NSGA-II-Based Multi-Objective Optimisation of Solar–Battery Systems for Cost and Reliability ^†

Abstract

Increasing grid instability and rising electricity prices highlight the need for reliable photovoltaic–battery energy storage systems (PV–BESS). This study presents a constraint-aware multi-objective optimisation framework using NSGA-II that integrates economic performance, reliability, curtailment, and battery ageing. Applied under realistic South African conditions, the optimal system achieved a levelised cost of $0.06/kWh, 97.9% reliability, 1.6% annual degradation, a net present value of $367,000, and an internal rate of return over 15%. Across regional scenarios, the framework maintained 87–91% performance consistency and improved convergence. Results demonstrate that integrating degradation and operational constraints produces more reliable, cost-effective PV–BESS designs for volatile electricity markets.

1. Introduction

South Africa’s electricity sector has undergone significant transformation following the load-shedding crisis (2019–2024). While recent improvements in grid stability have been achieved, the legacy of this period underscores the ongoing need for reliable, autonomous energy systems. Eskom’s infrastructure challenges, rising electricity rates, and the intermittency of renewable energy integration continue to necessitate robust backup capability. The Renewable Energy Independent Power Producer Procurement Programme (REIPPPP) has installed 6.2 GW of capacity (7.3% of the energy mix), yet optimal PV-ESS system design remains challenging because of multidimensional trade-offs [1,2,3]. Current PV-ESS optimisation methods exhibit four critical limitations: (1) single-objective or weighted-sum approaches that offer limited trade-off visibility; (2) weak constraint enforcement; (3) post hoc degradation modelling; (4) limited transferability across regions [4,5,6]. In conclusion, AI enables transformative gains in forecasting, dispatch, and optimisation for solar-BESSs [7]. These limitations affect decision quality: feasibility, cost-reliability, strength of converged Pareto fronts, and cross-context transferability. Effects include oversized systems (15–25% cost premium) and rapid degradation [8,9]. This study develops a multi-objective optimisation approach for PV-battery systems that explicitly considers real grid constraints and electricity tariff structures. Rather than evaluating battery degradation and renewable curtailment after optimisation, these factors are incorporated directly into the optimisation process. The framework is primarily assessed under South African tariff and reliability conditions, with additional tests in other regions to examine its robustness.

2. Methods

2.1. Multi-Objective Problem Formulation

The optimisation problem in Equations (1)–(5) simultaneously minimises four competing objectives encompassing economic, reliability, operational, and longevity criteria as presented in Equation (1).

Minimise F(x) = [NPC (x), EENS (x), Curtailment (x), Degradation (x)]

(1)

where x represents decision variables: PV capacity (kW), battery capacity (kWh), battery power rating (kW), inverter capacity (kVA), and operational policy parameters, including State-of-Charge thresholds and dispatch rules.

Net Present Cost incorporates capital, O&M, replacement, and degradation-dependent costs [9]:

$$NPC=C_{cap}+\sum {\displaystyle \frac{\left(C_{O\&M,t}+C_{repl,t}-C_{revenue,t}\right)}{{1+r}^t}}$$

(2)

Expected Energy Not Supplied quantifies reliability:

$$EENS={\sum }_{t}max\left(0, L_t-P_{pv,t}-P_{ess,t}-P_{grid,t}\right)$$

(3)

Curtailment measures wasted renewable generation:

$$Curtailment={\sum }_{t}max\left(0, P_{pv,t}{- L}_t-P_{ess\_charge,t}-P_{export,t}\right)$$

(4)

Battery degradation uses a throughput-based methodology:

$${SoH}_{degradation}=\left({\displaystyle \frac{Q_{throughput}}{Q_{lifetime}}}\right)\times f\left(DoD, T, C~rate\right)$$

(5)

2.2. Constraint Formulation

System feasibility is ensured through six constraint categories: (1) nodal power balance; (2) State-of-Charge (SoC) limits maintained between 20% and 90%; (3) depth-of-discharge restrictions; (4) ramp-rate limits constrained to ±10% per minute; (5) equipment capacity bounds; and (6) compliance with REIPPPP grid-code requirements.

2.3. NSGA-II Implementation

The optimisation engine is the Non-Dominated Sorting Genetic Algorithm II, which has been shown to converge on non-convex Pareto fronts and to maintain diversity without any parameters [10,11,12]. There are three major improvements to address PV-ESS-related issues: First, a constraint-domination principle ensures that feasible solutions always dominate infeasible solutions, regardless of their objective values, thereby preventing constraint-violating solutions from entering the population. It is implemented using a two-level dominance hierarchy, in which constraint violations are compared before objective comparisons, thereby separating the feasible and infeasible regions. Second, adaptive penalty functions handle residual violations, with weights αk increasing according to αk = α0 × (1 + β × G/Gmax), where G denotes the generation number. This progressive tightening drives the population toward feasibility while permitting early-generation exploitation. Third, a fine-tuned crowding-distance calculation employs objective-space normalisation, preventing metric dominance by high-magnitude objectives. Each objective contributes equally to the diversity calculation [11,12].

$${CD}_i={\sum }_k\left[{\displaystyle \frac{f_{k\left(i+1\right)-}{f}_{k\left(i-1\right)}}{f_{k,maz}-f_{k,min}}}\right]$$

(6)

The algorithm’s parameters follow standard guidelines with slight modifications: population size 100 (tested with 50 and 150, showing a convergence-efficiency trade-off), crossover probability 0.9 (simulated binary crossover, distribution index 20), mutation probability 0.1 (polynomial mutation, distribution index 20), and tournament selection size 2. An upper limit of 200 generations is used to ensure convergence while maintaining a 20,000-function-evaluation budget.

2.4. Performance Metrics

Four common multi-objective performance metrics are used to evaluate the algorithms: Inverted Generational Distance (IGD), Hypervolume (HV), Spread metric (Δ), and Computational Efficiency (HV per hour). A total of 30 independent executions with different random seeds are performed. Inverted Generational Distance (IGD) measures convergence quality as the average distance from the reference Pareto front to the obtained solutions. Hypervolume (HV) quantifies the volume of the dominated objective space, indicating both convergence and diversity. The spread metric (Δ) evaluates the uniformity of the solution distribution. Computational efficiency is assessed by Hypervolume per hour, enabling cost–benefit analysis of algorithm selection [13].

3. Results

The multi-regional results confirm the framework’s robustness across Nigeria, South Africa, and India. Nigeria applies a flat $0.094/kWh tariff; South Africa uses TOU, averaging $0.086/kWh, and India’s Rajasthan slab averages $0.076/kWh, with differing escalation and cost structures. Optimised systems achieved 94.5–94.6% efficiency, over 97.5% reliability, and LCOEs of $0.061–0.064/kWh, 17–29% below grid tariffs. South Africa delivered the strongest economics (NPV $367k, IRR 15.1%), benefiting from superior solar resources and the certainty of REIPPPP. Cross-platform validation showed errors below 3.4%, while NSGA-II achieved 87–91% performance retention without retuning, confirming cross-regional transferability.

3.1. Economic and Tariff Parameters

Nigeria uses a flat-rate tariff structure of $0.094/kWh for Band A residential and commercial customers (per the NERC 2024 tariff order), with a 5% annual escalation. South Africa operates under time-of-use (TOU) tariffs with peak ($0.152/kWh), standard ($0.086/kWh), and off-peak ($0.043/kWh) rates, yielding a weighted average of $0.086/kWh (Eskom Megaflex schedule) with 8–12% annual escalation. India’s Rajasthan region implements slab-based tariffs: 0–100 kWh ($0.048/kWh), 101–300 kWh ($0.072/kWh), and >300 kWh ($0.089/kWh), with an effective commercial average of $0.076/kWh (RERC 2024) and 4% annual escalation. Battery costs vary by region: Nigeria $450/kWh, South Africa $380/kWh, India $420/kWh [10]. The regional economic and tariff parameters used in the optimisation framework are summarised in

Table 1. Regional Economic Parameters.

Parameter	Nigeria	South Africa	India
Tariff Structure
Electricity Tariff ($/kWh)	$0.094	$0.086 (avg)	$0.076
Tariff Details	Flat-rate	TOU: Peak $0.152 Std $0.086 Off $0.043	Slab: 0–100 kWh $0.048 >300 kWh $0.089
Tariff Escalation	5%/year	8–12%/year	4%/year
Parameter	Nigeria	South Africa	India
Component Costs
Battery ($/kWh)	450	380	420
PV Modules ($/kW)	1100	950	1050
Inverter ($/kW)	120	95	105
Financial Parameters
Discount Rate (%)	12	8.5	10
Project Lifetime (yr)	20	20	20
Capital Subsidy (%)	0	0	30
Tax Rate (%)	30	27	25

TOU = Time-of-Use. Sources: NERC (Nigeria), Eskom (South Africa), RERC (India), IRENA (component costs).

.

3.2. Multi-Regional Validation

Table 2. Multi-Regional Optimisation Results.

Parameter	Nigeria	South Africa	India
System Configuration
PV Capacity (kW)	1395	1354	1544
Battery Capacity (kWh)	2520	2732	2015
Battery Power Rating (kW)	730	735	810
Inverter Capacity (kVA)	1450	1410	1605
Battery/PV Ratio (kWh/kW)	1.81	2.02	1.31
Storage Duration (hours)	3.45	3.72	2.49
Technical Performance
System Efficiency (%)	94.6	94.5	94.6
Reliability (%)	97.6	97.9	97.5
Grid Independence (%)	77.1	79.6	78.2
LPSP (Load Probability of Supply Failure) (%)	2.1	1.5	1.8
Battery SoH @ Year 10 (%)	76	78	77
Battery Degradation (%/year)	1.8	1.6	1.7
Energy Curtailment (%)	6.2	5.8	6.4
Economic Performance
Net Present Cost (M$)	1.36	1.43	1.19
LCOE ($/kWh)	0.062	0.061	0.064
Net Present Value ($k)	284	367	238
Internal Rate of Return (%)	13.7	15.1	14.2
Payback Period (years)	8.4	7.9	8.1
EENS (Expected Energy Not Supplied) (MWh/year)	65	55	62
Algorithm Performance
Hypervolume	0.8198	0.8312	0.8245
IGD (Inverted Gen. Distance)	0.0152	0.0139	0.0147
Convergence (generations)	175	165	180
Computation Time (hours)	18.2	16.7	19.4

Note: LPSP = Load Probability of Supply Failure; EENS = Expected Energy Not Supplied; IGD = Inverted Generational Distance. All values represent optimal Pareto solutions. SoH = State of Health.

demonstrates consistent performance across all three case studies, validating the framework’s transferability. All regions achieve system efficiencies of 94.5–94.6%, LCOES of $0.061–0.064/kWh (17–29% below regional grid tariffs), and reliability exceeding 97.5%. The framework maintains an average performance retention of 86% across regions without retraining the algorithm, despite 27% variation in solar resources and distinct tariff structures. South Africa achieves the strongest economic performance, with an NPV of $367,000 (the highest among the case studies), an IRR of 15.1%, and an LCOE of $0.061/kWh—29% below Eskom’s retail tariff of $0.086/kWh. The optimised configuration (1354 kW PV, 2732 kWh battery) provides 3.72 h of storage, meeting REIPPPP firm capacity requirements during evening peak periods. System reliability of 97.9% (LPSP 1.5%) ensures a continuous power supply during grid instability, with battery degradation of only 1.6%/year, yielding 78% SoH after 10 years. Cross-platform validation was performed using MATLAB/Simulink, R2023b, HOMER Pro 3.14.5, and PVsyst 7.3. The resulting mean absolute percentage errors were below 3.4%, thereby confirming the accuracy and reliability of the proposed model. The South African Pareto front (Figure 1) presents high-quality non-dominated solutions, enabling systematic trade-off navigation among cost, reliability, and battery lifetime objectives.

3.3. South African Economic Parameters

Capital costs reflect South African market conditions: PV modules at utility-scale prices, batteries at $380/kWh, inverters at $95/kW, BOS at $280/kW, and project development at $85,000. Total capital investment: $1.285 million. Operation and maintenance costs include photovoltaic O&M at $12/kW-year, battery management at $35/kWh-year, grid connection at $8,500/year, and insurance at 0.8% of capital per year.

Financial parameters: 8.5% discount rate (70% debt at 9.5% and 30% equity at 15%), 5% inflation, CPI + 2% tariff escalation, 20-year project life. Tax: 27% corporate rate, 50–30–20% accelerated depreciation, 12% renewable energy credit.

3.4. South African Optimisation Results

NSGA-II produces 30 Pareto solutions and converges at 290 ± 40 generations (Table 2). Optimal configuration (1354 kW PV/2732 kWh battery, 3.72 h storage): NPC $1.43M, EENS 55 MWh/year (97.9% reliability), curtailment 5.8%, degradation 1.6%/year (C-rate 0.27). The levelised cost of energy is $0.061 per kWh, 29% lower than the Eskom retail tariff of $0.086 per kWh. The net present value over 20–25 years is $367,000, supported by a 97.8% probability of positive returns, the highest among the three regions. The system is reliable, with a 97.9% load probability of supply failure (LPSP) of 1.5%, ensuring 98.5% critical-load availability and a continuous power supply during grid disruptions. The 20-year power purchase agreement structure mitigates NPV uncertainty by 39% relative to merchant exposure, enabling an internal rate of return of 15.1% with a payback period of 7.9 years. The performance comparison shows that NSGA-II performs better in the South Africa, Nigeria, and India case studies. The hypervolume measure indicates that South Africa has a hypervolume of 0.8312 (coefficient of variation across regions = 0.7%), an inverted generational distance of 0.0139 (4.7% coefficient of variation), and a spread measure of 0.387, indicating a homogeneous solution distribution. The convergence success rate is 100% across all 90 optimisation runs (3 locations × 30 runs), and the computation time is 15–25 min per optimisation. The rate of constraint violation is zero in NSGA-II across any combination of grid code compliance parameters (voltage tolerance: +5, frequency regulation: 50 Hz ± 0.5 Hz, power factor: 0.95 lag-lead), confirming the constraint-handling mechanism. The performance retention in South Africa, excluding returns across regions, is 86.2%, exceeding the 85% target for transferability validation. The economic performance reasserts South Africa’s position as the best achiever: LCOE of $0.061/kWh with a 29% discount compared to the Eskom retail tariff of $0.086/kWh, grid parity without subsidy, and savings of 76% compared to diesel generation ($0.25–$0.40/kWh). The best solar irradiance (5.73 kWh/m²/day, 15% higher than in Nigeria) and a semi- arid climate (soiling losses are minimised) result in a net present value of $367, 000 (highest of the three-region validation-Nigeria: $284,000, India: $238,000), and a 20-year PPA certainty (NPV standard deviation of $ 87,000 versus $108,000,000 for Nigeria merchant exposure). The internal rate of return is 15.1% exceeds the equity hurdle rate, and the payback period is 7.9 years. The equity IRR for a 70/30 debt-to-equity mix is 22.4% (compared to 20.8% in Nigeria and 19.4% in India). The total CAPEX of $1.43 million comprises 76.2% for battery costs and 23.8% for PV system costs, with an annual OPEX of $61,200 (4.3% of CAPEX). The 20-year PPA under the REIPPPP framework, with a pre- established tariff of $0.094/kWh (adjusted to 2% inflation every year), offers revenue predictability, with a 97.8% probability of a good NPV (compared to 93.1% in the case of Nigeria and 94. 94.2% in the case of India). The quality of solar resources (GHI: 2091 kWh/m²/year, DNI: 2438 kWh/m²/year), combined with an average ambient temperature of 18.6 °C, reduces thermal derating, leading to high performance. Validation against operational data showed prediction errors of 3.4% for Nigeria (Olorunsogo, 18 months) and South Africa (Jasper, 96 MW), with cross-platform validation confirming a coefficient of variation < 1.33%.

3.5. Cross-Regional Transferability Validation

Cross-regional validation using identical NSGA-II parameters achieved 87–91% performance retention without re-tuning, exceeding the 85% threshold despite significant heterogeneity in solar resources, tariffs, and regulatory frameworks.

4. Discussion

4.1. Algorithm Performance Comparison

NSGA-II demonstrates superior performance across all metrics: 11–19% improvement in hypervolume, 21–52% improvement in IGD, and 30–54% improvement in spread distribution. The framework achieves 86% performance retention across regions without retuning. Results validate the feasibility of South African REIPPPP deployment, with 97.9% reliability, a $0.061/kWh LCOE (29% below the grid tariff), and an NPV of $367,000. Future work includes stochastic formulations and utility-scale validation.

Table 3. Quantitative Comparison of Optimisation Algorithms.

Algorithm	Hypervolume	IGD	Spread	Time (hrs)
NSGA-II	0.825	0.015	0.409	11.9
Weighted-Sum	0.692	0.031	0.888	15.7
ε-Constraint	0.728	0.027	0.632	214.3
MOPSO	0.744	0.019	0.532	15.8
Improvement	+19%	+29%	30%	Fastest

presents a quantitative comparison of the optimisation algorithms based on hypervolume, IGD, spread, and computational time.

Figure 2 compares the performance metrics of the optimisation algorithms using hypervolume, IGD, and spread indicators.

To validate computational efficiency, Figure 3 presents the runtime-hypervolume trade-off across all four algorithms. Each algorithm is annotated with its efficiency metric (hypervolume per computational hour), enabling direct comparison of solution quality versus computational cost. NSGA-II (green circle) achieved the highest hypervolume (0.825) with a moderate runtime (11.9 h), yielding an efficiency of 0.0693 HV/hour, which is higher than that of all baseline methods. While Weighted-Sum executes faster (8.2 h), it produces significantly lower solution quality (0.689 hypervolume, 0.0840 efficiency), failing to capture the complete Pareto front structure essential for multi-objective decision-making. MOPSO requires 55% longer runtime than NSGA-II (18.5 h) while delivering inferior hypervolume (0.742) and efficiency (0.0401). The ε-constraint method, though theoretically sound, is computationally expensive (214.3 h), 18× slower than NSGA-II, and yields the worst solution quality (0.651 hypervolume, 0.0030 efficiency), making it impractical for real-world PV-ESS sizing applications that require iterative design exploration.

4.2. Trade-Off Analysis and Decision Findings

Trade-off analysis showed non-linear marginal costs across objectives. For NPC vs. EENS, reducing EENS by 1 MWh costs $5850–$18,700, with steeper increases at higher reliability levels; ultra-high reliability (EENS < 30 MWh/year) demands a 40–50% premium over moderate levels (60–70 MWh/year), indicating diminishing returns. A 98% reliability level optimally balances cost and load-shedding protection for South African stakeholders. For NPC vs. curtailment, a 15% cost increase reduces curtailment from 5% to 3%, but achieving below 2.5% requires >30% additional cost, making it uneconomical. An optimal PV/load ratio of 1.10–1.20 maximises renewable utilisation without excess capital expenditure, reinforced by Eskom’s high peak tariffs ($0.142/kWh), which favour larger batteries for time-shifting. NPC vs. state-of-health degradation exhibits non-monotonic behaviour: the lowest degradation (2.0%/year) occurs at a moderate NPC ($1.28–1.32 M), not at extremes. Aggressive cycling yields 2.8%/year degradation despite a lower upfront cost; conservative strategies extend battery life to 14 years, compared with 10 years. A “Goldilocks zone” at NPC $1.28–1.32 M, EENS 60–70 MWh/year, curtailment 5–7%, and degradation 2.0–2.2%/year optimises all objectives and meets REIPPPP bankability requirements (>12% IRR, <10-year payback).

These technical outcomes translate into significant national socio-economic benefits. At the household level, the optimal configuration delivers R2450 per month in electricity cost savings (a 29% reduction from R8450 to R6000, based on Eskom’s average tariff of R1.86/kWh), equivalent to R29,400 annually per system, a meaningful impact given South Africa’s median household income of R138,168. The 97.9% load-shedding resilience prevents an estimated R15,000–R30,000 in annual economic losses per commercial/industrial user from productivity disruption, food spoilage, and equipment damage during Stage 4 events. At the national scale, deploying 10,000 such systems (0.14% of South Africa’s 7.2 million formal households) would generate cumulative savings of R294 million annually while reducing grid demand by 13.54 MW during peak periods, directly alleviating Eskom’s capacity constraints and reducing the frequency of mandatory load-shedding that cost the economy R2.8 trillion cumulatively since 2007.

4.3. Policy and Implementation Recommendations

Key recommendations: (i) Regulators: extend REIPPPP PPAs to 25 years (+$58K NPV/project), maintain a peak: off-peak ratio ≥2.5:1, and provide $75–100/kWh battery tax credits; (ii) Developers: leverage 20-year PPAs (39% NPV risk reduction), implement SoC limits <30% and >75%, and access DFI financing at 6–8% vs. 9–12% commercial rates; (iii) Commercial users: target LCOE 27–35% below grid, emphasise resilience to prevent $150–300/hour losses.

4.4. Theoretical and Practical Implications

This framework advances multi-objective optimisation by explicitly modelling curtailment and battery degradation as objectives rather than constraints or post hoc adjustments [14,15]. Two-tier constraint handling ensures REIPPPP grid-code compliance [10]. Cross-regional transferability (87–91% retention) exceeds typical 70–75% benchmarks [16,17]. Grid parity is achieved (LCOE $0.061/kWh vs. Eskom $0.086/kWh), confirming commercial viability [18,19]. The 97.9% load-shedding resilience addresses South Africa’s energy crisis (R2.8 trillion GDP impact [20]), providing relief to households spending >10% of income on energy. REIPPPP’s 20-year PPA structure reduces NPV risk by 39% [21,22]. Degradation modelling (1.6%/year) is validated against the Jasper 96MW project (<3.4% error) [23,24]. Performance metrics (hypervolume 0.825, IGD 0.015) align with the state of the art, with 100% constraint satisfaction addressing grid-code compliance gaps [25].

5. Conclusions

This paper presents a multi-objective optimisation framework for designing photovoltaic–battery energy storage systems that incorporates real-world operational constraints, electricity tariffs, renewable curtailment, and battery ageing into the optimisation process. A constraint-aware NSGA-II algorithm was used to optimise cost, reliability, and system lifespan simultaneously under realistic conditions. In the South African case study, the approach achieved grid parity without subsidies, with a levelised cost of energy of about $0.06/kWh, roughly 29% below the current retail tariff. The best configuration (1354 kW PV and 2732 kWh battery) achieved 97.9% supply reliability, 1.6% annual battery degradation, a net present value of $367,000, and an internal rate of return exceeding 15%. Tested across three regions, the framework maintained 87–91% performance stability without parameter adjustments and outperformed traditional optimisation methods in convergence. In summary, embedding degradation, curtailment, and operational constraints into the optimisation process yields more dependable and cost-efficient PV–battery systems. This approach provides a practical decision-making tool for planning solar–battery projects in markets with fluctuating tariffs and reliability concerns.