esa-sceva/satcom-chunk-collection huggingface dataset:
# State Estimation for DC Microgrids using Modified Long Short-Term Memory Networks
## 2 Conventional WLS SE Method  
SE is an algorithm used to determine a close approximation of the system's actual state from raw measurement data that is corrupted by noise. The WLS method is one of the most common methods adopted for SE [1; 12]. The weight in the WLS method is introduced into the algorithm to emphasize trusted measurements and de-emphasize less-trusted measurements. The WLS method is based on a nonlinear optimization problem that minimizes the squared error of the measurement while taking into account the error variance of different measurement types. The general measurement model for SE is written as  
\[z=h(x)+e, \tag{1}\]  
where \(z\) is the measurement vector, \(x\) is the true state vector, \(h(x)\) is the function linking the measurement to the state, and \(e\) is the measurement error, which is assumed to have zero mean and variance \(\sigma_{N}^{2}\). The complex nodal voltage is a commonly used variable in SE [1]. The true state vector \(x\) is written as  
\[x=\left[\begin{array}{c}V_{N}\\ \theta_{N}\end{array}\right], \tag{2}\]where \(V_{N}\) is the voltage magnitude and \(\theta_{N}\) is the voltage angle.  
The WLS method calculates the system's state by iteratively minimizing the cost function  
\[J=\sum_{i=1}^{N}\frac{(z_{i}-h_{i}(x))^{2}}{\sigma_{i}^{2}}=[z-h(x)]^{T}\cdot W \cdot[z-h(x)], \tag{3}\]  
where \(\sigma_{i}^{2}\) is the error variance and \(N\) is the number of measurements [1, 12, 15]. The weighting matrix \(W\) is equal to the inverse of the error variance matrix \(R\in\mathbb{R}^{N\times N}\). The Gauss-Newton algorithm is used to minimize the cost function (Equation (3)). The algorithm is calculated iteratively, resulting in the normal equation being written as  
\[G\Delta x=H^{T}\cdot W\cdot(z-h(x)), \tag{4}\]  
where \(H\) is the Jacobian matrix of the measurement function \(h(x)\). The gain matrix \(G\) in Equation (4) is defined as \(G=H^{T}\cdot W\cdot H\). The state estimate \(x^{k+1}\) is calculated using the vector \(\Delta x\) for each iteration \(k\) by the following equation, which is written as  
\[x^{k+1}=x^{k}+\Delta x. \tag{5}\]  
The iteration process will stop and give the state estimate result when \(\Delta x^{T}\Delta x<\epsilon\) is reached, where \(\epsilon\) is the desired accuracy.
## 4 Performance Evaluation and Results
### Simulation Results  
The simulation results, which are presented in Table 1, confirm the efficiency of the PPS algorithm. The simulation, which was conducted by progressively reducing the stride from its largest value of 10-6 (same mechanism as that of the ordinary algorithm) to 2-2, showed a gradual increase in the matching rate. Although the decrease in \(C_{\text{pk}}\) may appear significant, the decrease in the adjusted matching rate, which is a statistical representation of Cpk, was not significant compared with the increase in the matching rate. Additionally, the computation time for the 2-2 stride decreased to one-fifth of the initial value, demonstrating that PPS effectively improves the matching rate, computation time, and scalability in multiple-slot row structures.  
Figure 5 shows how the clearance distribution of the assembled parts varies depending on the simulation environment. Compared to the ordinal algorithm, after applying PPS, the distribution of the center of design tolerance decreases, but the distribution near the center increases. Also, it was omitted because the figure is duplicated, but if you want to check the distribution change within the entire slot, you can refer to Figure 4a,b.
## 4 Simulation
### Impact of Node Density  
The connectivity of DTMSN is closely related to the density of sensor nodes. The following experiments show the network performance of the four protocols with different sensor node density.  
As shown in Figure 5(a), the epidemic protocol almost achieves the upper bound of the data delivery ratio when the node density is very low, since low node density means low transmission load and a small amount of wireless collisions. As the node density increases, the number of message copies increases dramatically in epidemic routing, which results in an increasing number of collisions and the reduction of the data delivery ratio. FAD shows slightly better data delivery ratio than MEDR when the node density is very low. This is due to poor connectivity resulting from very low node density, which influences the performance of MEDR. With the increment of node density, the connectivity of the network is enhanced, and thus the performance of MEDR improves rapidly. When the node number reaches 90, MEDR outperforms the other three protocols. As far as the performance of FAD is concerned, when the node density is very low, it's difficult for a node to meet another node with higher  
Figure 4: Impact of message generation ratio. **(a)** Average delivery ratio; **(b)** Average delay.  
Figure 5: Impact of node density. **(a)** Average delivery ratio; **(b)** Average delay.  
delivery probability to help forward messages; when the node density is high, FAD will generate large numbers of message copies, which expends the limited resources of bandwidth and buffer quickly. Thus the performance of FAD descends under the two cases above. In direct transmission scheme, mobile sensors just communicate with the sink node directly, so the performance of this protocol has almost nothing to do with the node density.
## 3 Mathematical Modeling
### Multi-Objective Optimization  
In engineering and scientific research, there are a large number of problems that need to be optimized and solved for more than one target at the same time, which is defined as multi-objective optimization. The optimal solution of a multi-objective optimization problem is often not unique, and the relationship between multiple optimization objectives is often conflicting, which means the optimal solutions of each objective cannot be obtained at the same time and the solutions are non-dominated for each other. To acquire the non-dominated solution, which is also defined as the Pareto frontier solution, the multi-objective optimization is carried out by using the nondominated sorting genetic algorithm-II (NSGA-II) method, first proposed by [PERSON] et al. (2018). NSGA-II is developed on NSGA and has been enhanced by decreasing the computation complexity of the nondominated sorting, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Component** & **Investment Cost/USD** & **Original Year** & **CEPCI** \\ \hline Compressor & \(\frac{71.1+\text{m}_{\text{C}}}{0.9-\text{m}_{\text{C}}}\cdot\frac{p_{\text{C} }}{17}\cdot ln\frac{p_{\text{C}}}{17}\) & 1996 & 381.7 \\ Turbine & \(\frac{479.34+\text{m}_{\text{T}}}{0.92-\text{m}_{\text{T}}}\cdot ln\frac{p_{ \text{C}}}{P_{\text{C}}}\cdot(1+exp(0.036\cdot T_{\text{i}}-54.4))\) & 1996 & 381.7 \\ Heat exchangers & \(Z_{\text{ref}}\cdot\left(\frac{A}{100}\right)^{0.6}\) & 2000 & 394.1 \\ Valve & \(114.5\cdot\text{m}_{\text{val}}\) & 2000 & 394.1 \\ Thermal storage tanks & \(600\cdot V_{\text{tan}k}^{0.78}+8000\) & 2009 & 521.9 \\ GST/LST & \(4042\cdot V_{\text{tan}k}^{0.506}\) & 2013 & 567.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Investment cost of each system component [22; 45; 47; 48; 49; 51]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Unit** & **Value** \\ \hline Electricity cost during charge period (\(c_{\text{wc}}\)) & USD/kWh & 0.05 \\ Maintenance factor & - & 1.06 \\ Interest rate & - & 0.12 \\ System life span & year & 20 \\ Density of molten salt & kg/m\({}^{3}\) & \(2090-0.636\cdot t(^{\circ}\text{C})\) \\ Heat transfer coefficient & W/(m\({}^{2}\cdot\)K) & Condenser: 2000 \\  & Evaporator: 1500 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Major input data applied in the proposed system [36; 45; 49; 52].
## 3 Modelling Framework
### The eXtended Finite Element Method (XFEM)  
The extended finite element method (XFEM) is an extension of the conventional finite element method based on the concept of partition of unity, which allows local enrichment functions to be easily incorporated into a finite element approximation. The cracks are represented by enrichment functions, which are added to the normal displacement interpolation function, see Equation (1) [21]. That allows the cracks to be represented within mesh elements without remeshing during the simulation. The XFEM has been implemented into  
Figure 2: The geometrical representation of the critical capsular clustering patterns ranging from (**a**) non-clustered to (**g**) fully clustered.  
commercial software such as Abaqus and has become one of the most popular methods for fracture [16].  
\[\mathrm{u}=\sum_{\mathrm{I}\in\mathrm{N}}\mathrm{N_{I}(x)}\left[\mathbf{u_{I}}+ \mathrm{H(x)}\mathbf{a_{I}}+\sum_{\alpha=1}^{4}\mathrm{F_{\alpha}(x)}\mathbf{b_{ I}^{\alpha}}\right] \tag{1}\]  
\(\mathrm{N_{I}(x)}\) are shape functions, \(\mathrm{u_{I}}\) is a nodal displacement vector, \(\mathrm{H(x)}\) are jump functions to represent the cracks, \(\mathrm{a_{I}}\) and \(\mathrm{b_{I}}^{\alpha}\) are nodal vectors of the enriched degree of freedom, and \(\mathrm{F_{\alpha}(x)}\) are crack-tip functions.
# Hierarchical Model Updating Method for Vector Electric-Propulsion Satellites
## 4 Results and Discussion
### The Result of Hierarchical Model Updating
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Variable**} & \multirow{2}{*}{**Real**} & \multicolumn{3}{c}{**The First Loop**} & \multicolumn{3}{c}{**The Second Loop**} \\ \cline{3-8}  & & **Estimated** & **Error** & **Relative Error** & **Estimated** & **Error** & **Relative Error** \\ \hline \(r_{x0}\), m & 7,027,732.4 & 7,027,728.25 & 4.14 & 0.00\% & 7,027,732.94 & 0.55 & 0.00\% \\ \(r_{y0}\), m & 61,327.76 & 61,324.52 & 3.23 & 0.26\% & 61,328.61 & 0.86 & 0.07\% \\ \(r_{z0}\), m & 535.20 & 531.00 & 4.20 & 39.26\% & 535.61 & 0.41 & 3.80\% \\ \(v_{x0}\), m/s & \(-65.72\) & \(-65.71\) & 0.01 & 0.43\% & \(-65.72\) & 0.00 & 0.06\% \\ \(v_{y0}\), m/s & 7530.43 & 7530.43 & 0.00 & 0.00\% & 7530.43 & 0.00 & 0.00\% \\ \(v_{z0}\), m/s & 65.72 & 65.72 & 0.00 & 0.01\% & 65.72 & 0.00 & 0.01\% \\ \(\mu_{GPS_{r}}\), m & 5.00 & 9.22 & 4.22 & 42.15\% & 4.62 & 0.38 & 3.85\% \\ \(\sigma_{GPS_{r}}\), m & 5.00 & 5.00 & 0.00 & 0.01\% & 5.00 & 0.00 & 0.02\% \\ \(\mu_{GPS_{r}}\), m/s & 0.50 & 0.50 & 0.00 & 0.00\% & 0.50 & 0.00 & 0.00\% \\ \(\sigma_{GPS_{r}}\), m/s & 0.50 & 0.50 & 0.00 & 0.00\% & 0.50 & 0.00 & 0.00\% \\ \hline RMSE & & & 0.00\% & & & 0.00\% & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The estimation of orbit state variables. Figure 3: The variation in state variables estimation relative error and telemetry RMSE. **(a)** Estimation of \(r_{0}\); **(b)** Estimation of \(v_{0}\); **(c)** Estimation of GPS errors; **(d)** RMSE of orbit telemetry variables. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Variable**} & \multicolumn{3}{c}{**RMSE**} \\ \cline{2-4}  & **Uncorrected** & **The First Loop** & **The Second Loop** \\ \hline \(\varphi_{m}\) & 15.372\% & 0.109\% & 0.0278\% \\ \(\theta_{m}\) & 9.215\% & 0.132\% & 0.0543\% \\ \(\psi_{m}\) & 13.216\% & 0.0874\% & 0.0573\% \\ \(\omega_{xGyno}\) & 12.027\% & 0.0529\% & 0.00999\% \\ \(\omega_{yGyro}\) & 5.157\% & 0.0548\% & 0.0102\% \\ \(\omega_{zGyno}\) & 15.192\% & 0.0594\% & 0.0272\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The RMSE of attitude telemetry variables. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Variable**} & \multirow{2}{*}{**Real**} & \multicolumn{3}{c}{**The First Loop**} & \multicolumn{3}{c}{**The Second Loop**} \\ \cline{3-8}  & & **Estimated** & **Error** & **Relative Error** & **Estimated** & **Error** & **Relative Error** \\ \hline \(\theta_{wing_{1},0,\,\,}^{\circ}\) & 90 & 89.14 & 0.86 & 4.29\% & 89.07 & 0.93 & 4.64\% \\ \(\theta_{wing_{2},0,\,\,}^{\circ}\) & 90 & 89.14 & 0.86 & 4.29\% & 89.07 & 0.93 & 4.64\% \\ \(\eta_{w}\) & 0.15 & 0.15 & 0.00 & 0.00\% & 0.15 & 0.00 & 0.00\% \\ \(k_{Canye}\) & 1 & 1.00 & 0.00 & 0.05\% & 1.00 & 0.00 & 0.02\% \\ \hline RMSE & & & 0.121311\% & & & 0.051712\% \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The estimation of power state variables.
## 3 Results and Analysis
### Spatial Distribution Characteristics of Soil Nutrients
#### 3.2.2 Soil Nutrient Spatial Distribution Status  
Having determined the most suitable semivariogram function model along with its parameters, we utilized the Kriging interpolation method to draft a spatial distribution map of soil nutrients within Fengtai County, as depicted in Figure 5. Surveying all seven indicators of soil nutrient content, it becomes apparent that Fengtai County exhibits a marked spatial differentiation in its soil nutrients. Some areas prominently showcase a block-like distribution, emphasizing the spatial variability within the county. Specifically, for soil pH, the northwest regions have an acidic disposition, the central regions lean towards being acidic, and the southeastern zones generally tend towards alkalinity. Organic matter, total nitrogen, and alkali-hydrolyzable nitrogen have largely synchronous distribution patterns. Nutrient-rich sectors are predominantly in the central-western zones, whereas the northeastern areas are nutrient-scarce. We also observe sporadic pockets of lower values across the study landscape. Areas with high concentrations of available phosphorus are sporadically situated in the central-northeast zone, with some traces on the western boundary. In contrast, the southern regions predominantly showcase low-value zones. The distribution of rapid-acting potassium is notably fragmented over limited stretches. The northwest and southeast harbor the low-concentration zones, while the high-concentration areas are primarily interspersed in the northeast and southwest. A general trend can be seen where the slow-acting potassium content rises from the northern to the southern regions. However, an unmistakable high-value triangular zone dominates the central area, with a conspicuous low-value tract stretching from the southwest to the northeast.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & & & & & \\ \cline{3-8}
**Index** & **Model** & \multirow{2}{*}{**Mean**} & \multirow{2}{*}{**Root Mean**Square**} & \multirow{2}{*}{**Standardized**} & \multirow{2}{*}{**Root Mean**Square**} & \multirow{2}{*}{**Average**} \\  & & & & & & **Standardized** \\ \hline \multirow{3}{*}{Available K} & Spherical & 0.00315 & 45.63573 & 0.00331 & 0.99972 & 45.06263 \\  & Exponential & 0.61080 & 46.35305 & 0.01352 & 1.01579 & 45.35491 \\  & Gaussian & \(-\)0.02717 & 45.92718 & 0.00309 & 1.02823 & 44.09285 \\ \hline \multirow{3}{*}{Slowly released K} & Spherical & \(-\)0.09869 & 57.64919 & 0.00093 & 1.00379 & 57.30993 \\  & Exponential & 0.08808 & 56.98518 & 0.00298 & 0.98078 & 57.87014 \\ \cline{1-1}  & Gaussian & \(-\)0.60303 & 57.67448 & \(-\)0.00651 & 0.99826 & 57.64336 \\ \hline \hline \end{tabular}
\end{table}
Table 3: _Cont._
# Hybrid Model-Based Analysis of Underground Articulated Vehicles Steering Characteristics
## 3 Validation of Dynamic Model  
### Field Test Setup  
The field test UAV was a remote control (load-haul-dump vehicle; LHD) with a capacity of 4 m\({}^{3}\) in; the test system is shown in Figure 6. All the sensors and modules were commercial products, and we only designed the installation frameworks. A serial port was used to connect with the remote control to record the signal of driving and steering, which was used as the input of the simulation model. An LMS SCADAS (\(\lx@sectionsign\)1, SCM05) was used to record the data during the field test, and the sampling frequency was set as 100 Hz [23]. Two inertial measurement units (IMUs) (\(\lx@sectionsign\)2, \(\lx@sectionsign\)3, SC-AHRS-200A) were installed in the front and rear frames, respectively. Two kinds of encoders were used to measure the articulated angle of the frame (\(\lx@sectionsign\)4, MIRAN WOA-C) and the speed of transmission (\(\lx@sectionsign\)5, SCHMEASAL IFL 5-18M-10P), respectively. A gyroscope (\(\lx@sectionsign\)6, Biewei) with high precision was installed to provide the reference values. In addition, two pressure sensors (\(\lx@sectionsign\)2, AE-H) were installed near the rodless and rod cavities of left cylinder, and two displacement sensors (\(\lx@sectionsign\)8, WFS-100-P15-11R5) were installed at the two steering cylinders, respectively.  
Usually, characteristics of the route for an LHD are special in underground mines, and always include a square turn and change of operation direction. Thus, the route of the field test was planned as shown in Figure 7, where the direction of the arrow represents the direction of the LHD and the solid and hollow lines correspond to the forward and backward directions, respectively. The key dimensions were L1 = 50 m, L2 = 20 m. The road of the whole field test site was uneven, and the uncertainty of the road surface would have made it difficult to verify the model during the whole test process, therefore,  
Figure 5: Hybrid model relationship of hydraulic steering system and frames.  
we intercepted a piece of typical test data in which the speed of vehicle was constant and the road was relatively flat to verify the model.
# A New Method for Haul Road Design in Open-Pit Mines to Support Efficient Truck Haulage Operations
## 4 Case Study
### Optimization and Modification of Road Layout  
Figure 13 shows the result of analyzing the 2D haul road layout with a slope limit of 10\({}^{\circ}\) (approximately 18%) through LCPA. It shows the haul road centerline, which connects the work sites at the bottom of Pit 1 and Pit 2 to the entrances of the waste dump and the stockpile, respectively. The haul road interconnects the lower work site of the mine and the upper earth surface in a spiral shape. The two haul roads departing from the lower work site of Pit 1 and Pit 2, respectively, are combined at one point near the earth surface and share the same path. A zigzag-shaped path was analyzed due to the limitation of the raster-based LCPA method.  
Figure 14 shows the simplification of a zigzag-shaped haul road layout using the Douglas-Peucker algorithm. The tolerance used in the simplification work was set as 5 m, which is identical to the resolution of the DEM.  
The radius of curvature was calculated using Equation (4). The maximum speed when the truck runs on a downward slope without a load was set at 40 km/h, as in Table 3. Furthermore, the super-elevation was assumed to be 4% (approximately 2\({}^{\circ}\)), and the friction coefficient was set at 0.3, assuming a sandy road surface. The substitution of the parameter values in the equation enabled us to calculate the radius of curvature as 37.6 m. Figure 15 shows the haul road layout modified by considering the radius of curvature.  
Figure 13: Haul road layout in the Pit 1 and Pit 2 areas analyzed by LCPA. The paths connect the loading point in Pit 1 with (**a**) the stockpile and (**b**) the waste dump and the loading point in Pit 2 with (**c**) the stockpile and (**d**) the waste dump.
## 4 Performance Evaluation
### Results of Knowledge Transfer between the Apartment Buildings
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Task**} & \multirow{2}{*}{**CNN**} & \multicolumn{5}{c}{**\(t\) (s)**} \\ \cline{3-8}  & & **0.152** & **0.501** & **1.00** & **1.50** & **2.00** & **3.00** \\ \hline \(\mathcal{T}_{\textbf{LAPT II}|\textbf{APT I}}\) & VGG16 & 0.5778 & 0.6244 & 0.5865 & 0.5952 & **0.6617** & 0.2238 \\  & & (0.8273) & (0.8520) & (0.8178) & (0.8356) & **(0.8800)** & (0.3044) \\  & SoundNet & 0.2557 & 0.2318 & 0.5789 & 0.5636 & 0.5421 & 0.4995 \\  & & (0.3727) & (0.3274) & (0.7838) & (0.7661) & (0.7848) & (0.7396) \\ \cline{2-8} \(\mathcal{T}_{\textbf{LAPT I}|\textbf{APT I}}\) & VGG16 & 0.5296 & 0.7522 & 0.7482 & 0.7682 & 0.6122 & **0.7684** \\  & & (0.7285) & (0.8046) & (0.7957) & (0.7970) & (0.7656) & **(0.8093)** \\  & SoundNet & 0.2048 & 0.2196 & 0.6009 & 0.5196 & 0.5691 & 0.4876 \\  & & (0.3231) & (0.3385) & (0.6537) & (0.5977) & (0.6481) & (0.5674) \\ \cline{2-8} \(\mathcal{T}_{\textbf{p,APT II}|\textbf{APT I}}\) & VGG16 & **0.8037** & 0.7842 & 0.7384 & 0.6786 & 0.6976 & 0.5678 \\  & SoundNet & 0.4528 & 0.4741 & 0.5747 & 0.6941 & 0.7943 & 0.6443 \\ \cline{2-8} \(\mathcal{T}_{\textbf{p,APT I}|\textbf{APT I}}\) & VGG16 & **0.8903** & 0.8309 & 0.7654 & 0.7057 & 0.6850 & 0.6006 \\  & SoundNet & 0.2171 & 0.2969 & 0.7650 & 0.7513 & 0.7094 & 0.7290 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Inter-floor noise classification knowledge transfer results with \(t\) variation. The values are F1 score of the source type knowledge transfer results. The values in the parentheses are the F1 scores when the HH and HD categories are assumed to be the same category. The underlined values are rearranged localization knowledge transfer results to their corresponding floors. Figure 6: Results of inter-floor classification and classification knowledge-transfer tasks evaluated using F1 score. HD and HH were considered the same category for source type classification results. (**a**) Cross-validation and test results of the source type classification tasks in Table 3. (**b**) Cross-validation and test results of localization tasks in Table 4. (**c**) Rearranged localization results to their corresponding floor. (**d**) Results of the knowledge transfer tasks in Table 5.
# Analytical Modeling of the Maximum Power Point with Series Resistance
## 5 Numerical Results
Finally, it is worth mentioning that the overlap between \(V_{\text{mpp}}^{\text{mod}}\) and \(V_{\text{mpp}}^{\text{ref}}\) is particularly good with the perovskite (PSC) and the amorphous silicon (a-Si) solar cells due to their large bandgaps. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Device** & \(E_{\text{g}}\) **(eV)** & \(V_{\text{oc}}\) **(V)**1  & **ERE (\%)** \\ \hline InP & 1.34 & 0.939 & 31.15 & 0.365 \\ GaAs & 1.42 & 1.107 & 29.60 & 14.510 \\ CdTe & 1.51 & 0.876 & 30.25 & 10\({}^{-4}\) \\ CIGS & 1.08 & 0.734 & 39.58 & 1.750 \\ a-Si & 1.69 & 0.896 & 16.36 & 1.96 \(\times 10^{-7}\) \\ PSC 2  & 1.60 & 1.042 & 20.40 & 0.002 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for selected single-junction solar cell technologies. The ERE values were estimated from Equation (27). Figure 4 displays the base-10 logarithm of the relative discrepancy (in %) between \(P_{\text{mpp}}^{\text{mod}}\) and \(P_{\text{mpp}}^{\text{ref}}\) as a function of the bandgap energy and the ERE for \(r=2\)\(\Omega\cdot\text{cm}^{2}\). The relative discrepancy between \(P_{\text{mpp}}^{\text{mod}}\) and \(P_{\text{mpp}}^{\text{ref}}\) for the six solar cell technologies presented in Table 1 is also shown. The white dotted lines represent levels of fixed relative discrepancy. To compute this figure, the ERE values of the solar cells presented in Table 1 were estimated by making use of:  
\[\text{ERE}=\exp\left[\frac{V_{\text{oc}}-V_{\text{oc}}^{\text{rad}}}{V_{\text{ t}}}\right], \tag{27}\]  
where \(V_{\text{oc}}^{\text{rad}}\) can be calculated from Equation (1) by assuming that \(i_{0}\) results only from radiative recombination. Figure 4 shows that, for a given series resistance, the accuracy of Equation (15) increases with increasing bandgap and decreases with decreasing ERE. Note that the \(P_{\text{mpp}}\) of all the investigated cases was predicted with a discrepancy below 0.07%. Finally, Figure 5 displays the base-10 logarithm of the relative discrepancy (in %) between \(P_{\text{mpp}}^{\text{ref}}\) and \(P_{\text{mpp}}^{\text{mod}}\) (Figure 5a) and between \(P_{\text{mpp}}^{\text{ref}}\) and \(P_{\text{mpp}}^{\text{app}}\) (Figure 5b) as a function of the series resistance for the six devices presented in Table 1. Here, it can be seen that the discrepancy between \(P_{\text{mpp}}^{\text{ref}}\) and \(P_{\text{mpp}}^{\text{app}}\) is around one order of magnitude larger than between \(P_{\text{mpp}}^{\text{mod}}\) and \(P_{\text{mpp}}^{\text{ref}}\). \(P_{\text{mpp}}^{\text{app}}\) could still predict \(P_{\text{mpp}}\) in most of the studied cases with errors below 1% for values of the series resistance up to 5 \(\Omega\cdot\text{cm}^{2}\).