2.1.1. Modeling and Simulation Equipment

The modeling and simulation of the biopile and the photovoltaic system were performed using MATLAB software, which allows users to manipulate matrices and display graphs and data, among other things. MATLAB also enables users to implement algorithms and design user interfaces. It can communicate with various programming languages, including C, C++, Java, Python, and others.

2.1.2. Hardware Components of the Prototype Residential Virtual Power Plant

Figure 1. Components of a residential VPP combining PV and a biofuel cell.

To compare the biosensor model’s simulation data with field data, a prototype virtual residential power plant based on the Internet of Things (IoT) was designed. It combines two energy sources: a solar panel and a plant-based biopile. This IoT device incorporates hardware components, including IoT sensors, microcontrollers, and a nano computer (Figure 1).

2.1.3. Software Used in the Residential Virtual Power Plant

On the software side, the VPP uses the Mosquitto MQTT broker to collect data acquired from each sensor via a microcontroller that contains an MQTT client. It uses Node-RED as an open-source visual programming tool based on data flows collected from the sensors, and PostgreSQL/PostGIS as a geographic database into which the sensor data is inserted after processing.

The NASA POWER web application was used to extract solar radiation data from the study area in order to compare it with data from the pyranometer installed on-site.

2.2.1. Physical and Mathematical Model of Energy Sources: Biopile/Photovoltaics

1)Physicalandmathematicalmodelofthebiopile

The biopile is modeled as a one-dimensional (1D) soil continuum of depth d = 0.05   m , spatially discretized across N = 21 nodes spanning from the anode ( x = 0 ) to the cathode ( x = d ), with a nominal electrode cross-sectional area A e = 10 − 3   m 2 (evaluated below as a calibrated parameter). Six spatially distributed state fields are resolved along this continuum, temperature T ( x , t ) , volumetric water content θ ( x , t ) , dissolved oxygen concentration C O ( x , t ) , soluble substrate concentration S ( x , t ) , proton concentration C H ( x , t ) , and planktonic microbial biomass X ( x , t ) , coupled with three zero-dimensional (0D) lumped interfacial boundary states: anodic biofilm areal density X b ( t ) , residual magnesium anode quantity n M g ( t ) , and passivation film thickness δ f i l m ( t ) . The complete dynamic system state vector y ( t ) ∈ ℝ 129 thus comprises 6 × 21 + 3 = 129 state variables.

a)Electrochemicalsub-model

Anode and cathode equilibrium potentials are governed by the Nernst relation [8]:

where E a n o d e 0 = − 2.356   V and E c a t h o d e 0 = + 0.401   V represent the standard reduction potentials for the sacrificial magnesium/carbon electrode couple. In the low-overpotential regime, charge-transfer resistances ( R c t ) are formulated via the linearized Butler-Volmer equation [8]:

where the exchange current density j 0 follows an Arrhenius temperature dependence with an activation energy of E a , j 0 = 30   kJ ⋅ mol − 1 . To account for bio-catalytic enhancement, the anodic exchange current density j 0 , a is scaled by up to a factor of 5 as a function of the anodic biofilm density X b , saturated via a half-saturation constant K b . Ohmic transport within the unsaturated soil pore water is governed by Archie’s empirical law [9], expressed in conductivity form and modified by an Arrhenius thermal correction factor ( E a , σ = 15   kJ ⋅ mol − 1 ):

where ϕ is the soil porosity, θ is the volumetric water content, τ is tortuosity, and m , n are empirical cementation and saturation exponents, respectively.

Given the strictly one-dimensional geometry and the lack of lateral current pathways, the quasi-static electrical sub-problem yields an exact closed-form solution derived from Kirchhoff’s Voltage Law (KVL) around the anode-electrolyte-cathode-Rext loop:

This closed-form formulation replaces prior iterative finite-volume numerical schemes, providing an exact analytical expression that collapses strictly into the algebraic series/parallel topology equations under uniform, steady-state, and saturated conditions.

b)Transport,kinetics,andthermalsub-models

Substrate transport (S) is governed by Fickian diffusion with an Arrhenius-corrected diffusion coefficient (DS) and a Monod-type microbial consumption reaction [10]:

incorporating a first-order biomass decay rate ( − k d X ). To account for plant activity, rhizodeposition is introduced as a distributed substrate source within the root zone, spatially weighted by a Gaussian root-density distribution Ψ ( x ) centered at depth x r o o t with spatial variance x r o o t :

Dissolved oxygen ( C O ) diffuses with diffusivity D O , subject to a volumetric sink driven by aerobic substrate oxidation ( 0.2 R M o n o d ), a cathodic electrochemical flux boundary condition − j c a t h o d e / ( z F ) , and atmospheric exchange h O ( C O , a t m − C O ) at the top cathode surface. Proton dynamics ( C H ) with diffusivity D H are driven at the electrode boundaries by Faradaic anodic and cathodic current fluxes, j c a t h o d e / F and − j c a t h o d e / F , respectively.

Soil moisture transport ( θ ) is modeled via a diffusive Richards-type equation [11], subjected to a sinusoidal diurnal surface-humidity flux boundary condition (period 86,400 s) and a uniform root-extraction sink term S s i n k . At the anodic interface (x = 0), the growth of the attached biofilm areal density ( X b ) follows a surface-restricted, Arrhenius-corrected Monod kinetic expression:

Thermal transport follows a 1D heat conduction equation parameterized by

thermal conductivity ktherm and volumetric heat capacity ρ c p , driven by internal Joule heating σ ( ∂ x ∂ ϕ ) 2 , exothermic bio-oxidation heat generation

Δ H S R M o n o d , and a surface boundary condition combining net absorbed solar radiation ( 1 − a l b e d o ) I s o l a r and convective air cooling h T ( T s u r f − T a i r ) . Finally, the depletion of the sacrificial magnesium anode follows Faraday’s law of electrolysis:

while the passivation layer dynamics are governed by a stoichiometric precipitation-dissolution mass balance.

c)Reductiontoalumpedalgebraiccellmodel

To verify the numerical implementation of the spatially resolved formulation, an asymptotic reduction to its lumped algebraic limit was performed. Under the four uniform and saturating hypotheses of a spatially uniform thermal field ( T = T r e f = 298.15   K ), constant soil moisture at reference saturation ( θ = θ r e f ), non-limiting substrate supply ( S = 1.0   mol ⋅ m − 3 ), and a fully mature anodic biofilm ( X b = 100 × K b ), the distributed continuum field equations collapse to a single lumped operating point. Under these bounds, the spatial integral for internal resistance simplifies to scalar expressions for open-circuit voltage ( E O C V ) and total internal resistance ( R i n t ), governed directly by Equations (3)-(4). Direct numerical evaluation of the full 1D spatially resolved code under these exact hypotheses with a default load resistance R e x t = 50   Ω yields E O C V = 3.0852   V , R i n t = 1484.04   Ω , I = 2.011   mA , and V c e l l = I R e x t = 0.1006   V . These numerical outputs match the analytical scalar solution identically, confirming the structural consistency of the spatial discretization and ODE solver implementation by demonstrating that the continuum model preserves exact mathematical convergence to its lumped theoretical limit prior to experimental validation.

d)Parameterclassification

Model parameters are systematically classified below according to their functional role and derivation origin, categorized as directly measured, literature-derived, or calibrated/adjusted values (Table 1). To maintain full methodological transparency, parameters designated as calibrated or adjusted are explicitly identified as fitted metrics and are not presented as independently measured physical properties.

Table 1.Classification of biopile parameters.

2)Photovoltaicbranchmodel

The photovoltaic (PV) branch evaluated in the diurnal response analysis is modeled using a single-diode open-circuit voltage relationship, which captures the logarithmic dependence of the potential on solar irradiance (G) [12]:

where N s denotes the number of series-connected solar cells, n is the diode ideality factor, V t = k B T / q is the thermal voltage, G S T C = 1000 W⋅m⁻² is the standard test condition (STC) irradiance, and V o c , S T C represents the nominal open-circuit voltage at STC. A hard lower bound ( G m i n ) models the operational threshold below which the maximum power point tracking (MPPT) power electronics disengage.

This instantaneous algebraic formulation contrasts the fundamental time scales with the branch of the biosystem related to the physical and mathematical model, whose dynamic response is governed by biological and transport time constants spanning several days. This structural distinction provides the physical mechanism underlying the asymmetry of the diurnal response analyzed in the experimental analysis and dynamic simulation of a photovoltaic system.

2.2.2. Development of a Residential Virtual Power Plant Combining Photovoltaic Technology and a Plant-Based Biopile

1)HardwareandsoftwaredesignoftheresidentialVirtualPowerPlant

The residential virtual power plant, built on an IoT architecture (Figure 2), consists of a photovoltaic component, a plant-based biopile component, and a battery.

Figure 2. Block diagram of the IoT system for remote monitoring of hybridization (PV, Biopile).

2)DesignofthedataStorageandprocessingsystemfortheresidentialVirtualPowerPlant

Figure 3. Data Storage and processing processes for the IoT remote monitoring device.

Data management is crucial in the residential virtual power plant with remote monitoring, as it enables the collection, storage, and processing of information from sensors. The process, from initial data acquisition to storage in a database, uses the MQTT protocol (Figure 3).

This general method shows that the sensors collect data on the renewable energy source, including temperature and energy production. This data is then structured into MQTT messages, each with a specific topic corresponding to the type of information to be transmitted (Table 2).

Table 2. Description of the publishing/subscription system.

Figure 4. Storage and processing of hybridization data (PV, Plant Biopile).

Having designed plant-based biofuel cells made of spirally arranged ginger with spikes and electrodes (magnesium and carbon), the next step is to produce the IoT remote monitoring system for the micro-generator. This system combines a photovoltaic panel and a plant-based microbial fuel cell to charge a battery and power electrical loads. During the production phase, all sensors send their messages to the MQTT server (Mosquitto), also known as the MQTT broker. Subscribers (clients) receive the MQTT messages relevant to their topics of interest and extract the useful data. This extracted data is then stored in a database, with tables structured by topic to facilitate information management. Node-RED acts as a software layer that simplifies the development, management, and automation of data flows and interactions between IoT devices in a remote monitoring system. It provides an efficient way to connect hardware and software components and offers a visual interface for creating and managing these flows (Figure 4).

3)SizingofBiopiles

Figure 5. General method for sizing a plant-based biobattery microgenerator.

Solar energy and plant-based (phyto-microbial) fuel cells are renewable energy sources that are intermittent and depend on weather conditions, soil conditions, and plant cultivation conditions. Optimization ensures a continuous energy supply by adapting production to demand. Thus, this part of the research aims to contribute to scientific development by providing concrete applications for the design of plant-based biobatteries, taking into account both variations in solar energy and the changing conditions under which these biobatteries may be used.

By creating a mixed series/parallel configuration of a set of plant-based biosolar cells, it is possible to obtain a significant energy source capable of powering solar-powered electrical loads such as light bulbs, televisions, etc.

The method for optimizing plant-based biosolar cells is illustrated in Figure 5 above.

All numerical results presented and developed in the mathematical modeling of the biobattery were generated using the custom multiphysics framework. The spatial discretization of the six dynamic fields distributed across the 21-node domain uses a second-order centered finite-difference scheme. The boundary conditions described in the biopile model are implemented using one-sided finite-difference closures, without ghost nodes, to ensure flux conservation at reactive spatial boundaries.

The resulting stiff system of 129 time-dependent ordinary differential equations (ODEs) was solved using the Radau implicit Runge-Kutta scheme (SciPy implementation) with relative and absolute integration tolerances set to 10⁻⁹ and 10⁻⁶, respectively. A parallel reference code was implemented using MATLAB’s ode15s stiff solver with variable step size and order for structural cross-validation. Both numerical implementations share the same basic equations, spatial discretizations, and parameter sets. Although the stability of the solution was maintained for all simulations, a formal study of mesh and tolerance independence remains a goal for future, more in-depth computational work.

The quasi-static electrical subproblem is solved algebraically at each step of the time evolution, directly from the instantaneous values of the spatial state. This direct algebraic evaluation eliminates numerical iteration errors and the risk of non-convergence. This formulation is based on the physical assumption that the electrical relaxation kinetics are decoupled from and significantly faster than the thermal, mass transport, and biological timescales—an assumption consistent with the estimated RC time constants of the system (R_int ≈ 1.5 kΩ) for transport dynamics spanning several hours and biological dynamics spanning several days.

3.2.1. Empirical Basis for Selecting Plant Species for a Biopile

The classic microbial fuel cell transforms the chemical energy of organic matter into electricity by coupling with a living plant. This plant provides organic matter to the soil via photosynthesis and excretes substances through its roots, which soil microorganisms consume, thus releasing electrons collected by conductive electrodes [13].

The plant species were selected during a preliminary empirical screening campaign conducted on the physical prototype prior to the formulation of the model. Four candidate plant species were evaluated in parallel under identical instrumentation and monitoring conditions, yielding open-circuit potentials of 1.391 V for Solanumlycopersicum (tomato) and 1.643 V for Costusspicatus (spiral ginger), 1.352 V for Capsicum annuum (chili pepper), and 1.501 V for aloe vera (Figure 6). Costusspicatus exhibited the highest open-circuit response and was selected as the botanical matrix for all subsequent characterizations. Consequently, the rhizodeposition parameters ( β , ε p h o t o , f A P A R ) defined in the model are specific to the species Costus spicatus and do not represent generalized P-MFC values.

Figure 6. Measuring the energy produced by plants.

3.2.2. Empirical Bases for Biobile Electrodes

Anode and cathode materials are crucial for oxidation and reduction reactions. Anodes must maximize cell adhesion and electron transfer while being conductive, biocompatible, and chemically stable. Metallic materials used in anodes must be corrosion-resistant and non-toxic.

A subsequent screening evaluated four electrode pairs within Costusspicatus. Costus spicatus matrix: a magnesium anode paired with a graphitic carbon cathode (1.0 - 2.5 V), a magnesium anode paired with a charcoal cathode (1.0 - 1.5 V), a copper anode paired with a charcoal cathode (0.5 - 1.3 V), and an iron anode paired with a charcoal cathode (0.5 - 1.0 V) (Table 3). These voltage ranges are consistent with the values reported for comparable galvanic metal-carbon sacrificial pairs in ground energy recovery systems [1][7]. The magnesium/graphitic carbon pair was selected for the final architecture to maximize the output potential, thereby forming the basis for the standard thermodynamic reduction potentials ( E a n o d e 0 , E c a t h o d e 0 ) presented in the biobattery model. The range of 1.0 - 2.5 V established during this preliminary screening serves as the empirical open-circuit reference evaluated in this study.

Table 3.Electrode configuration of a biobattery.

The measured voltage of a pot without a plant, using a substrate similar to that of the plant pots, was 0.916 V, while a pot containing sterilized sandy soil showed 0.840 V (Figure 7). These results were obtained under identical experimental conditions, including the same mass, water volume, electrode size, anode-cathode distance, and sunlight and temperature conditions.

Figure 7. Measurement of the energy produced by control pots without plants using the same Mg/graphite electrodes.

3.2.3. Residential Virtual Power Plant Prototyping

Figure 8 shows the residential virtual power plant (VPP) used to collect data via sensors and store it in a nanocomputer. The VPP consists of a photovoltaic solar panel, a biofuel cell array connected in parallel/series, a solar charge controller, a battery, and sensors concealed on-site in strategic locations, including planters, junction boxes, etc. This device is built on the architecture of the Internet of Things (IoT).

Figure 8. Presentation of the residential virtual power plant.

3.3.1. Analysis of Experimental Data Sources

The model’s performance was evaluated using two complementary empirical datasets collected from the physical prototype during a field campaign conducted on October 15, 2023, in Côte d’Ivoire.

The first source includes the architectural specifications and operating parameters defined during system assembly, prior to model development. These parameters include: 1) the nominal electrical specifications of the individual cells (0.5 - 1.5 V, operating range of 0.2 A), 2) the sizing diagrams for the stack topology for four series/parallel configurations (Figures 9-12), 3) open-circuit test parameters for the bioelectrodes (1.0 - 2.5 V for the selected magnesium/graphitic carbon pair), and 4) dashboard time series covering the photovoltaic (PV) voltage, biosensor voltage, battery status, ambient temperature, relative humidity, global solar irradiance, brightness, and precipitation.

In these log files, the PV branch shows nominal diurnal voltage fluctuations between 15.5 V and 20.0 V before experiencing a sharp drop at dusk starting at 18:57:50 and reaching 0.00 V at 19:25:16, whereas the potential of the biopile branch remains nearly stable at 2.50 V with a nominal current of 0.2 A throughout the recording, serving as the experimental open-circuit voltage E O C V reference evaluated in the biopile model.

The second empirical source consists of raw, high-frequency digital IoT sensor logs exported directly from the data acquisition hardware (INA219 current/voltage/power monitors), synchronized with local soil/microclimate instrumentation and hourly satellite-derived solar irradiance records (NASA-POWER, ALLSKY_SFC_SW_DWN). To maintain complete methodological transparency, the spatial and parametric alignment between the exported raw spreadsheet channels and the rendered dashboard charts is treated as logically inferred.

Figure 9. Design of plant-based biopile: 4S3P.

Figure 10. Design of plant-based biopile: 8S3P.

Figure 11. Design of plant-based biobatteries: 6S2P.

Figure 12. Sizing of plant-based biopiles: 12 S2P.

3.3.2. Model Performance Evaluation

The model evaluation is organized according to three distinct levels of evidential rigor in order to explicitly distinguish between mathematical consistency, parametric calibration, and structural validation.

First, the reduction test performed within the model verifies internal mathematical consistency by comparing the 1D spatial implementation to its global algebraic limit, without relying on empirical data.

Second, the unmeasured geometric and circuit parameters ( A e and R e x t ) are calibrated directly from specific empirical operating points; the agreement observed in these cases represents a parametric fit rather than an independent validation and is referred to as such throughout the document.

Third, the circuit topology and the model’s emerging diurnal dynamics are evaluated against the system’s nominal specifications and raw telemetry records, without tuning the parameters to these specific target outputs, providing a structural evaluation limited to topological scaling and transient behavior. The crucial point is that the experimental data used for parametric calibration are not subsequently reused to claim independent validation of the same output.

A point-by-point statistical comparison of the time series (e.g., RMSE, Pearson correlation coefficient r) between the transient simulation and the raw sensor recordings was deliberately avoided due to the stationarity of the signal. The voltage/current recordings, whether single-cell or multi-cell, remain stationary at the sensor resolution throughout the 5.79-hour monitoring window, which makes the correlation coefficient statistically degenerate relative to a reference signal with near-zero variance. Consequently, the model’s performance is evaluated using the absolute and relative biases between the time-averaged empirical operating points and the model’s predictions, thereby providing a physically meaningful measure of error for stationary field data.

3.3.3. Loaded Operating Point: Single-Cell Branch

The raw data from the INA219 digital sensor for the single-cell monitoring channel (N = 6418 samples from the 5.79-hour field session, from 15:08:29 to 20:55:00) shows steady-state behavior, with an average cell voltage of V = 1.5025 ± 0.0076 V (range: 1.48 - 1.53 V) and an average output current of I = 0.1961 ± 0.0193 A, resulting in an implied external load resistance of R e x t = V / I = 7.66   Ω . This empirical voltage (1.5025 V) corroborates the nominal upper limit of the per-cell potential (1.5 V) used in this study, based on an independent, directly recorded dataset rather than a value transcribed from specification tables.

Figure 13 illustrates this empirical trajectory as well as the model’s response. The measured voltage and current profiles exhibit quasi-stationarity over the entire 5.79-hour monitoring period, indicating that the branch operated under a fixed external load, without dynamic disturbances. Physically, this invariant response is consistent with a static resistive load connected to an electrochemical matrix whose biological and transport dynamics evolve on characteristic time scales of the order of days rather than hours. A representative model output ( A e = 0.5   m 2 , corresponding to the upper limit of the parameter scan range) is plotted to visually illustrate the basic electrical behavior. Crucially, this graphical comparison highlights the two-order-of-magnitude discrepancy between the predictions based on default parameters ( A e = 1.0 × 10 − 3   m 2 ) and the field measurements, thereby establishing the physical justification for the calibration sweep of the effective area.

Figure 13. Single-cell branch: empirical trajectory and model response.

The raw INA219 signal (blue, full 5.79-hour session) compared to the model prediction for R e x t = 7.66   Ω . and A e = 0.5   m 2 , (red dashed line, an illustrative data point from the A e sweep in the table below). The actual signal is virtually flat, given the instrument’s resolution; the model curve corresponds to one of the swept A e values, not to a fitted trajectory. For a default electrode area ( A e = 1.0 × 10 − 3   m 2 ) under a fixed load R e x t = 7.66   Ω ., the model predicts a quasi-steady-state output of I = 2.07 mA and V = 0.0158 V, underestimating the empirical operating point by two orders of magnitude due to an uncalibrated scaling of the internal resistance. To evaluate the model’s sensitivity and identify the effective active area required to reproduce field performance, a parametric sweep was performed while keeping all other kinetic and transport parameters constant (Table 4).

Table 4. Parametric scanning based on kinetic and transport parameters.

The voltage bias ( Δ V = V m o d e l − V m e a s u r e d ) transitions through zero between A e = 0.10   m 2 ( Δ V = − 0.43   V ) and A e = 0.20   m 2 ( Δ V = + 0.10   V ), indicating that an effective active area of A e ≈ 0.18   -   0.20   m 2 (corresponding to an approximate 40 cm × 45 cm buried electrode surface) reconciles the model prediction with the empirical operating point. This result is presented explicitly as a parametric calibration rather than an independent structural validation, as A e functions as a lumped scaling parameter that absorbs combined uncertainties from exchange current density ( j 0 ) and soil electrolyte conductivity ( σ w a t e r ). Across the 500-fold variation in A e ( 1.0 × 10 − 3   m 2 to 0.50 m²), the predicted current scales nonlinearly by a factor of ~144 (from 2.07 mA to 297.77 mA), adhering directly to the theoretical inverse ohmic scaling law R i n t ∝ 1 / A e and confirming that the underlying electrical formulation operates as derived regardless of the specific calibrated A e value.

3.3.4. Circuit Topology: Series/Parallel Stacking Law

Applying Kirchhoff’s circuit laws to a stack comprising n_s series-connected cells and n_p parallel branches yields the fundamental electrical scaling relation:

where u ∈ [ 0.5 , 1.5 ] V and i = 0.2   A represent the nominal per-cell operating bounds established during preliminary characterization. Equation (11) calculates the nominal topological scaling according to Kirchhoff’s laws, compared to the design specifications of the 4S3P, 8S3P, 6S2P, and 12S2P configurations. This allows for verification of the consistency of the data in Table 5, which shows the measured and verified operating points for the 6S2P configuration against the designed model.

Table 5.Comparison of energy output between the model and the physical system.

ND: Not Defined.

The 6S2P configuration data automatically recorded in the database during the production period made it possible to create Figure 14 below.

Figure 14. Experimental characterization of the electrical behavior of the biofuel cell_6S2P.

The maximum measured battery voltage corresponds precisely to n_s × 1.5 V for all four configurations, indicating that the individual cells are operating near their upper thermodynamic limit in open circuit. This result is independently corroborated by the measurement of the field strength of a single cell (1.50 V). The measured battery currents correspond to the theoretical model n_p × 0.2 A for all four configurations. For the 6S2P array, a current of 0.3 A is mathematically consistent with the measured power and voltage limits: 8.5 V × 0.3 A = 2.55 W exactly reproduces the documented range of 2.0 to 3.6 W. This demonstrates that Kirchhoff’s scaling law accurately describes the array topology.

3.3.5. Experimental Analysis and Dynamic Simulation of a Photovoltaic System and a Biopile

The Biopile_6S2P channel shows an average voltage of 7 ± 2 V and a constant current of 0.3000 A, both below the nominal limits of 9 V and 0.4 A. Channel 2 is a potentiometer combination, confirming the 6S2P topology. The single cell of the biopile and the PV branch were simulated, and their time stability is recorded in Table 6.

Table 6. Simulated temporal stability of a single biofuel cell and PV branch.

Figure 15. Dynamic divergence between the simulated photovoltaic voltage and the quasi-open-circuit potential of the biopile.

Figure 15 compares the simulated trajectory of a photovoltaic generator with the model’s near-open-circuit biofuel cell voltage. It highlights a qualitative difference in the simulated responses. The associated field recordings show a decrease in photovoltaic current and a near-steady biofuel cell voltage around 2.50 V, while the current tends towards zero. A stable voltage close to open circuit does not demonstrate available power under load after the drop in photovoltaic production; the data therefore only confirm voltage persistence, not a usable power supply complement.

Simulated day/night contrast: the PV model voltage drops to zero, while the modeled quasi-open-circuit biofuel cell voltage varies by approximately 0.46% over the interval (Table 6). This simulation does not demonstrate either the load output power or the available power of the biofuel cell.

The different response times result from the instantaneous relationship of the model’s PV diode and the slowness of biological and transport processes. This is a mechanistic deduction drawn from the model’s structure. Field telemetry shows voltage trends consistent with a qualitative day/night contrast, but because the biofuel cell channel is close to open circuit and its galvanic and microbial contributions are not dissociated, the measurements establish neither plant-induced causal complementarity nor usable power after sunset.

The mechanism governing this dynamic contrast is structural rather than artificial: V_PV. It is expressed as an instantaneous algebraic function of the solar flux via the open-circuit relationship of an ideal diode, whereas V_biopile depends on the local Nernstian equilibrium. In the model’s architecture, solar irradiance is coupled to the bioelectrochemical potential only indirectly, through a cascading sequence of biological and transport phenomena on a daily timescale (photosynthetic exudation → substrate diffusion → microbial kinetics and biofilm growth). These biochemical transitions are governed by reference kinetic rate constants ( μ m a x , r e f et μ b , r e f ) associated with characteristic relaxation times of the order of 10⁵ to 10⁶ s (approximately 1 to 10 days).

Over the simulated 5.73-hour interval, the model produces a rapid drop in photovoltaic voltage and a voltage variation in the biofuel cell of approximately 0.46% (Table 6). This behavior is simulated and does not constitute independent validation of the biofuel cell dynamics; therefore, the quasi-open voltage should not be interpreted as the power delivered.

The predicted quasi-open-circuit voltage of the biobattery (3.07 - 3.09 V) consistently exceeds the empirical quasi-open-circuit potential (2.50 V) by a difference of ΔV = 0.585 V. This potential difference stems from the ideal equilibrium formulation adopted in the model: the Nernst equation calculates an uncompensated thermodynamic ceiling that omits anode passivation phenomena at rest, parasitic losses at mixed potential, and non-ideal activity coefficients within the complex electrolytic matrix of the soil.

Figure 13 shows the empirical voltage and current recordings across the photovoltaic (PV) system on October 15, 2023, which serve as the direct experimental counterpart to the dynamic simulation trajectory. The telemetry data reveal two dominant physical regimes: sustained voltage oscillations in the afternoon, ranging from 15.5 to 20.0 V, followed by a rapid, nearly vertical drop in potential, which began at 6:57:50 p.m. and ceased at 7:25:16 p.m.

PV telemetry records voltage fluctuations in the afternoon, followed by a drop beginning around 18:57:50 (Figure 16). The sunset threshold used in the simulation was selected with reference to this observed transition. The comparison therefore documents the calibration basis of the forcing parameter; it is not an independent validation of the sunset timing or the PV model. Figure 16 establishes the empirical baseline against which the simulated temporal asymmetry (Figure 15) and the parameterized cutoff of solar forcing (sunset) are calibrated and structurally verified.

Figure 16. Experimental voltage and current measurements of the photovoltaic system.

The PV channel, which is the device’s PV monitoring channel, shows a voltage that fluctuates between approximately 15.5 and 20 V during the afternoon, then drops sharply starting at approximately 18:57:50, reaching 0 V around 19:25:16.

Figure 17. Experimental characterization of the electrical behavior of the biobattery.

Figure 17 shows the biofuel cell monitoring channel during the same session. The terminal voltage remains around 2.50 V while the current tends towards 0 A during the monitoring period (15:08:29 - 20:51:00), including the drop in photovoltaic voltage. This recording describes the persistence of a voltage close to that of an open circuit; it does not demonstrate the usable power of the biofuel cell after the drop in photovoltaic production.

Physically, this undisturbed profile confirms that the biobattery operated in a quasi-open-circuit condition (I ≈ 0 A), with the voltage measured across the terminals reflecting the actual, uncompensated bioelectrochemical equilibrium potential. Furthermore, it demonstrates that the characteristic relaxation times of the underlying biological and mass-transfer dynamics are sufficiently long such that a single diurnal solar transition does not induce any measurable disturbance in the cell’s potential. Finally, Figure 14 provides the main empirical evidence for day/night functional asymmetry, serving as a reference for establishing the structural—rather than parametric—validity of the differential-algebraic model formulation.

The biopile channel allows monitoring of the biopile in a physical device, showing a flat voltage of 2.5 V and a low current tending toward 0 A between 3:08 PM and 8:51 PM. This data constitutes major evidence of the qualitative day/night contrast, which is structurally reproduced by the simulation presented in this study.

3.3.6. Solar Forcing: Comparison between Model Assumptions, Actual Irradiance Data, and Data on the Transition to Darkness

The synthetic solar forcing profile was compared to hourly satellite irradiance, on-site measured photovoltaic voltage, and optical ADC recordings from October 15, 2023. The pyranometer was shaded after approximately 3:30 PM and therefore did not serve as a quantitative reference for irradiance. Because the satellite data are hourly, they cannot resolve sub-hourly transient variations. The sunset threshold was determined from the on-site observed photovoltaic transition and is a calibrated input, not an independent validation.

Figure 18. Comparison of satellite-measured solar irradiance and analytical cosine forcing.

Figure 18 compares the hourly satellite irradiance to the analytical forcing profile. Their evening transition times differ by approximately one hour: the satellite data indicate zero at 6:00 PM, while the forcing profile uses 6:57:40 PM. The hourly satellite data cannot resolve this sub-hourly transition. Since the model threshold was selected from the photovoltaic transition observed on site, this comparison describes a calibrated input data and not an independent validation.

The satellite irradiance record indicates a diurnal peak of 649.6 W∙m⁻² at 11:00:00, decaying through 267.9 W∙m⁻², 143.4 W∙m⁻², and 37.4 W∙m⁻² at 15:00:00, 16:00:00, and 17:00:00, respectively, before dropping to zero at the 18:00:00 bin. Interpreted in isolation, this satellite time-series places the darkness transition approximately one hour prior to the model’s parameterized sunset ( t s u n s e t ≈ 18.96   h , or 18:57:40).

The on-site photovoltaic voltage begins to decrease around 6:57:50 p.m., and the optical analog-to-digital converter’s reading reaches its lowest noise threshold between 7:23 p.m. and 7:30 p.m. These observations made it possible to determine the model’s sunset threshold at 6:57:40 p.m. Their agreement with this threshold constitutes calibration evidence, not an independent test. The spatial resolution of the zero irradiance interval at 6:00 p.m. in the satellite data is too coarse to assess the transition occurring in less than an hour.

On-site synchronized sensors provide the empirical basis used to select the sunset threshold. Since these same observations influenced the choice of boundary conditions, they cannot independently validate it. Hourly satellite data provide only a coarser contextual comparison.