | ||
| All rates doubled | 9.0% | $0.15 | $0.40 | 55.09 | ||
| Carbon fee $0.45 | 6.0% | $0.12 | $0.45 | 56.61 | ||
| Carbon fee $0.55 | 6.0% | $0.12 | $0.55 | 58.98 | ||
| Carbon fee $0.75 | 6.0% | $0.12 | $0.75 | 60.52 | ||
| Cheaper loan | 4.0% | $0.15 | $0.40 | 49.76 |
At a 9% loan the optimizer builds nothing. At 4% it builds almost 30,000 m² of panels, and the cost of energy falls from 55 to 50¢. The carbon fee is the main lever, but close to the tipping point, cheaper financing alone can flip the design.
A single run answers one set of prices. We ran the optimizer under random prices in two stages: first to see which technology wins, then to measure how uncertain the cost of the chosen design is.
For each scenario we drew the three rates uniformly across their ranges, optimized all 12 panel and storage pairs, and kept the cheapest. That is 1,200 optimizations in total.
The winning designs fell into three groups, set by the grid price and the carbon fee:
With the design fixed to high-efficiency panels and compressed-air storage, we drew each rate from a lognormal distribution, since real rates are positive and skewed to the right. Each one is centered on the middle of its range, with a spread that puts the ends of the range about two standard deviations out. The three rates were drawn independently.
| Rate | Range | Median | Coefficient of variation |
|---|---|---|---|
| Loan rate | 4% to 9% | 6.5% | 0.19 |
| Grid price | $0.09 to $0.15 | $0.12 | 0.125 |
| Carbon fee | $0.10 to $0.90 | $0.50 | 0.40 |
In 90% of scenarios the cost of energy falls between 41 and 71 ¢/kWh. Most land between 45 and 60¢, with a slight skew to the right: low financing and low carbon fees give costs in the low 40s, while expensive scenarios pass 70¢, about 75% more.
The carbon fee dominates. As it rises, the optimizer pays more up front to avoid it, so the cost climbs steeply until the system is built out. The loan rate matters moderately, since it sets the cost of every panel and kWh of storage. The grid price barely registers, for two reasons. Its typical spread was only about ±1.5¢ around 12¢, against roughly ±20¢ around 50¢ for the carbon fee, and both apply to the same grid kWh. And whenever the carbon fee is high, the optimizer avoids the grid, which removes the grid price from the cost too.
Panel area behaves like a yes-or-no decision. With the narrower Monte Carlo bounds, nearly every run ends at about 1,000 m² or about 30,000 m². A high carbon fee with a moderate loan rate means full investment; a low fee with an expensive loan means the floor and grid power.
Storage is the tuning knob. It often sits at its floor too, but when it is built it mostly lands between 10,000 and 20,000 kWh, sized to how much daytime solar has to be shifted into the night.
Next step. Our model gives surplus solar no value. If the town could sell excess energy back to the grid, even below the retail price, oversizing would stop being a penalty. That would push more designs toward a full build and make the cost less sensitive to the carbon fee and the loan rate.
The MATLAB code behind the results. SolarMicro_opt runs the hourly simulation and the SQP optimization for one set of rates.
% use solar to meet demand when available; prioritize charging storage with any excess if (Pd < Ps) Psd = Pd; Psx = min([Ps - Pd, max_P_I_X * cap_E_X, (maxCharge * cap_E_X - E) / Efficiency_X]); % if solar falls short, use storage to cover the remaining demand if (Pd > Ps) Psd = Ps; Pxd = min([Pd - Ps, max_P_O_X * cap_E_X, (E - minCharge * cap_E_X) * Efficiency_X]); %if solar + storage still cant meet demand, draw remaining energy from the grid if Psd + Pxd < Pd Pud = Pd - Psd - Pxd;
clear; clc; % clear command window solarTypes = 1:3; % 1 = Low Efficiency, 2 = Standard Eff, 3 = High Eff storageTypes = 1:4; % 1 = Li-Ion, 2 = Flywheel, 3 = PSH, 4 = CAES % Step 2: Define fixed financial rates (baseline case), we took the average of each range utilRate = 0.12; % $/kWh from the grid ghgRate = 0.40; % $/kWh to offset GHG emissions loanRate = 0.065; % Interest rate on capital investment rates = [utilRate; ghgRate; loanRate]; % Step 3: Initialize results matrix results = []; % Each row will be: [storageType, solarType, cost, area, storage] % Step 4: Loop through all combinations for st = solarTypes for xt = storageTypes try % Call the optimization function [f_opt, ~, v_opt] = SolarMicro_opt(rates, 0, xt, st); results(end+1,:) = [xt, st, f_opt, v_opt(1), v_opt(2)]; end end end % Step 5: Display colNames = {'StorageType', 'SolarType', 'Cost_c_per_kWh', 'Area_m2', 'Storage_kWh'}; resultTable = array2table(results, 'VariableNames', colNames); disp('=== Technology comparison ==='); disp(resultTable);
clear; clc; N = 100; % Number of iterations solarTypes = 1:3; % 1=low, 2=standard, 3=high-eff storageTypes = 1:4; % 1=Li-Ion, 2=Flywheel, 3=PSH, 4=CAES results = []; %starts with initialize the list for i = 1:N % Random financial parameters in their valid ranges, choosing uniformly utilRate = 0.09 + rand() * (0.15 - 0.09); % Uniform between 0.09 and 0.15 ghgRate = 0.10 + rand() * (0.90 - 0.10); % Uniform between 0.10 and 0.90 loanRate = 0.04 + rand() * (0.09 - 0.04); % Uniform between 0.04 and 0.09 rates = [utilRate; ghgRate; loanRate]; best_cost = inf; best_result = []; for st = solarTypes for xt = storageTypes try [f_opt, ~, v_opt] = SolarMicro_opt(rates, 0, xt, st); global avgPud_per_h % import global grid use if f_opt < best_cost best_cost = f_opt; best_result = [utilRate, ghgRate, loanRate, st, xt, f_opt, v_opt(:)', avgPud_per_h]; end catch fprintf('Error for combo xt=%d, st=%d\n', xt, st); continue; end end end results(end+1,:) = best_result; end % Display the results colnames = {'utilRate','ghgRate','loanRate','solarType','storageType','cost_c/kWh','Area_m2','Storage_kWh','GridUse_kWh/h'}; disp(array2table(results, 'VariableNames', colnames));
%solarMico_montecarlo.m NMcs = 200; % loan utility ghg rates_min = [0.04 ; 0.09 ; 0.10]; rates_max = [0.09 ; 0.15 ; 0.90]; rates_med = (rates_max +rates_min)/2 ; rates_cov = (rates_max - rates_min)/4 ./ rates_med; %this might not be exact R = [1.0 , 0 , 0 ; % CORRELATION MATRIX, must be positive definite (when eigenvalues are positve), dont change the diagobal, maybe change the zeros 0 , 1.0, 0 ; 0 , 0 , 1.0 ]; %make sure eigenvaluea are positve if any (eig(R) <= 0) error('your correlation values make for a non positive definitie correlation matrix') return; end rates = corr_logn_rnd(rates_med, rates_cov, R, NMcs); %3 x Nmcs f_opt =zeros(1,NMcs); %optimization cost cents / KW.h for each mcs v_opt = zeros(2,NMcs); %optimization design variable for each mcs for k = 1:NMcs this_rate = rates(:,k); % Get the k-th random rate set: [loanRate; utilRate; ghgRate] fprintf('Trial %d | Loan = %.4f | Utility = %.4f | GHG = %.4f\n', ... k, this_rate(1), this_rate(2), this_rate(3)); [f_opt(k), g_opt, v_opt(:,k)] = SolarMicro_opt(this_rate, k); fprintf(' Cost = %.4f /kWh | Solar = %.1f m | Storage = %.1f kWh\n\n', ... f_opt(k), v_opt(1,k), v_opt(2,k)); end save('SolarMicro_montecarlo.mat', 'f_opt', 'v_opt', 'rates'); %what next?? % find statistics for ... %CDF of f_opt? CDF of v_opt(1)? CDF of v_opt(2)? plot CDFci (confidence %intervals) %correlation between f_opt and rates(3,:) ... %etc %the point is to understand how uncertainty changes the outcome, it gives %out a bunch of optimal solutions