Techno-Economic Photovoltaic-Battery Energy Storage System Microgrids with Diesel Backup Generator: A Case Study in Industrial Loads in Germany Comparing Load-Following and Cycle-Charging Control

Abstract

This paper compares two common dispatch policies—Load-Following (LF) and Cycle-Charging (CC)—for a photovoltaic Battery Energy Storage System (PV–BESS) microgrid (MG) with a 12 kW diesel generator, using a full-year of real 15 min PV and load data from an industrial use case in Germany. A forward time-step simulation enforces the battery State-of-Energy (SoE) window (total basis [20, 100] %, DoD = 80%) and computes curtailment, generator use, and unmet energy. Feasible designs satisfy a Loss of Power Supply Probability (LPSP) ≤ 0.03. Economic evaluation follows an Equivalent Annual Cost (EUAC) model with PV and BESS Capital Expenditure/Operation and Maintenance (CAPEX/O&M) (cycle life dependent on DoD and 15-year calendar life), generator costs, and fuel via SFC and diesel price. A value of lost load (VOLL) can be applied to unserved energy, with an optional curtailment penalty. Across the design space, a clear cost valley appears toward moderate storage and modest PV, with the baseline optimum at ≈56 kWp PV and 200 kWh BESS (DoD = 80%). Both policies meet the reliability target (in our runs LPSP ≈ 0), and their SoE trajectories are nearly identical; CC only lifts the SoE slightly after generator-ON events by using headroom to charge, while LF supplies just the residual deficit. Sensitivity analyses show that the optimum is most affected by diesel price and discount rate, with smaller shifts for ±10% changes in SFC. The study provides a transparent, reproducible workflow—grounded in real data—for controller selection and capacity planning.

1. Introduction

1.1. Background and Motivation

The rapid uptake of renewables in industry calls for time-resolved planning of PV–BESS microgrids that can meet reliability targets at minimum cost. The asynchronous nature of solar resource and industrial demand makes sizing and dispatch non-trivial under realistic converter and battery limits [1,2,3,4]. In this work, we analyze an industrial, inelastic load with real 15 min data and include a 12 kW diesel generator in the dispatch to capture realistic backup behavior.

1.2. Reliability and Cost Metrics

The indicator Loss of Power Supply Probability (LPSP) is adopted in our paper as the adequacy metric—defined here as the energy fraction of demand not served over the study horizon—and we evaluate economics via Equivalent Annual Cost (EUAC) with DoD-dependent cycle life and a calendar cap. Feasible designs satisfy LPSP ≤ 0.03. This follows the mainstream view that reliability must be co-optimized with cost rather than appended post hoc [1,3,5,6,7,8,9].

In many microgrid sizing studies, LPSP is used as a planning-level adequacy indicator, and admissible values are often chosen in the range of 1–5%, balancing investment cost and reliability for remote or commercial/industrial applications [1,2,3,4,5,6,7,8]. In line with this practice, we adopt a target of LPSP ≤ 0.03 as the baseline constraint for the industrial microgrid considered here.

From an engineering perspective, LPSP = 0.03 corresponds to a situation in which, in the worst case, up to 3% of the annual energy demand could remain unmet. In the optimal configurations identified in this study, the simulated LPSP is in fact below this limit, indicating that reliability is not a binding constraint at the optimum and that the system operates more reliably than the imposed threshold. Qualitatively, tightening the constraint (e.g., to LPSP ≤ 0.01) would require somewhat larger PV and/or battery capacities, leading to higher equivalent uniform annual cost (EUAC), while relaxing it (e.g., to LPSP ≤ 0.05) would allow slightly smaller capacities and lower EUAC at the expense of more frequent or longer energy deficits. This trade-off is consistent with the behavior observed in other LPSP-based microgrid sizing studies [1,2,3,4,5,6,7,8].

While LPSP is convenient and widely adopted as a scalar reliability indicator, it also has several limitations that should be kept in mind when interpreting the results. First, LPSP is an energy-based metric: it aggregates all unserved energy over the year and does not directly reflect the number of interruption events, their individual duration, or their temporal clustering. Two designs with the same annual LPSP may therefore exhibit very different outage patterns. Second, LPSP does not account for power quality aspects such as voltage and frequency deviations, which are assumed to be managed by lower-level control and protection schemes. Third, the metric does not distinguish between unserved energy during low-critical hours and unserved energy during highly critical industrial processes; such distinctions require supplementary risk or resilience indices.

For these reasons, LPSP is used here primarily as a first-order adequacy constraint for long-term capacity planning. To provide a richer picture of operational performance, we complement it with state-of-energy (SoE) trajectories, seasonal SoE statistics, and generator-usage indicators, which together offer additional insight into how the LF and CC strategies behave under different seasonal stress conditions.

Dispatch Policies (Definitions)

• Load-Following (LF): A deficit-triggered policy in which the diesel generator (GEN) supplies only the residual demand that cannot be met by PV and the admissible battery window. When the usable battery energy is exhausted (i.e., E_bat → E_min), the GEN turns ON and does not charge the battery; it simply follows the residual load.
• Cycle-Charging (CC): A deficit-triggered policy in which, once the GEN is ON, it operates at (near) rated power to both cover the residual demand and charge the battery with any headroom up to Emax.

1.3. Related Work (Thematic Synthesis)

• Dispatch strategies and controller selection. Comparative frameworks for microgrid dispatch (Load-Following, Cycle-Charging, etc.) emphasize selection based on energy balance constraints and operational objectives; we follow this stance when contrasting LF vs. CC on real data [10,11].
• Charge/discharge management and SoC windows. Reviews synthesize Maximum Power Point Tracking (MPPT)/DC-DC and inverter modeling with SoC window enforcement—elements we explicitly implement [12]. Hourly/quarter-hour strategies with set-points mitigate chattering and bound deep cycling [6,13].
• Techno-economic sizing and sensitivity. Full LCC/EUAC accounting with sensitivity to fuel price and discount rate motivates our parametric sweeps; typical results reveal a cost valley toward moderate storage with adequate PV [7,8,14,15,16].
• Context and microgrid practice. Remote/mini-grid studies clarify reliability requirements and generator interaction; our dataset-driven approach mirrors this evidence [9,15,16].
• Optimization practice. Meta-heuristics (e.g., GA/PSO) are widely used for capacity search, but transparent parametric sweeps on real time series remain highly interpretable and reproducible. Evidence across studies shows that battery and fuel cost assumptions are the dominant drivers of EUAC variance—hence our emphasis on battery cost/DoD/cycle life sensitivities [4,17,18,19]. Recent reviews also recommend embedding reliability directly in the objective, e.g., by including the deficit probability (LPSP) during optimization—precisely the stance adopted in our paper [17].
• Policy and sector coupling. Incorporating VOLL/curtailment penalties or carbon pricing moves the cost–reliability frontier and can justify larger PV/storage; flexible sinks (PtH, EV, H₂) valorize surplus at fixed LPSP [8,20]. Because our summer curtailment is non-trivial, adding a PEM electrolyzer—well suited to fast RES ramps—would likely shift the EUAC minimum toward higher PV while keeping LPSP unchanged, in line with the synthesis of RES-to-H₂ integration challenges in [21].
• Microgrid/DC sharing. Sharing/aggregation in DC microgrids reduces both LPSP and cost compared to isolated systems [22].
• Role separation PV vs. BESS (application evidence). In a stand-alone PV EV-charging use case, more PV primarily depresses daytime LPSP, while larger storage suppresses nighttime LPSP; this separation echoes our A/B/C/D case analysis and supports high-PV/moderate-storage pairings at lowest annualized cost [23].

1.4. Contributions and Novelty of This Work

In summary, this work contributes to the existing literature on Load-Following (LF) and Cycle-Charging (CC) in several ways.

First, it uses a full-year industrial load profile measured at 15 min resolution, combined with PV generation derived from PVGIS TMY data for Karlsruhe, instead of synthetic or hourly averaged signals. This allows us to capture short-term variability that is often smoothed out in previous LF/CC studies.

Second, the techno-economic objective is formulated as an equivalent uniform annual cost (EUAC) that explicitly embeds depth-of-discharge (DoD)-dependent battery cycle life together with a calendar life cap, so that operational stress and long-term replacement cost are consistently linked.

Third, the microgrid sizing problem is solved through a transparent parametric scan of PV and battery capacities under an explicit reliability constraint (LPSP ≤ 0.03), providing complete feasibility and cost maps rather than only a single optimal point obtained by a black-box optimizer.

Finally, the two dispatch strategies are evaluated not only in terms of cost and LPSP, but also through detailed state-of-energy (SoE) trajectories and generator use indicators, which help explain why their optimal designs converge in this industrial case and under which operating conditions their differences become relevant for practitioners. In addition, a systematic sensitivity analysis on diesel price, discount rate, and specific fuel consumption clarifies which economic parameters most strongly shift the optimal design.

1.5. Paper Organization

Section 2 details the datasets and modeling framework: meteorological inputs and PV modeling, the industrial load profile, the DC-coupled PV–BESS–diesel architecture, the dispatch policies (Load-Following and Cycle-Charging), and the techno-economic setup (objective, constraints, and parameterization). Section 3 presents and discusses the results: LPSP feasibility maps and EUAC surfaces, identification of the cost-optimal design, SoE analyses (annual and seasonal), the LF vs. CC comparison, robustness checks with VOLL/curtailment penalties, sensitivity (tornado) analyses, and the generator-sizing rationale. Section 4 concludes with the main findings, practical implications, limitations, and directions for future work (e.g., multi-year weather, stochastic uncertainty, and surplus valorization via flexible loads).

2. Materials and Methods

2.1. Microgrid Context and Control Scope

Figure 1 sketches a common classification of microgrids by power type, supervisory control, operation mode, phase, and application. The system studied here is a DC-coupled PV–BESS microgrid supplying an industrial, inelastic AC load with an optional 12 kW diesel generator. In the terms of the figure, it is islanded, three-phase, industrial/commercial, and operated by a centralized energy management system (EMS) [24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62].

Figure 2 summarizes the hierarchical control stack as follows [63]:

• Primary control (fast, local) governs current/voltage and droop characteristics of the grid-forming inverter and generator.
• Secondary control restores frequency/voltage and handles grid synchronization.
• Tertiary control (EMS) optimizes power flows over minutes–hours, i.e., dispatch and scheduling.

This paper is squarely in the tertiary layer: it compares between Load-Following (LF) and Cycle-Charging (CC) using real 15 min data and sizes PV and storage under the LPSP ≤ 0.03 reliability constraint using an EUAC objective. Primary/secondary dynamics, protection, and power quality issues are assumed to be satisfied by the grid-forming converter and generator governors; the model is quasi-steady-state and enforces only the battery SoE window and converter efficiencies [64,65,66].

Beyond tertiary energy management, voltage regulation in active distribution networks benefits from local Distributed Energy Resource (DER) controllers that explicitly limit loop interactions among devices. Recent work has shown that carefully tuned local laws can stabilize the voltage profile and avoid adverse controller coupling without relying on high-bandwidth communications. This supports our modeling choice to abstract primary/secondary voltage control as ‘satisfied by design’, letting the paper focus on tertiary LF/CC dispatch and sizing [67,68].

Recent advances in distributed cooperative anti-windup show that DER controllers can preserve voltage quality even under reactive power saturation by coordinating local action to prevent integrator windup and oscillations. Such results justify our assumption that primary/secondary controls maintain admissible voltages while our analysis concentrates on tertiary scheduling and cost–reliability trade-offs [69].

Sensitivity-based Low-Voltage (LV) network models with DERs provide linearized relationships suitable for fast voltage assessment and control design. This stream of work underpins our assumption that primary/secondary layers can enforce voltage limits with modest computational burden while we evaluate year-long LF/CC dispatch on 15 min data [70].

2.2. Analyzing the System’s Flowchart

The microgrid adopts a DC-coupled architecture: the PV array and the battery share a common DC bus, the inelastic AC load and the optional diesel generator (GEN) are on the AC bus. A bidirectional DC/AC inverter–charger exchanges power between the buses. Meteorological inputs—GHI (W/m²), ambient temperature (°C), and wind speed (m/s)—drive the PV model and determine the instantaneous PV output. The described configuration is schematically illustrated in Figure 3.

Solar irradiance is converted to DC power. A DC/DC MPPT stage (buck/boost) regulates the operating point at maximum power and adapts the array voltage to the DC bus; losses in this charging path are captured by the charge controller efficiency ${\eta }_{cc}$. The resulting PV DC power $P_{PV,dc}$ can either (i) serve the AC load via the inverter or (ii) charge the battery when surplus exists.

We adopt a 12 kW (AC) diesel genset because the observed peak demand (~11.5 kW) must be fully covered during low-irradiance periods so that residual loads are always served. Beyond simply matching the measured peak, this rating also provides a modest short-term dynamic margin: in practice, industrial loads exhibit inrush currents and step changes when motors, compressors, or other equipment start, so a small headroom (on the order of 5–10%) above the nominal peak is desirable to avoid excessive frequency dips and nuisance trips. Choosing a significantly smaller genset (e.g., 8–10 kW) would risk under-dimensioning under such transient conditions and could compromise power quality, whereas a much larger unit would unnecessarily increase CAPEX and fixed O&M and would tend to operate at very low-load factors during backup periods, reducing fuel efficiency and potentially accelerating wear. The 12 kW rating therefore represents a balanced and defensible baseline choice for the studied load profile, offering sufficient dynamic robustness while keeping annualized generator costs within a realistic range.

The battery is connected to the DC bus (via a bidirectional DC/DC interface). Its State-of-Energy is defined on a total basis within the admissible window $\left[E_{min},E_{max}\right]$, with $E_{max}=B_{nec}$ and $E_{min}=B_{nec}\left(1-DoD\right)$. Discharge losses are modeled by the one-way battery efficiency ${\eta }_{bat}$, while charging losses (from PV or generator) are modeled by ${\eta }_{cc}$. When PV surplus exists, the controller stores energy up to $E_{max}$; if PV is insufficient, the battery discharges down to $E_{min}$ to support the load.

A bidirectional DC/AC inverter–charger supplies regulated AC power to the load with efficiency ${\eta }_{inv}$ and enables energy exchange between the DC resources (PV/battery) and the AC bus. The diesel generator connects to the AC bus to provide backup power. Under Load-Following (LF), the generator, when ON, covers only the residual deficit; under Cycle-Charging (CC), if headroom remains while the generator is ON, the inverter–charger routes power to charge the battery (GEN → BESS).

While our case study targets an industrial load, the modeling choices (EUAC objective, reliability constraint, quarter-hour resolution) are consistent with techno-economic assessments of PV–BESS systems for commercial/telecom facilities [71], indicating method transferability beyond the present profile.

This choice is also consistent with evidence that storage-assisted dispatch helps avoid inefficient low-load operation of diesel units, improving fuel economics and maintenance outcomes [72].

2.3. Energy Production Estimation

We generate a time-resolved PV output series using site-specific meteorological inputs—global horizontal irradiance (GHI), ambient temperature, and wind speed—together with the PV system’s technical parameters. At each simulation step Δt = 15 min, the GHI is transposed to the plane of array and combined with a standard cell temperature model; the resulting DC power is then mapped to AC using the inverter efficiency curve. This workflow yields a physically consistent PV production trace that is natively aligned with the simulation timeline.

The load is treated as inelastic and largely location-agnostic, reflecting industrial demand driven by operations rather than weather. We use the “LG-07” profile from an open dataset [73] of small-to-medium industrial facilities in Southern/Western Germany; Baden-Württemberg is among the most represented regions. All time series are harmonized to the same time zone and 15 min resolution to avoid phase errors, including those from daylight saving transitions. While the meteorological data and the load need not originate from the exact same site, we ensure comparable climates and proper time alignment.

For a concrete and reproducible weather source, we select Karlsruhe, Germany (49.0094° N, 8.4044° E). Hourly PVGIS TMY data (GHI, temperature, wind) are retrieved for Karlsruhe and resampled to 15 min. Karlsruhe offers a representative Central European climate (annual irradiance ≈ 1.1–1.2 MWh/m²) that stresses both winter deficits and summer surpluses. This configuration—PV from PVGIS-TMY at Karlsruhe and the LG-07 load, both at 15 min resolution—underpins all simulations and metrics reported in the paper, including SoE trajectories and LPSP.

2.4. Analysis of Load Profiles

The load profile represents the electrical demand of a building or system over time, typically constructed from historical consumption data and power measurements. Expressed in kilowatts (kWs) across defined time intervals, it provides a detailed view of how electricity is used throughout the day, week, or year.

Such profiles are essential for understanding and forecasting demand patterns, optimizing energy use, managing operational costs, evaluating the performance of energy systems, and identifying opportunities for efficiency improvements. In the context of Distributed Energy Resource (DER) planning—such as photovoltaic battery systems (PVBATs)—a detailed load profile is critical for accurate sizing and system optimization.

This study utilizes real-world load profile data sourced from Braeuer (2020) [73], available via Zenodo. The dataset contains electricity demand time series from 50 small- and medium-sized industrial enterprises (SMEs) in Germany, sampled every 15 min over one calendar year. Each data point corresponds to the average active power demand (in kW) during that interval.

The load profile selected for simulation in this work is “LG-07”, representing one of the 50 SME facilities. It forms a time series of 35,040 inelastic demand points over the year. This profile was also previously evaluated in related techno-economic research [74].

Figure 4 illustrates the annual consumption pattern of LG-07, which totals approximately 56.594 MWh, with an average active power demand of 6.641 kW. The dataset captures both peak and off-peak behaviors, as well as seasonal variations, making it suitable for reliable DER system design and performance evaluation.

The profile reflects daily human activity, starting the day with a consistent baseload of around 2 kW from 02:00 to 07:00 in the morning throughout the year, except for one day in mid-August. The rest of the daily consumption maintains a plateau of approximately 8 kW. It also exhibits seasonal variations, with peak consumption rising to approximately 11.5 kW during the summer months, in contrast to the rest of the year.

Table 1. Statistic summary of the LG-07 load profile with mean, standard deviation (STD), min, and max expressed in kW.

Count	Mean	STD	Min	25%	50%	75%	Max
35,040	6.461	2.547	0.000	4.688	7.313	8.275	11.550

lists the summary statistics of this load profile.

Figure 5, on the other hand, shows the power demand distribution among the daily hours over a year. Electricity consumption during daytime, from 10 pm to 23:59 am, shows a low level of variability, whereas during the early morning hours, from 1 pm to 5 pm, demand varies significantly despite the relatively constant medians.

Figure 6 shows the electricity demand of a typical small-to-medium industrial facility in Germany (dataset “LG-07”, Braeuer 2020 [73]), at 15 min resolution. For visualization, we aggregate the 15 min samples into hour-of-day bins along the y-axis and day-of-year along the x-axis; the color scale reports load power (kW). The plot reveals clear daily cycles and a weak seasonality, with a stable baseload during working hours. This realistic, inelastic profile is the reference demand used in all sizing and reliability simulations.

2.5. System’s Operation Algorithm

The simulation model of the microgrid is governed by a time-step energy balance that classifies each hourly condition into one of four operational cases: A, B, C, or D. These cases represent the dynamic interaction between PV generation, battery State-of-Energy (SOE), and load demand, and they are used to calculate energy flows, battery updates, and supply deficits. The complete dispatch algorithm and the transitions between the four cases and the two dispatch policies (LF and CC) are summarized in Figure 7.

The symbols are as follows (per time-step t of length Δt [h]):

• ${\mathrm{P}}_{\mathrm{P}\mathrm{V}}\left(\mathrm{t}\right)$: PV power on the DC bus after MPPT [kW].
• ${\mathrm{P}}_{\mathrm{L}\mathrm{o}\mathrm{a}\mathrm{d}}\left(\mathrm{t}\right)$: AC load power [kW].
• ${\mathsf{\eta }}_{\mathrm{c}\mathrm{c}}$: PV/GEN → battery charge efficiency (one-way).
• ${\mathsf{\eta }}_{\mathrm{b}\mathrm{a}\mathrm{t}}$: battery discharge efficiency (one-way).
• ${\mathsf{\eta }}_{\mathrm{i}\mathrm{n}\mathrm{v}}$: DC/AC inverter efficiency (one-way).
• ${\mathrm{B}}_{\mathrm{n}\mathrm{e}\mathrm{c}}$: nominal battery energy [kWh].
• DoD∈(0,1]: admissible depth-of-discharge.
• ${\mathrm{E}}_{\mathrm{m}\mathrm{i}\mathrm{n}}={\mathrm{B}}_{\mathrm{n}\mathrm{e}\mathrm{c}}·\left(1-\mathrm{D}\mathrm{o}\mathrm{D}\right),{\mathrm{E}}_{\mathrm{m}\mathrm{a}\mathrm{x}}={\mathrm{B}}_{\mathrm{n}\mathrm{e}\mathrm{c}}$.
• ${\mathrm{E}}_{\mathrm{b}\mathrm{a}\mathrm{t}}$: battery energy state [kWh].
• ${\mathrm{P}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{m}\mathrm{a}\mathrm{x}}$: diesel generator rated power [kW] (12 kW).

Derived AC/energy quantities

$${\mathrm{E}}_{\mathrm{s}\mathrm{u}\mathrm{r}}\left(\mathrm{t}\right)=\mathrm{m}\mathrm{a}\mathrm{x}(0,\left({\mathrm{P}}_{\mathrm{P}\mathrm{V}}\left(\mathrm{t}\right)-{\displaystyle \frac{{\mathrm{P}}_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\left(\mathrm{t}\right)}{{\mathsf{\eta }}_{\mathrm{i}\mathrm{n}\mathrm{v}}}}\right)\mathsf{\Delta }\mathrm{t})$$

$${\mathrm{E}}_{\mathrm{d}\mathrm{e}\mathrm{f}}\left(\mathrm{t}\right)=\mathrm{m}\mathrm{a}\mathrm{x}(0,\left({\displaystyle \frac{{\mathrm{P}}_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\left(\mathrm{t}\right)}{{\mathsf{\eta }}_{\mathrm{i}\mathrm{n}\mathrm{v}}}}-{\mathrm{P}}_{\mathrm{P}\mathrm{V}}\left(\mathrm{t}\right)\right)\mathsf{\Delta }\mathrm{t})$$

$$\mathrm{R}\left(\mathrm{t}\right)={\mathrm{E}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{E}}_{\mathrm{b}\mathrm{a}\mathrm{t}}\left(\mathrm{t}\right) (\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{m} \mathrm{i}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{b}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{y})$$

The SoE on a total basis:

$$\mathrm{S}\mathrm{o}\mathrm{E}(\mathrm{t})=100{\displaystyle \frac{{\mathrm{E}}_{\mathrm{b}\mathrm{a}\mathrm{t}}\left(\mathrm{t}\right)}{{\mathrm{B}}_{\mathrm{n}\mathrm{e}\mathrm{c}}}}\%$$

Case A—Surplus, battery full (curtailment)

Condition

$${\mathrm{E}}_{\mathrm{s}\mathrm{u}\mathrm{r}}\left(\mathrm{t}\right)>0,\mathrm{R}\left(\mathrm{t}\right)=0$$

Updates

$${\mathrm{E}}_{\mathrm{w}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{e}}\left(\mathrm{t}\right)={\mathrm{E}}_{\mathrm{s}\mathrm{u}\mathrm{r}}\left(\mathrm{t}\right),$$

$${\mathrm{E}}_{\mathrm{b}\mathrm{a}\mathrm{t}}\left(\mathrm{t}+1\right)={\mathrm{E}}_{\mathrm{m}\mathrm{a}\mathrm{x}}.$$

Case B—Surplus, battery has room (store, possibly with partial curtailment)

Condition

$${\mathrm{E}}_{\mathrm{s}\mathrm{u}\mathrm{r}}\left(\mathrm{t}\right)>0,\mathrm{R}\left(\mathrm{t}\right)>0.$$

Updates

$${\mathrm{E}}_{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d}}\left(\mathrm{t}\right)=\mathrm{m}\mathrm{i}\mathrm{n}({\mathsf{\eta }}_{\mathrm{c}\mathrm{c}}{\mathrm{E}}_{\mathrm{s}\mathrm{u}\mathrm{r}}\left(\mathrm{t}\right),\mathrm{R}\left(\mathrm{t}\right)),$$

$${\mathrm{E}}_{\mathrm{w}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{e}}\left(\mathrm{t}\right)={\mathrm{E}}_{\mathrm{s}\mathrm{u}\mathrm{r}}\left(\mathrm{t}\right)-{\displaystyle \frac{{\mathrm{P}}_{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d}}\left(\mathrm{t}\right)}{{\mathsf{\eta }}_{\mathrm{c}\mathrm{c}}}}(\ge 0),$$

$${\mathrm{E}}_{\mathrm{b}\mathrm{a}\mathrm{t}}\left(\mathrm{t}+1\right)={\mathrm{E}}_{\mathrm{b}\mathrm{a}\mathrm{t}}\left(\mathrm{t}\right)+{\mathrm{E}}_{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d}}\left(\mathrm{t}\right).$$

Case C—Deficit, battery can supply fully (no GEN)

Condition

$${\mathrm{E}}_{\mathrm{d}\mathrm{e}\mathrm{f}}\left(\mathrm{t}\right)\le {\mathsf{\eta }}_{\mathrm{b}\mathrm{a}\mathrm{t}}({\mathrm{E}}_{\mathrm{b}\mathrm{a}\mathrm{t}}\left(\mathrm{t}\right)-{\mathrm{E}}_{\mathrm{m}\mathrm{i}\mathrm{n}})$$

Updates

$${\mathrm{E}}_{\mathrm{b}\mathrm{a}\mathrm{t}}\left(\mathrm{t}+1\right)={\mathrm{E}}_{\mathrm{b}\mathrm{a}\mathrm{t}}\left(\mathrm{t}\right)-{\displaystyle \frac{{\mathrm{E}}_{\mathrm{d}\mathrm{e}\mathrm{f}}\left(\mathrm{t}\right)}{{\mathsf{\eta }}_{\mathrm{b}\mathrm{a}\mathrm{t}}}},$$

$${\mathrm{E}}_{\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{r}\mathrm{t}}\left(\mathrm{t}\right)=0,$$

$${\mathrm{P}}_{\mathrm{g}\mathrm{e}\mathrm{n}}=0$$

Case D—Deficit exceeds usable battery (battery hits $E_{min}$)

Define the residual after exhausting the usable window:

$${\mathrm{E}}_{\mathrm{r}\mathrm{e}\mathrm{s}}\left(\mathrm{t}\right)={\mathrm{E}}_{\mathrm{d}\mathrm{e}\mathrm{f}}\left(\mathrm{t}\right)-{\mathsf{\eta }}_{\mathrm{b}\mathrm{a}\mathrm{t}}\left({\mathrm{E}}_{\mathrm{b}\mathrm{a}\mathrm{t}}\left(\mathrm{t}\right)-{\mathrm{E}}_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)\left(>0\right).$$

Set: ${\mathrm{E}}_{\mathrm{b}\mathrm{a}\mathrm{t}}(\mathrm{t}$⁺)$={\mathrm{E}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ (battery at the floor before using GEN).

(i) Load-Following (LF)

The generator covers only the residual load: no charging.

$${\mathrm{E}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{a}\mathrm{v}\mathrm{a}\mathrm{i}\mathrm{l}}\left(\mathrm{t}\right)={\mathrm{P}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{m}\mathrm{a}\mathrm{x}}\mathsf{\Delta }\mathrm{t}$$

$${\mathrm{E}}_{\mathrm{g}\mathrm{e}\mathrm{n},\to \mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\left(\mathrm{t}\right)=\min \left({\mathrm{E}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{a}\mathrm{v}\mathrm{a}\mathrm{i}\mathrm{l}}\left(\mathrm{t}\right),{\mathrm{E}}_{\mathrm{r}\mathrm{e}\mathrm{s}}\left(\mathrm{t}\right)\right)$$

$${\mathrm{E}}_{\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{r}\mathrm{t}}\left(\mathrm{t}\right)={\mathrm{E}}_{\mathrm{r}\mathrm{e}\mathrm{s}}\left(\mathrm{t}\right)-{\mathrm{E}}_{\mathrm{g}\mathrm{e}\mathrm{n},\to \mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\left(\mathrm{t}\right)(\ge 0)$$

$${\mathrm{E}}_{\mathrm{b}\mathrm{a}\mathrm{t}}\left(\mathrm{t}+1\right)={\mathrm{E}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$$

(ii) Cycle-Charging (CC)

The generator runs (typically at rated) to serve the residual and use headroom to charge the battery.

$${\mathrm{E}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{a}\mathrm{v}\mathrm{a}\mathrm{i}\mathrm{l}}\left(\mathrm{t}\right)={\mathrm{P}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{m}\mathrm{a}\mathrm{x}}\mathsf{\Delta }\mathrm{t}$$

$${\mathrm{E}}_{\mathrm{g}\mathrm{e}\mathrm{n},\to \mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\left(\mathrm{t}\right)=\min \left({\mathrm{E}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{a}\mathrm{v}\mathrm{a}\mathrm{i}\mathrm{l}}\left(\mathrm{t}\right),{\mathrm{E}}_{\mathrm{r}\mathrm{e}\mathrm{s}}\left(\mathrm{t}\right)\right)$$

$${\mathrm{E}}_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}\left(\mathrm{t}\right)={\mathrm{E}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{a}\mathrm{v}\mathrm{a}\mathrm{i}\mathrm{l}}\left(\mathrm{t}\right)-{\mathrm{E}}_{\mathrm{g}\mathrm{e}\mathrm{n},\to \mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\left(\mathrm{t}\right)(\ge 0)$$

$${\mathrm{E}}_{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{e},\mathrm{G}\mathrm{E}\mathrm{N}}\left(\mathrm{t}\right)=\mathrm{m}\mathrm{i}\mathrm{n}({\mathsf{\eta }}_{\mathrm{c}\mathrm{c}}{\mathrm{E}}_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}\left(\mathrm{t}\right),{\mathrm{E}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{E}}_{\mathrm{m}\mathrm{i}\mathrm{n}})$$

$${\mathrm{E}}_{\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{r}\mathrm{t}}\left(\mathrm{t}\right)={\mathrm{E}}_{\mathrm{r}\mathrm{e}\mathrm{s}}\left(\mathrm{t}\right)-{\mathrm{E}}_{\mathrm{g}\mathrm{e}\mathrm{n},\to \mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\left(\mathrm{t}\right)(\ge 0)$$

$${\mathrm{E}}_{\mathrm{b}\mathrm{a}\mathrm{t}}\left(\mathrm{t}+1\right)={\mathrm{E}}_{\mathrm{m}\mathrm{i}\mathrm{n}}+{\mathrm{E}}_{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{e},\mathrm{G}\mathrm{E}\mathrm{N}}\left(\mathrm{t}\right)$$

Adequacy metric LPSP [75,76,77,78,79]:

$$LPSP= {\displaystyle \frac{{\sum }_{i=1}^T{E}_{shortage}}{{\sum }_{i=1}^T{E}_{demand}}}$$

• $E_{shortage}$: Energy demand not met by the system.
• $E_{demand}$: Total energy demand.

2.6. Model Validation and Parameter Benchmarking

To ensure that the adopted techno-economic model remains consistent with practical hybrid PV–BESS–diesel implementations, the main sub-models and parameter choices were benchmarked against typical values reported for commercial and industrial systems.

For the PV subsystem, the specific annual yield obtained from the PVGIS-based simulations for Karlsruhe is approximately 1100 kWh/kWp·yr. This range is consistent with typical performance of rooftop and small industrial PV installations in Southern Germany and confirms that the combination of TMY irradiance, transposition, and temperature modeling does not overestimate the available solar resource.

Regarding the conversion and storage chain, the adopted inverter efficiency (η_inv ≈ 0.95) and charge controller efficiency (η_cc ≈ 0.98) represent system-level values that are compatible with contemporary commercial hardware for DC-coupled PV–BESS systems. The one-way battery discharge efficiency (η_bat ≈ 0.85) is also representative of stationary Li-ion battery energy storage systems once inverter and auxiliary losses are taken into account. Battery aging is captured at an aggregated level through a depth-of-discharge (DoD)-dependent cycle life function combined with a 15-year calendar life cap, which is appropriate for long-term techno-economic planning. More detailed electro-thermal or data-driven degradation models, such as Random Forest-enhanced electro-thermal aging approaches for Li-ion batteries, can be integrated in future work for applications where intra-cycle dynamics and temperature gradients play a dominant role.

For the diesel generator, the nominal rating of 12 kW(ac) and the specific fuel consumption of 0.28 L/kWh are aligned with manufacturer data and published performance ranges for small industrial gensets in the 10–20 kW class. The baseline diesel price of EUR 1.20/L reflects recent fuel cost levels in European commercial and industrial contexts. As discussed later in the sensitivity analysis, this parameter is deliberately varied over a wide band to capture the impact of fuel price volatility on the optimal design.

Finally, recent research on fast-charging strategies based on advanced Li-ion degradation models, risk-averse stochastic operation of ship microgrids under uncertain weather routing, and add-on tools for improving unit commitment with enhanced renewable and reserve predictions provide valuable benchmarks for more sophisticated operational models. In contrast, the present work deliberately focuses on a transparent, planning-oriented representation of the PV–BESS–diesel microgrid with two simple dispatch rules (LF and CC), using real 15 min data and an explicit reliability constraint. These choices keep the framework tractable and make the sizing results easier to interpret and reproduce.

3. Results and Discussion

3.1. Interpretation of the SOE Profile over the Year

Figure 8 presents the full-year battery State-of-Energy (SoE) profile for the economically optimal configuration—approximately 140 PV modules (~56 kWp) and 200 kWh battery capacity, with 80% depth-of-discharge (DoD)—simulated at 15 min resolution. The SoE is shown on a total basis, with an admissible operating window between 20% and 100%. Throughout the year, the trace never falls below the 20% threshold, confirming that the controller consistently respects the minimum DoD constraint.

A clear seasonal pattern emerges as follows: from spring to late summer, the battery frequently reaches high SoE levels during midday hours, often approaching full charge. In contrast, during autumn and winter, the SoE remains closer to the lower bound due to reduced PV availability.

This behavior aligns with the distribution of operating cases. Case C (PV charging the battery) dominates midday during the sunnier months. Case B (PV and battery supplying the load) is more common during mornings, evenings, and throughout winter. Case A appears when PV generation closely matches the load. Case D (generator charging) only occurs under Cycle-Charging (CC) during extended low-irradiance periods in winter.

Both control strategies—Load-Following (LF) and Cycle-Charging (CC)—yield nearly identical annual SoE trajectories. Under CC, the SoE tends to be slightly higher after generator start-up events, as the controller uses available headroom to recharge the battery. However, the difference is marginal and mainly observable during the shoulder seasons. Importantly, both policies maintain the SoE above the 20% limit and ensure reliability, with the annual Loss of Power Supply Probability (LPSP) remaining ≤ 0.03, effectively near-zero.

From a techno-economic perspective, the selected configuration (56 kWp PV, 200 kWh BESS, of which 160 kWh is usable at 80% DoD) lies near the EUAC minimum for this DoD level. It provides sufficient intraday energy buffering to meet the reliability criterion, while also minimizing summer curtailment. The top and bottom panels of Figure 8 illustrate the SoE evolution under Load-Following and Cycle-Charging, respectively. The dashed line at 20% indicates the lower SoE limit. As expected, the SoE drops toward the floor on winter nights, forms midday plateaus in spring, and frequently reaches 90–100% in summer. The close alignment of both control strategies confirms that the system behaves robustly under either policy.

Figure 9 and Figure 10 show the diurnal–seasonal posture of the battery State-of-Energy (SoE) under Load-Following (LF) and Cycle-Charging (CC) for the optimal design (56 kWp PV, 200 kWh BESS, DoD = 80%). The SoE is reported on a total basis with admissible window [20, 100] %. In both policies, the SoE rises with solar availability, forming a pronounced midday ridge that expands from spring through late summer and recedes in winter; winter nights remain near the 20% floor, whereas summer noons approach 100%. Differences between LF and CC are modest: when the generator is ON, CC uses any remaining headroom to recharge the battery, which yields slightly higher midday plateaus and quicker rebounds after low-SoE events, especially in the shoulder months; in periods with little generator use (late spring–summer), the two maps are nearly indistinguishable. Both policies respect the 20% floor and satisfy the LPSP ≤ 0.03 target; CC marginally reduces time spent near the floor by bundling necessary generator use into fewer, longer ON events.

Figure 11 reports the monthly State-of-Energy (SoE) for the economically optimal configuration—56 kWp PV (Npv≈140) and 200 kWh battery with DoD = 80%—under Load-Following (LF) and Cycle-Charging (CC). The SoE is plotted on a total basis with the admissible window [20, 100]% as designed, and the monthly minima touch but never drop below 20%. The seasonal pattern is clear: monthly averages rise from ≈48–50% in winter to ≈67–72% in late spring–summer, then decline towards ≈30% in December. Importantly, the LF and CC averages are nearly identical (differences typically < 0.5 pp) because CC only uses generator headroom to charge when the genset is already ON; in months with little generator use (spring–summer), the two policies become indistinguishable. Thus, both policies meet the reliability target with a very similar energy posture; CC may leave a slightly higher SoE immediately after ON events, whereas LF tends to spend marginally more time near the floor but often with slightly lower generator energy overall.

Figure 12 shows the full-year SoE with generator use. Solid orange and dashed blue lines show the battery SoE under Load-Following (LF) and Cycle-Charging (CC), respectively. The black hatched bands indicate periods when the diesel generator is ON (either covering the residual deficit in LF, or covering the deficit and charging the battery in CC). The SoE is plotted on a total basis (20–100%) with DoD = 80%; the dashed horizontal line marks the SoE_min = 20% limit. The two policies yield nearly identical SoE trajectories over the year because CC only exploits headroom to charge when the generator is already ON. System sizing: 56 kWp PV, 200 kWh battery, Pmax, GEN = 12 kW; native 15 min time-step.

Solid orange and dashed blue lines report the battery State-of-Energy (SoE) under Load-Following (LF) and Cycle-Charging (CC), respectively. Black strips near the 20% floor mark generator-On intervals (lower strip for LF, slightly offset above for CC). The SoE is shown on a total basis (20–100%), with the dashed horizontal line indicating SoEmin = 20% (DoD = 80%). In January, the SoE frequently touches the floor and the generator runs in short blocks; when it does, CC uses the available headroom to recharge, producing quicker rebounds than LF. April shows higher plateaus and sporadic generator use during cloudy sequences. In July, abundant PV keeps nighttime troughs shallow and no generator is needed. October exhibits mixed behavior; CC tends to consolidate fewer, longer activations that top up the battery. Overall, both policies achieve the reliability target, while CC bundles unavoidable starts and leaves the system at a marginally higher SoE after each ON event.

3.2. Seasonal Weekly Operation Under Load-Following and Cycle-Charging (Case A/B/C/D)

Winter is the most demanding regime: short, weak PV windows produce narrow Case C and extended Case B overnight. LF displays repeated gray GEN-ON clusters over several days; because the generator only follows the residual, the SoE often sits near the admissible floor (20–35%) until PV returns. CC handles the same deficits with prominent red Case D bands: after covering the load, spare headroom is routed to the battery, lifting the SoE into the ~40–55% range post-event and reducing residence at low-SoE. Both policies respect the DoD limit (no violation of the 20% floor) and satisfy the reliability target, but CC clearly mitigates deep cycling and improves resilience during the darkest weeks.

This shoulder period features alternating sunny and overcast days. Midday Case C intervals raise the SoE to about 60–85%, but evening declines are deeper than in summer, and extended Case B (Battery → Load) operation appears on cloudy days. The LF panel shows clustered gray GEN-ON bands—especially around 3 April and 7/8 April —indicating short generator interventions to cap residual deficits without increasing SoE. In contrast, under CC control, these activations merge into red Case D blocks: once ON, the generator both supplies the load and recharges the battery, so the SoE rebounds to roughly 45–60% after each event. The two panels therefore highlight the operational contrast: LF tracks net demand with a leaner SoE profile, whereas CC maintains a higher post-event buffer and reduces time spent near the 20% lower limit.

Strong irradiance drives a clean diurnal pattern: Case A (PV → Load) and Case C (PV → Battery) dominate every day, pushing the battery to high plateaus (SoE ≈ 80–100%) around noon and leaving shallow night troughs (≈35–45%). In LF, generator activity is almost absent—only a brief gray overlay during a short cloudy spell—because PV plus modest discharge (Case B) suffice. In CC, the same weather yields virtually identical power flows; the few generator starts produce thin red Case D slivers (GEN → Battery), so the SoE exits those events slightly higher than in LF. Overall, summer operation is PV-led, curtailment appears only when the battery reaches Emax, and the reliability target is met with minimal diesel contribution under both policies.

The autumn shoulder shows alternating clear and cloudy sequences that expose the LF/CC contrast. Early in the week (≈2–4 October) irradiance is weak: Case B (Battery → Load) occupies long stretches and the SoE tracks the lower window (≈25–40%). In LF, gray GEN-ON overlays appear in several short bursts; the genset only covers the residual deficit, so the battery remains near E_min until PV returns. In CC, the same starts consolidate into red Case D bands: after serving the load, spare headroom charges the battery (GEN → Battery), lifting the SoE to ~45–60% and shortening subsequent B intervals. Later in the week (≈5–7 October) clearer days bring broad Case C (PV → Battery) windows and higher midday SoE plateaus—locally approaching full—while nights show moderate B with little or no generator action in either policy. Overall, autumn operation alternates between PV-led days and deficit-dominated spells; CC leaves a larger post-event energy buffer and reduces time near the 20% floor, whereas LF attains the same adequacy with a leaner SoE and more fragmented starts.

This whole analysis through the year is presented to

Table 2. Analysis of the SoE at LF and CC per month.

Month	LF—Operation and SoE	CC—Operation and SoE	Key Takeaway
January	Short days, Case C windows compressed, long B at night, SoE often 20–35%, frequent GEN-ON clusters in overcast spells.	Same starts as LF but with D charging while GEN is ON, SoE exits events at ~40–55%, reducing time near Emin.	CC reduces deep excursions vs. LF, reliability respected.
February	Noon PV widens, night minima ~ 30%, GEN still needed but in shorter bursts.	Higher post-GEN SoE due to D, back-to-back starts less likely, trajectories otherwise similar to LF.	Convergence begins; CC gentler overnight posture.
March	Shoulder month: regular midday Case C plateaus (~60–80% SoE), gentler evening descents; GEN calls drop.	Fewer, longer starts, brief D segments consolidate charging and raise evening SoE.	CC further reduces cycling frequency.
April	Longer days, SoE commonly recovers to 70–90%, GEN sporadic.	Short D patches visible when GEN runs, lifting SoE post-event, policies nearly indistinguishable otherwise.	Weak policy difference under strong PV.
May	Abundant PV, noon often near full, night troughs 35–45%, Case A prevalent, GEN rare.	D almost disappears (few GEN starts); SoE high and stable.	PV dominance; dispatch choice immaterial.
June	Strong resource, clear saw-tooth, 90–100% plateaus, GEN virtually absent.	Operationally identical to LF, negligible D.	Summer reliability driven by PV.
July	Similar to June; occasional clouds deepen B at night but far from Emin.	Rare D after brief starts; negligible impact on monthly posture.	Both policies overlap.
August	Days start shortening, midday C persists, troughs deepen modestly, GEN marginal.	A handful of D segments during multi-cloud sequences top SoE and cut future starts.	CC preserves a slightly higher stance.
September	Shoulder returns: narrower midday plateaus, minima 30–40% common, short GEN stripes reappear.	Consolidated D episodes keep SoE higher overnight; fewer consecutive starts.	Policy contrast becomes visible again.
October	More time at 20–30% SoE; GEN-ON clustering increases; C windows contract.	Prominent D blocks lift SoE after each start, reducing deep excursions and start probability.	CC improves resilience vs. LF.
November	Weak irradiance, long nights, extended B and frequent GEN clusters, SoE ~ 25–35% for many mornings.	D common, GEN charging leaves SoE 40–55%, moderating stress.	CC clearly preferable operationally.
December	Most demanding month: sparse C, long B, regular GEN; 20% floor never violated (DoD = 80%).	Dense D blocks document active top-ups; LPSP ≤ 0.03 satisfied with less time near Emin than LF.	CC reduces low-SoE residency at same reliability.

below.

Taken together, the SoE plots in this section provide a consistent picture of how LF and CC operate the battery and the generator over the year. The annual SoE timelines in Figure 8 and the seasonal SoE heatmaps in Figure 9 and Figure 10 confirm that both policies respect the prescribed 20–100% window throughout the year. Winter and late-autumn months are clearly the most demanding, with frequent excursions towards the lower SoE bound, whereas summer months exhibit extended plateaus close to full charge driven by abundant PV generation.

The monthly average SoE plot in Figure 11 shows that, for the optimal design (56 kWp PV, 200 kWh, DoD = 80%), the mean SoE profile is very similar under LF and CC across all months. CC maintains a slightly higher average SoE during the winter and shoulder periods, reflecting the fact that the battery is actively recharged during GEN-ON intervals, whereas the two policies become essentially indistinguishable in summer when PV dominates the energy balance. This is consistent with the qualitative description in Table 2, where CC is seen to reduce the time spent near the minimum SoE level in the most stressed months.

The SoE–generator overlay in Figure 12, the compact seasonal overlays in Figure 13, and the detailed winter, spring, summer, and autumn weeks in Figure 14, Figure 15, Figure 16 and Figure 17, together with the month-by-month operational mosaics in Figure 18 and Figure 19, further illustrate that the generator operates in very similar windows under both policies. CC tends to consolidate generator operation into slightly longer ON periods that include short GEN → BESS charging segments, while LF follows the net load more strictly. As a result, the differences in annual generator operating hours and fuel consumption between LF and CC remain modest, and the two strategies are essentially equivalent in terms of reliability and cost for the considered industrial profile. CC mainly provides a more comfortable SoE posture during prolonged low-irradiance episodes, whereas LF minimizes generator energy use in a marginal way.

3.3. Techno-Economic Objective and Annualization

In order for the techno-economic study to take place, an objective function is defined to minimize the equivalent annual cost under reliability [80,81]. This function is shown below:

$$\begin{array}{ll}{min}_{N_{PV},B_{nec},DoD}J& \\ & ={EUAC}_{PV}+{EUAC}_{Bat}+{EUAC}_{Gen}+VOLL·{\displaystyle \sum _t}{E}_{short}\left(t\right)\\ & +c_{curt}·{\displaystyle \sum _t}{E}_{waste}(t)\end{array}$$

$$LPSP\le \tau , E_{min}\le E_{bat}\le E_{max},\forall t$$

with $E_{min}=B_{nec}(1-DoD)$, $E_{max}=B_{nec}$, threshold $\tau =0.03,$ and standard annualization for $EU AC$.

Where

$${EUAC}_{PV}={CAPEX}_{PV}·{CRF(r,n}_{PV})+{O\&M}_{PV}·{CAPEX}_{PV}$$

(1)

$${EUAC}_{Bat}={CAPEX}_{Bat}·{CRF(r,n}_{Bat})+{O\&M}_{Bat}·{CAPEX}_{Bat}$$

(2)

$$\begin{gathered} {EUAC}_{Gen}={CAPEX}_{Gen}·{CRF(r,n}_{Gen})+{O\&M}_{Gen}·{CAPEX}_{Gen} \\ +Fuel{ · SFC · E}_{Gen} \end{gathered}$$

(3)

$$CRF(r,n)={\displaystyle \frac{r{(1+r)}^n}{{(1+r)}^n-1}}$$

(4)

Recent global evidence shows that installed costs for utility-scale PV continued to fall through 2023, with IRENA reporting global weighted average installed costs on the order of <USD 1000/kW in 2023 and a long-run downward trend driven by module, balance-of-system, and soft-cost reductions. For commercial and industrial rooftop systems in advanced economies, installed costs typically sit in the same ballpark or somewhat higher than utility-scale depending on scale and soft costs; adopting EUR 800/kWp for a ~128 kWp rooftop array (320 × 0.40 kWp) is therefore conservative but well within contemporary ranges [82].

For operations and maintenance (O&M), European benchmarks for commercial PV consistently place fixed O&M in the “low tens of EUR/kW-yr”, which corresponds to roughly 1–2% of CAPEX per year for present-day systems. The European Commission’s JRC O&M benchmarking review for PV plants documents typical fixed O&M magnitudes at the system level that align with this percentage range; using 1.2%/yr as a baseline for modeling is therefore consistent with practice [83].

For Li-ion BESS at commercial and industry utility scales, recent U.S. national laboratory syntheses, which are widely used as global benchmarks, place installed (not just cell/pack) costs in the low hundreds of USD per kWh and falling. NREL’s recent cost updates and the 2025–2050 “Battery Cost Perspectives” work show current-to-near-term installed BESS costs ranging roughly USD 150–350/kWh (system-level- and hours class-dependent), while PNNL’s cost/performance guides present similar ranges for stationary Li-ion systems. Taking EUR 250/kWh as a 2023–2025 central value for a 2–4 h system is therefore squarely within contemporary ranges and keeps the analysis on mainstream ground [84,85,86,87].

A 7% real discount rate is standard in international generation cost comparisons and is widely used by IEA/NEA in the Projected Costs of Generating Electricity series and its companion LCOE tools [88,89].

Reliability is already enforced as a hard constraint (LPSP ≤ 0.03). Adding a positive VOLL in the objective while simultaneously capping LPSP would penalize the same phenomenon (unserved energy) twice. The baseline therefore minimizes EUAC subject to the adequacy constraint, keeping the economics and the reliability treatment orthogonal [90,91].

The system is islanded with no grid export or alternative sink (PtH/EV/H₂). Curtailment thus carries no opportunity cost in the baseline, setting c_curt = 0 reflects that “spilled” PV has zero revenue and avoids injecting an artificial bias toward larger storage.

Using zero penalties yields a clean reference that exposes the geometry of the cost–reliability trade-off driven by the load/solar profiles and converter/battery limits, without relying on subjective penalty magnitudes.

All monetary values are expressed in constant 2023 EUR, excluding VAT/levies; no subsidies are assumed.

The economic parameters in

Table 3. Baseline economics (consistent in the whole paper).

Block	Parameter	Value/Assumption
PV	Specific yield (Karlsruhe, PVGIS TMY)	≈1100 kWh/kWp·yr
	CAPEX	EUR 800/kWp
	Fixed O&M	1.2% of CAPEX per year
	Lifetime (economic)	25 years
Converter	Inverter efficiency (ηinv)	0.95
	Charge controller efficiency (ηcc)	0.98
Battery (Li-ion)	CAPEX	EUR 250/kWh
	Fixed O&M	1.5% of CAPEX per year
	Calendar life cap	15 years
	One-way discharge efficiency (ηbat)	0.85
	Depth-of-discharge used (DoD)	80% (usable window = 0.8·Bnec)
Diesel Generator	Rated power	12 kW(ac)
	Specific fuel consumption (SFC)	0.28 L/kWh
	Diesel price (base)	EUR 1.20/L
	Generator lifetime	15 years
	Generator CAPEX (CAPEXGen)	EUR 500/kW (→EUR 6000 at 12 kW)
	Generator O&M (O&MGen)	3%/yr of CAPEXGen
Economics	Discount rate (real)	7%
	Objective	EUAC + VOLL·ΣEshort + ccurt·ΣEwaste
	VOLL	not fixed
	Curtailment penalty ccurt	0 (baseline)
Simulation	Time-step Δt	15 min
	Load series	LG-07 (Braeuer 2020 [73]), 35,040 samples
Reliability	Constraint	LPSP ≤ 0.03
Design grid	PV size Npv	140…320 modules (0.40 kWp each)
	Battery Bnec	200…700 kWh
	DoD	{20%, 50%, 80%}

are selected to be representative of small commercial and industrial hybrid PV–BESS–diesel projects in Europe in the 2023–2025 timeframe. The baseline PV CAPEX of EUR 800/kWp and BESS CAPEX of EUR 250/kWh, together with fixed O&M rates of 1.2% and 1.5% of CAPEX per year, respectively, fall within the ranges reported for rooftop PV and stationary Li-ion storage in recent techno-economic assessments of small commercial and industrial systems [87,88,89]. These values are intentionally conservative so that the resulting EUAC estimates do not rely on overly optimistic cost assumptions.

For the diesel generator, the specific fuel consumption of 0.28 L/kWh and the baseline diesel price of EUR 1.20/L are consistent with manufacturer data and cost levels for small industrial gensets in the 10–20 kW class [87,88,89]. The generator CAPEX of EUR 500/kW and the 3%/yr O&M rate are likewise typical for small-scale diesel units in hybrid microgrid applications. Under these assumptions, fuel and O&M contributions remain significant in the EUAC, especially when generator operation becomes frequent.

With respect to the battery, an 80% depth-of-discharge (DoD) window (SoE on a total basis between 20% and 100%) is adopted as the reference case, reflecting common practice in stationary Li-ion BESS installations [ref-DoD]. In the model, this is combined with a 15-year calendar life cap and a DoD-dependent cycle life function, so that increased utilization of the admissible window is consistently linked to long-term replacement cost. Alternative DoD limits (e.g., 50%) are also explored in the parametric study and are found to require larger nominal storage capacities for the same reliability target.

The sensitivity analysis in Section 3.6 perturbs these baseline values over ranges that reflect historical variability and plausible future uncertainty. PV and BESS CAPEX are varied by ±20% to capture regional Engineering, Procurement, and Construction differences and market trends, the diesel fuel price is varied by ±50% to represent recent volatility, and the specific fuel consumption is perturbed by ±10%. In addition, the real discount rate is scanned between 5% and 10%. These ranges are deliberately chosen around Table 3 baselines so that the robustness of the optimal design can be evaluated under realistic economic fluctuations.

3.4. Synthesis of Optimization Objectives and Reliability Treatments in Microgrid Planning

A large body of work formulates microgrid sizing as a coupled cost–reliability problem. Most studies minimize an annualized or life cycle cost (EUAC/LCC/NPC/LCOE) while either constraining reliability through LPSP (e.g., LPSP ≤ τ) or co-optimizing cost and LPSP on a Pareto front; several monetize adequacy directly by adding VOLL·EENS (cost of unserved energy) to the objective. Recent extensions further penalize curtailment and battery aging so that the optimum reflects not only adequacy but also operational externalities. To situate our objective function (EUAC + VOLL·∑E_short + c_curt·∑E_waste, with LPSP ≤ 0.03) within this landscape,

Table 4. Objective function patterns in the literature for islanded/hybrid microgrids.

Reference	Tech Scope	Objective Terms	Reliability Term	Extra Penalties	Optimizer/Style
[92]	PV–WT–BESS (islanded)	NPC/LCC (annualized)	LPSP (constraint/target)	–	Parametric/iterative
[93]	PV–WT	Cost (sizing)	Reliability index + SoC bounds	Converter modeling	Deterministic search
[3]	SAPS/HRES (review)	LCC/EUAC frameworks	LPSP coupled with cost	–	Review/taxonomy
[4]	PV–WT– BESS	Annualized cost (Pareto)	LPSP (co-objective)	–	GA (MOEA)
[94]	PV–WT– BESS	Cost or LPSP	LPSP as objective/constraint	–	Analytical + sim
[95]	Islanded HRES	Cost (NPC/LCOE)	LPSP threshold	–	Heuristic/DE/PSO (typical)
[96]	Stand-alone PV/HRES	Cost metrics (LCOE/LCC)	LPSP workflows	–	Review
[97]	PV–WT– BESS	NPC/LCOE (Pareto)	LPSP (co-objective)	–	MOEA
[98]	PV–WT–BESS–Diesel	Annualized cost	LPSP cap	–	MOEA/deterministic
[99]	Microgrid planning	Capex + Opex + VOLL·EENS	EENS (monetized)	–	MILP/decomposition
[100]	Islanded MG	Operating cost (+capex in variants)	Reliability via constraints, VOLL variants	–	MILP/control-aware
[101]	Islanded MG	Operating cost + battery aging cost	–	DoD/throughput-aware degradation	Firefly/heuristics
[102]	MG scheduling	Cost + aging cost (linear)	–	Degradation model	MILP-ready
[103]	Poly-generation MG	Cost + aging + curtailment cost	–	Dumping/aging penalties	MILP
[80]	Islanded MG	Cost + curtailment penalties	–	Time-varying curtailment weights	MILP
[104]	Hybrid MGs (case studies)	LCC + resilience economics	–	Empirical validation	Case study
[105]	100% RES stand-alone MG	Investment + EENS cost (CERL)	LOLP/EENS monetized (no hard cap)	–	Analytical/Monte Carlo

synthesizes representative formulations from the literature, highlighting the objective terms, how reliability enters (constraint or co-objective), any curtailment/aging penalties, and the optimization style.

Our objective follows the mainstream cost–reliability formulation: minimize annualized cost (EUAC) while enforcing adequacy via an LPSP cap, or equivalently by monetizing unmet energy through VOLL. This is aligned with prior work that (i) minimizes EUAC/NPC subject to LPSP ≤ τ or jointly with LPSP in a Pareto setting, (ii) embeds VOLL·Eshort in the objective to internalize reliability, and (iii) optionally penalizes curtailment and battery aging to reflect operational externalities.

3.5. Parametric Sizing Results: LPSP Feasibility and EUAC Minimum (DoD = 20–80%, GEN = 12 kW)

• DoD = 20%: no feasible point with LPSP ≤ 0.03 in the scanned grid, the usable window $E_{usable}=DoD·B_{nec}$ is too small to bridge deficits.
• DoD = 50%: a feasible band emerges: cost contours show a valley toward high PV and mid storage; increasing PV reduces deficits more “cheaply” than very large storage, while too much storage raises EU AC with limited additional benefit.
• DoD = 80%: the feasible region widens markedly; the larger usable energy window depresses LPSP without necessarily increasing battery EUAC (calendar life-limited). The cost valley shifts toward moderate storage at higher PV.

Time resolution and energy balance. We simulate one full-year at Δt = 15 min. At each step we compute AC PV power PPV(t) from a scaled unit PV series (Karlsruhe-like specific yield 1100 kWh/kWp times the array size. Load Pload (t) is the LG-07 series from the Excel file. The DC battery state Ebat evolves with charge controller efficiency ηcc = 0.98 and discharge efficiency ηbat = 0.85. The admissible SOC window is controlled by the depth-of-discharge (DoD): Emin = Bnec(1 − DoD), Emax = Bnec.

To identify the most reliable and cost-effective microgrid configuration, a parametric simulation was performed where the number of PV modules (Npv) and the nominal battery energy capacity (Bnec) were varied. For each combination, the Loss of Power Supply Probability (LPSP) was computed based on real hourly load demand data (LG-07 profile) and location-specific PV generation (Karlsruhe, Germany).

In practice, the sizing problem is solved via a deterministic parametric grid search over the discrete design space (Npv, Bnec). For each candidate pair, we run a full-year simulation at 15 min resolution, compute LPSP and EUAC, discard all configurations with LPSP > 0.03, and retain the feasible design with the minimum EUAC as the chosen optimum.

The sizing problem minimizes EUAC subject to a reliability constraint LPSP ≤ 0.03. The 3-D surface shows LPSP (Npv, Bnec), with the black isocontour at 0.03 marking the feasibility frontier; points below the surface/contour are more reliable (smaller LPSP), points above are less reliable (larger LPSP).

• The starred point at (Npv = 140 modules ≈ 56 kWp, Bnec = 200 kWh) lies just inside the feasible region—i.e., LPSP ≈ 0.025, comfortably ≤ 0.03. In this study, this starred point (56 kWp PV, 200 kWh BESS, DoD = 80%) is the chosen optimum used in all subsequent techno-economic and SoE analyses.
• The red vertical section at Npv = 140 (the “56 kWp cut”) shows the crossing of the target: at Bnec = 200 kWh we are already below 0.03; any smaller BESS would push the section above the 0.03 line (infeasible), while any larger BESS only reduces LPSP further without improving the constraint.
• Because adequacy is non-binding (LPSP ≈ 0) and generator energy at the optimum is modest, EUAC exhibits a shallow minimum near the smallest feasible capacities. Hence the cost-minimizing design lies close to the LPSP boundary, at N_pv = 140 (56 kWp) and B_nec = 200 kWh.
• The LF vs. CC panels confirm this: under CC the 0.03 contour shifts slightly downward (marginal reliability gain), but at N_pv = 140 both policies are already below 0.03 around B_nec ≈ 200 kWh. Thus, cost—not reliability—dominates the choice locally, so the same pair remains optimal (and CC merely keeps a slightly higher SoE after GEN-ON, as shown elsewhere).

Figure A1 shows that moving left/down from (56, 200) violates the reliability constraint, while moving right/up only raises EUAC with negligible reliability benefit. Therefore, (56 kWp, 200 kWh) is the cost-optimal feasible point.

From a structural point of view, the location of the optimum can be understood by inspecting the annual energy balance and the shape of the industrial load. The LG-07 profile has an annual demand of approximately 56.6 MWh, with a pronounced daytime plateau around 8 kW and peak demands up to about 11.5 kW during the summer months. At the optimal design, the PV array of 56 kWp combined with a specific yield of ≈1100 kWh/kWp·yr in Karlsruhe produces on the order of 61–62 MWh/yr, i.e., broadly comparable to the annual load. The 200 kWh battery with an 80% DoD window provides about 160 kWh of usable storage, which corresponds to roughly 2.5–3 h of average load coverage. This capacity is sufficient to buffer intraday mismatches between PV production and demand, but not large enough to eliminate all multiday winter deficits; the 12 kW generator therefore remains necessary to cover extended low-irradiance periods.

Within this context, moving to smaller capacities than 56 kWp PV and 200 kWh BESS causes winter deficits to increase and the LPSP constraint to be violated. Moving to larger capacities, on the other hand, only marginally reduces generator operating hours and fuel use, because the generator is already used infrequently at the baseline optimum. As a result, the EUAC surface exhibits a shallow valley whose minimum lies close to the smallest capacity pair that satisfies the LPSP ≤ 0.03 constraint. This behavior explains why the LF and CC optima converge around the same design point: both strategies exploit the same fundamental balance between annual PV energy, storage buffer, and backup power, with CC mainly affecting the short-term SoE posture rather than the location of the cost minimum.

The underlying mechanism is expected to generalize to similar inelastic industrial profiles in temperate climates, where demand exhibits a stable daytime plateau and moderate seasonality, and where the generator is sized close to the annual peak load. In such cases, the cost-optimal design typically corresponds to a PV array that produces annual energy on the same order as the load, combined with a battery sized to bridge daily and short multiday deficits, but not to eliminate all winter shortfalls. For industrial sites in higher-irradiance regions, the same reasoning suggests that the optimum would shift towards somewhat larger PV and potentially smaller storage (in relative terms), as more of the adequacy burden is carried by solar generation with fewer generator hours.

Cost surface and policy comparison. Figure 20 and Figure 21 compare the annualized cost surface EUAC (Npv, Bnec) under Load-Following (LF) and Cycle-Charging (CC). In both policies the minimum lies near the smallest feasible capacities, around Npv ≈ 140 (56 kWp) and Bnec ≈ 200 kWh. At the baseline design, the annualized cost is JLF ≈ 13.86 k€/yr and JCC ≈ 14.19 k€/yr (Δ ≈ +0.33 k€/yr, +2.4% for CC). The sizing recommendation is therefore unchanged—the optimum remains close to 56 kWp + 200 kWh—while the dispatch policy mainly affects operating posture (CC leaves a higher SoE after GEN-ON by using headroom to charge) rather than feasibility. Both policies satisfy the adequacy target (LPSP ≈ 0), so reliability is non-binding.

3.6. Robustness of the Optimum and Generator-Sizing

As a robustness check, we also explored non-zero penalty coefficients. Increasing VOLL above zero (e.g., industrial-grade values) or assigning a small curtailment penalty c_curt > 0 shifts the minimum toward larger PV and/or BESS by internalizing reliability margins and surplus valorization. These scenarios do not change the qualitative conclusion that, under present costs and with a 12 kW generator, the baseline optimum sits near Npv = 140 (56 kWp) and Bnec = 200 kWh; they only widen the economic basin toward higher capacities.

Figure 22a,b report one-at-a-time tornado sensitivities around the baseline (56 kWp, 200 kWh). Each bar shows ΔJ (EUR/yr) relative to the baseline when a single driver is perturbed while others are held fixed (discount rate r: 5–10%; BESS and PV CAPEX: ±20%; diesel price: ±50%; SFC: ±10%). In both dispatch policies, the discount rate dominates via the capital recovery factor, followed by BESS CAPEX and diesel price; PV CAPEX is somewhat less influential, and SFC is the smallest lever because generator energy at the optimum is modest. The ranking and magnitudes are very similar for LF and CC. CC exhibits a slightly higher baseline EUAC (≈14.19 k€/yr vs. 13.86 k€/yr for LF) due to additional fuel used when charging the battery during generator-On periods.

4. Conclusions

This study applied a unified techno-economic objective—based on the equivalent uniform annual cost (EUAC), incorporating depth-of-discharge (DoD)-dependent battery cycle life, a 15-year calendar life, a 7% discount rate, and optionally, the Value of Lost Load (VOLL) for unmet energy. Using the LG-07 industrial load profile and PV generation scaled to Karlsruhe conditions (~1100 kWh/kWp·year), the optimal configuration was found to be approximately 56 kWp of PV and 200 kWh of battery storage (BESS) at 80% DoD.

This configuration satisfies the reliability constraint of LPSP ≤ 0.03. Monthly case distributions and full-year State-of-Energy (SoE) traces indicate that prolonged deep battery depletion does not occur. Sensitivity analysis highlighted diesel price and discount rate as the most influential parameters affecting EUAC. When battery costs are modeled based solely on throughput, or when higher VOLL values are assumed, the cost-optimal design shifts toward larger storage capacity—and, in some scenarios, increased PV sizing.

Excess PV energy can be effectively utilized through flexible demand side applications such as electric vehicle (EV) smart charging, power-to-heat (PtH), and hydrogen production via electrolysis. These strategies reduce curtailment, enhance economic performance, and maintain system reliability:

• EV smart charging can be scheduled to align with PV production peaks or to support SoE recovery during low-storage periods. Bidirectional charging, vehicle-to-building (V2B) can also offer short-term peak shaving, effectively turning part of the mobility demand into controllable load.
• Power-to-heat (PtH), such as high-COP heat pumps or electric boilers coupled with thermal storage, can convert midday surplus into usable heat at low marginal cost, displacing fossil fuel-based heating and leading to near-zero curtailment during summer months. In broader applications, PtH concepts can also be integrated into district heating networks, where surplus PV electricity is converted into heat and distributed to multiple consumers via centralized systems. Such integration could further enhance renewable energy utilization, reduce fossil-based heating demand, and support local decarbonization targets.
• Hydrogen production via electrolysis acts as a deeper sink for extended surpluses (“power-to-molecules”), provided part-load and start–stop limitations of the electrolyzer are respected. The produced hydrogen can serve as on-site fuel or an exportable energy carrier.

In the economic model, these flexible uses can either be implicitly represented (e.g., by setting a non-zero curtailment cost, ccurt > 0) or explicitly modeled through added revenues or avoided-cost terms. In both cases, they expand the viable design space toward higher PV sizing (greater surplus capture), while also reducing the required battery storage for short-term balancing—assuming flexible loads are activated only when adequacy is ensured (LPSP ≤ 0.03).

In terms of applicability, the present results are most directly relevant to inelastic industrial loads in temperate, moderate-irradiance climates similar to Karlsruhe, where annual PV production is of the same order as the yearly demand and a diesel generator is sized close to the site peak load. In such settings, the cost-optimal design typically combines a PV array whose yearly output roughly matches the annual load with a battery sized to bridge daily and short multiday deficits, while a diesel unit covers extended winter shortfalls. In higher-irradiance regions, the same sizing logic would be expected to shift the optimum towards somewhat larger PV capacities and relatively smaller storage (in relative terms), as a larger share of adequacy can be carried by solar generation with fewer generator hours. Conversely, in lower-irradiance or more variable climates, the role of storage and diesel backup becomes more pronounced, and the economic optimum is likely to move towards larger BESS capacities and more frequent generator support.

From an engineering perspective, the present results suggest several practical guidelines for choosing between Load-Following (LF) and Cycle-Charging (CC). For inelastic industrial loads with a stable daytime plateau and moderate seasonality in temperate, moderate-irradiance climates (such as the Karlsruhe-based case considered here), both strategies lead to very similar sizing recommendations and annualized costs: LF tends to minimize fuel use by letting the generator track only the residual demand, whereas CC slightly increases fuel consumption but leaves the battery at a higher SoE after generator events, thereby reducing time spent near the lower SoE bound and mitigating deep cycling. In such settings, the choice between LF and CC can therefore be based on operational priorities—for example, whether minimizing fuel use or reducing low-SoE operation and start–stop frequency is more critical for the operator. For load profiles with stronger evening peaks or more pronounced variability (e.g., some commercial or mixed-use facilities), CC is expected to offer a clearer robustness benefit by consolidating generator operation into fewer, longer ON periods that also recharge the battery, at the expense of some additional fuel. In very high-irradiance climates, where generator use becomes rare and PV plus storage carry most of the adequacy burden, the distinction between LF and CC largely fades and capacity sizing dominates performance; conversely, in harsher, lower-irradiance climates with frequent deficits and high diesel prices, the LF/CC choice becomes more relevant for operating expenditures, and the trade-off between fuel savings and battery cycling needs to be evaluated explicitly.

In conclusion, the baseline optimal configuration (56 kWp PV, 200 kWh BESS) proves robust. Surplus valorization through EVs, PtH, and hydrogen improves EUAC outcomes and supports moderate PV oversizing. These findings suggest promising avenues for future work on multi-sink coordination and market-integrated system operation.