Adaptive Fuzzy–Differential Reliability Framework for PV–ESS–SLC Energy Systems Using Gaussian Fuzzy DEMATEL

Adaptive Fuzzy Differential Framework for Reliability-Aware Energy Efficiency Control in PV– ESS–SLC Systems Author: Rana Amirova Azerbaijan Technical University [email protected], Baku, Azerbaijan Email: [email protected].az ORCID: https://orcid.org/0009-0005-6412-4736 Abstract This work examines the behaviour and reliability of a PV–ESS–SLC energy system by combining a fuzzy extension of its Boolean structure with a reliability-weighted DEMATEL analysis. The fuzzy model allows the system to be evaluated under partial operation, uncertainty and varying energy conditions. Reliability indicators for the PV unit, storage system and load controller are derived from fuzzy min–max rules and then embedded into a Gaussian fuzzy assessment of component interactions. The resulting weighted DEMATEL model highlights which elements stabilize the system and which ones remain sensitive to fluctuations. The approach offers a practical and interpretable way of analyzing reliability in modern smart-energy setups. Keywords: Fuzzy Boolean model, reliability-aware ccontrol, Gaussian Fuzzy DEMATEL, PV–ESS– SLC Energy System, Adaptive Fuzzy Differential Framework 1. Introduction Energy systems built on solar generation, battery storage and digital load management behave in a highly dynamic environment where sunlight, storage condition and consumption patterns constantly change. Because of this variability, decision mechanisms based solely on fixed thresholds or crisp logic often fall short in representing the true behaviour of the system. To address these limitations, this study develops a modelling approach that starts with a Boolean description of the PV–ESS–SLC structure and expands it into a fuzzy framework that can represent partial availability, uncertainty and intermediate states. Using this fuzzy model, reliability indicators are formed for each main component of the system. These indicators are then combined with a Gaussian fuzzy representation of component-to-component influence, producing a reliability-weighted version of the DEMATEL method. This allows us to observe how strongly each block affects the others and how their reliability shapes the overall stability of the system. The proposed framework provides a clear and compact way to interpret system behaviour without relying on heavy assumptions, making it suitable for practical energy-management scenarios. 2. Methodology Fuzzy Extension of the Boolean PV–ESS–SLC Model To move from a crisp on/off description toward an uncertainty-aware control structure, we extend the previous Boolean model of the PV–ESS–SLC energy system to a fuzzy framework [5]. In the Boolean formulation, the variables: • x - Solar PV • y - Energy Storage System (ESS), • z - Smart Load Controller (SLC) take values in {0,1} and the global system function is 𝐹 (𝑥,𝑦,𝑧) =𝑥+(𝑥⋅𝑦) +( 𝑥⋅𝑦⋅𝑧) This expression encodes three main operating modes: • xfull operation with active solar generation (optimal mode), • (𝑥⋅𝑦)- PV inactive but ESS supplying the system, • ( 𝑥⋅𝑦⋅𝑧)-minimal operation where only the load controller is active and non-critical loads are isolated. [5-7] 2.1. Fuzzification of System Variables In the fuzzy model, we interpret 𝑥,𝑦,𝑧 as degrees of availability or activation rather than binary states: • 𝑥∈ [0,1]: normalized availability of solar generation (e.g., based on irradiance and PV output), • 𝑦∈[0,1]: normalized availability of the ESS (e.g., state of charge and health), • 𝑧∈[0,1]: degree of activation/authority of the smart load management system. Thus, instead of a crisp output 𝐹∈{0,1}, we define a fuzzy system performance degree 𝜇𝐹(𝑥,𝑦,𝑧)∈[0,1] which reflects how well the energy system is able to supply and manage loads under uncertain and partial conditions. We adopt standard fuzzy connectives: • Fuzzy negation: 𝑁(𝑎)=1−𝑎 • Fuzzy AND (t-norm): T (a, b) =min (𝑎,𝑏), • Fuzzy OR (s-norm): S (a, b) =max (𝑎,𝑏) With these operators, the Boolean structure is lifted to the fuzzy domain as 𝜇𝐹(𝑥,𝑦,𝑧)=𝑆(𝑥, 𝑇(𝑁(𝑥), 𝑦), 𝑇(𝑁(𝑥), 𝑁(𝑦), 𝑧) 𝜇𝐹(𝑥,𝑦,𝑧)=max (𝑥,min(1−𝑥,𝑦),min (1−𝑥,1−𝑦,𝑧) Interpretation of the three fuzzy terms: • 𝜇𝑃𝑉=𝑥: degree to which the system operates in full PV-driven mode; • 𝜇𝐸𝑆𝑆=min(1−𝑥,𝑦) degree to which the system operates in ESS-driven backup mode when PV is weak or absent; • 𝜇𝐿𝐶=min(1−𝑥,1−𝑦,𝑧) degree to which the system operates in minimal, load-controlled survival mode, where non-critical loads are isolated. The overall fuzzy operation level is the maximum of these three contributions. 2.2. Fuzzy Min–Operator Analysis with Respect to Each Variable To identify which blocks are structurally critical for guaranteeing at least a minimal level of service, we extend the Boolean min-operator analysis to the fuzzy case. Following the Boolean idea, the “minimum with respect to a variable” is defined as the t-norm aggregation over its extreme values. For a variable 𝑣, we write 𝑚𝑖𝑛𝑣𝜇𝐹=𝑇(𝜇𝐹(𝑣=1,.),𝜇𝐹(𝑣=0,.)) Fuzzy 𝑚𝑖𝑛𝑣𝜇𝐹(𝑥,𝑦,𝑧): We first evaluate the system for 𝑥=1 and 𝑥=0. If PV is fully available (𝑥= 1)⇒ 𝜇𝐹(1,𝑦,𝑧)=𝑚𝑎𝑥 (1,𝑚𝑖𝑛(0,𝑦),𝑚𝑖𝑛 (0,1−𝑦,𝑧)=1.İf PV is completely unavailable (𝑥= 0)⇒𝜇𝐹(0,𝑦,𝑧)=𝑚𝑎𝑥 (0,𝑚𝑖𝑛(1,𝑦),𝑚𝑖𝑛 (1,1−𝑦,𝑧))=𝑚𝑎𝑥 (𝑦,𝑧) Using the fuzzy AND 𝑇=𝑚𝑖𝑛, we obtain⇒ 𝑚𝑖𝑛𝑥𝜇𝐹(𝑥,𝑦,𝑧)=min (𝜇𝐹(1,𝑦,𝑧),𝜇𝐹(0,𝑦,𝑧))= max (𝑦,𝑧) Even if the PV block is completely absent, the minimal guaranteed performance of the system is determined by the combined availability of ESS and the Load Controller. The fuzzy quantity max (𝑦,𝑧) indicates that either a sufficiently charged ESS or an active load controller can maintain a non-zero level of service. Fuzzy 𝑚𝑖𝑛𝑦𝜇𝐹(𝑥,𝑦,𝑧):Analogously, we evaluate for 𝑦=1 and 𝑦=0. If ESS is fully available (𝑦= 1), the system can always compensate for missing PV or rely on load management, giving 𝜇𝑦(𝑥,1,𝑧)= 𝑚𝑎𝑥 (𝑥,𝑚𝑖𝑛(1−𝑥,1),𝑚𝑖𝑛(1−𝑥,0,𝑧)=1. If ESS is completely unavailable (y=0), the system relies on PV or the Load Controller, leading to 𝜇𝑦(𝑥,0,𝑧)=𝑚𝑎𝑥 (𝑥,𝑚𝑖𝑛(1−𝑥,0),𝑚𝑖𝑛(1− 𝑥,1−0,𝑧)=𝑚𝑎𝑥 (𝑥,𝑧). The minimal guaranteed performance with respect to ESS is governed by PV availability and the Load Controller. Even if the ESS is absent, the system still operates in a fuzzy sense as long as PV or the Load Controller is present to some degree. Figure1. The 3D surface plot of the system performance 𝜇𝐹(𝑥,𝑦,𝑧)) as a function of PV (x) and ESS (y) activity levels, assuming a fully active Load Controller (z = 1). Fuzzy 𝑚𝑖𝑛𝑧𝜇𝐹(𝑥,𝑦,𝑧):For the Load Controller we obtain μz(x,y,1)=max (x,min(1−x,y),min(1−x,1−y,1)≥max (x,y) μz(x,y,0)=max (x,min(1−x,y),min(1−x,1−y,z)≥max (x,y) If either PV or ESS is available to a sufficient degree, the system does not critically depend on the Load Controller to maintain a basic energy supply. The Load Controller becomes more important in scenarios where both PV and ESS are weak or uncertain. 2.3. Fuzzy Max–Operator Analysis For completeness, we also define the fuzzy maximum with respect to a variable using 𝑡ℎ𝑒 𝑠− 𝑛𝑜𝑟𝑚 𝑆=𝑚𝑎𝑥: 𝑚𝑎𝑥𝑣𝜇𝐹(𝑥,𝑦,𝑧)=𝑆(𝜇𝐹(𝑣=1,.),𝜇𝐹(𝑣=0,.)) Because 𝜇𝐹(𝑣=1,.)=1, for each of the three variables, we obtain 𝑚𝑎𝑥𝑥𝜇𝐹(𝑥,𝑦,𝑧)=𝑚𝑎𝑥𝑦𝜇𝐹(𝑥,𝑦,𝑧)=𝑚𝑎𝑥𝑧𝜇𝐹(𝑥,𝑦,𝑧)=1 For each block (PV, ESS, Load Controller), there exists an operational configuration where this block is fully active and the others are at least partially available, yielding maximum system performance. This is consistent with the stepwise security principle: each block can contribute to a fully operational mode when it is highly available. 2.4. Role of the Fuzzy Model for Adaptive Control This fuzzy lifting of the Boolean model provides: 1. A quantitative measure of system performance 𝜇𝐹(𝑥,𝑦,𝑧) under partial and uncertain activation of PV, ESS and the Load Controller; 2. Fuzzy min– and max–based indicators 𝑚𝑖𝑛𝑣𝜇𝐹, 𝑚𝑎𝑥𝑥𝜇𝐹(𝑥,𝑦,𝑧 that quantify how critical each variable is for maintaining minimal or maximal operation; 3. A natural interface to an adaptive fuzzy differential controller and ANFIS-based learning, where 𝑥,𝑦,𝑧become fuzzy state variables and 𝜇𝐹 is used as a reliability-aware performance index. This structure therefore forms an ideal bridge from a simple Boolean safety logic to an adaptive, uncertainty-aware fuzzy control framework for PV–ESS–SLC systems. 3. From Fuzzy Boolean Model to DEMATEL Factors The Boolean-to-fuzzy transformation yields the following key components of the PV–ESS–SLC system: • Fuzzy system performance: 𝜇𝐹(𝑥,𝑦,𝑧)=max (𝑥,min(1−𝑥,𝑦),min (1−𝑥,1−𝑦,𝑧) • Minimal-operation indicators: 𝑚𝑖𝑛𝑥𝜇𝐹=max (𝑦,𝑧), 𝑚𝑖𝑛𝑦𝜇𝐹=max(𝑥,𝑧), 𝑚𝑖𝑛𝑧𝜇𝐹== max (𝑥,𝑦) These expressions serve as component-level reliability indices: 𝑅𝑃𝑉=𝑚𝑖𝑛𝑥𝜇𝐹=max(𝑦,𝑧),𝑅𝑅𝑆𝑆=𝑚𝑖𝑛𝑦𝜇𝐹=max(𝑥,𝑧),𝑅𝐿𝐶=𝑚𝑖𝑛𝑧𝜇𝐹=max (𝑥,𝑦) Based on these indices, DEMATEL factors are defined as: • C1: PV availability & reliability • C2: ESS availability & reliability • C3: Load Controller reliability • C4: Overall system performance (𝜇𝐹,) 3.1. Fuzzy DEMATEL Structure Following the classical DEMATEL procedure: 1. Expert evaluation of pairwise influences using linguistic terms (0, Low, Medium, High, Very High) 2. Conversion of linguistic assessments to fuzzy numbers 3. Construction of fuzzy direct-relation matrix 𝐴󰆻= [𝑎𝑖𝑗] 4. Normalization to obtain 𝐷. 5. Computation of total relation matrix: 𝑇=𝐷 (İ−𝐷)−1Influence metrics: 𝐷𝑖=∑𝑡𝑖𝑗𝑗 (influence given), 𝑅𝑖=∑𝑡𝑗𝑖𝑗 ( influence received) 𝐷𝑖+𝑅𝑖: importance, 𝐷𝑖+𝑅𝑖: cause/effect index 4. Gaussian Fuzzy Sets for Reliability and Influence Gaussian fuzzy sets are adopted due to their smooth transitions and suitability for representing uncertainty in PV output, ESS performance, and load-control behavior. For reliability classes: 𝜇𝐿𝑜𝑤(𝑟)=exp (−(𝑟−𝑐𝐿)2 2𝜎𝐿2), 𝜇𝑀𝑒𝑑(𝑟)=exp (−(𝑟−𝑐𝐿 𝑀)2 2𝜎𝑀 2), 𝜇𝐻𝑖𝑔ℎ(𝑟)=exp (−(𝑟−𝑐𝐻)2 2𝜎𝐻 2), with: • 𝑟∈[0,1] – reliability index ( 𝑅𝑃𝑉, 𝑅𝐸𝑆𝑆, 𝑅𝐿𝐶) • centers𝑐𝐿,𝑐𝑀,𝑐𝐻 – (0.2, 0.5, 0.8), • spreads𝜎𝐿, 𝜎𝑀,𝜎𝐻 – (0.1–0.15). 5. Reliability-Weighted DEMATEL Reliability values obtained from the fuzzy Boolean model are integrated into the DEMATEL framework to emphasize the operational stability of each component. Nominal-case reliability indices: • 𝑅𝑃𝑉=𝑚𝑖𝑛𝑥𝜇𝐹=max (𝑦,𝑧) • 𝑅𝑅𝑆𝑆=𝑚𝑖𝑛𝑦𝜇𝐹=max (𝑥,𝑧) • 𝑅𝐿𝐶=𝑚𝑖𝑛𝑧𝜇𝐹=max (𝑥,𝑦) Practical interpretations: • 𝑅𝑃𝑉− normalized average PV output • 𝑅𝐸𝑆𝑆− function of SOC and SOH • 𝑅𝐿𝐶 - percentage of successful load-shedding actions The reliability weighting matrix is defined as: W=diag (𝑅𝑃𝑉,𝑅𝐸𝑆𝑆,𝑅𝐿𝐶,𝑅𝑠𝑦𝑠𝑡𝑒𝑚,…) This matrix is applied to the DEMATEL total relationship matrix to obtain reliability-aware causal influence patterns. Figure2(a). 3D surface of PV reliability index 𝑅𝑃𝑉. Figure2(b). 3D surface of ESS reliability index 𝑅𝐸𝑆𝑆 (xz). Figure2(c). 3D surface of LC reliability index 𝑅𝐿𝐶 (x, y). 5.1. Reliability-Weighted Total Relation Matrix Once the DEMATEL total relation matrix TTT is obtained, reliability weighting is applied: 𝑇𝑅=𝑊𝑇𝑊 –This operation scales both outgoing and incoming influences according to the reliability of each component. As a result: • Components with high reliability exert stronger influence in the weighted matrix. • Components with low reliability have reduced influence in both cause-and-effect directions, making “vulnerable components” more clearly identifiable. 6. Gaussian DEMATEL-Based Reliability Interpretation For each factor, the reliability-weighted influence metrics are computed as: 𝐷𝑖𝑅=∑𝑇𝑖𝑗𝑅 𝑗 (row sum – influence given), 𝑅𝑖𝑅=∑𝑇𝑗𝑖𝑅 𝑗 ( influence received) 𝐷𝑖𝑅+𝑅𝑖𝑅: importance, 𝐷𝑖𝑅+𝑅𝑖𝑅: cause (+) / effect (−). These indices can then be classified using Gaussian fuzzy sets, for example: • High causal, high reliability → core driver • High effect, low reliability → critical vulnerable component • Medium effect, medium reliability → transition node Figure3. 3D surface representation of the reliability-weighted DEMATEL total relation matrix TRT^{R}TR for the PV–ESS–LC–System structure. 6.1. Component-Level Reliability Indices From the fuzzy Boolean model, each component is assigned a normalized reliability index 𝑅𝑖∈[0,1], 𝑖∈{𝑃𝑉,𝐸𝑆𝑆,𝐿𝐶,𝑆𝑦𝑠} where 𝑅𝑖=1 indicates full reliability and 𝑅𝑖=0 indicates total unreliability. For the PV–ESS–SLC architecture: 𝑅𝑃𝑉: reliability of the PV generation block 𝑅𝐸𝑆𝑆: reliability of the energy storage system 𝑅𝐿𝐶:reliability of the load controller 𝑅𝑆𝑦𝑠: overall system performance index derived from 𝜇𝐹. These indices may be obtained either from the fuzzy Boolean min–max operators or from operational statistics. Examples: 𝑅𝑃𝑉≈𝐸[𝑥] (Normalized average PV output), 𝑅𝐸𝑆𝑆≈𝐸[𝑦] (based on SOC and SOH), 𝑅𝐿𝐶≈𝐸[𝑧] (fraction of successful load-shedding actions), 𝑅𝑆𝑦𝑠≈𝐸[𝜇𝐹(𝑥,𝑦,𝑧)] (fraction of successful loadshedding actions) These values form the reliability vector and weight matrix: 𝑅=(𝑅𝑃𝑉 𝑅𝐸𝑆𝑆 𝑅𝐿𝐶 𝑅𝑠𝑦𝑠),𝑊𝑅=(𝑅𝑃𝑉 0000 𝑅𝐸𝑆𝑆 0000 𝑅𝐿𝐶 0000 𝑅𝑆𝑦𝑠), and the reliability-weighted relation matrix is: 𝑇𝑅=𝑊𝑅𝑇𝑊𝑅 6.2. Reliability Interaction Matrix To separately represent “reliability support” among components, the following interaction matrix is used: 𝑀𝑅= ( 𝑅𝑃𝑉 𝑅𝐸𝑆𝑆→𝑃𝑉 𝑅𝐿𝐶→𝑃𝑉 𝑅𝑆𝑦𝑠→𝑃𝑉 𝑅𝑃𝑉→𝐸𝑆𝑆 𝑅𝐸𝑆𝑆 𝑅𝐿𝐶→𝐸𝑆𝑆 𝑅𝑆𝑦𝑠→𝐸𝑆𝑆 𝑅𝑃𝑉→𝐿𝐶 𝑅𝐸𝑆𝑆→𝐿𝐶 𝑅𝐿𝐶 𝑅𝑆𝑦𝑠→𝐿𝐶 𝑅𝑃𝑉→𝑆𝑦𝑠 𝑅𝐸𝑆𝑆→𝑆𝑦𝑠 𝑅𝐿𝐶→𝑆𝑦𝑠 𝑅𝑆𝑦𝑠 ) Interpretation examples: • 𝑅𝑃𝑉→𝐸𝑆𝑆 : “To what degree does ESS reliability enhance overall system stability?” • 𝑅𝐸𝑆𝑆→𝑆𝑦𝑠: “To what degree does ESS reliability enhance overall system stability?” The off-diagonal elements may be derived from two sources: 1. Fuzzy Boolean min–max logic (theoretical structure). Example: when PV is inactive, minimal system reliability becomes 𝑀𝑎𝑥(𝑦,𝑧) thus ESS and LC support levels can be derived analytically. 2. Expert or data-driven assessments (practical filling). Each𝑅𝑖→𝑗 is rated as Low–Medium–High–Very High and converted via Gaussian fuzzy processing. 6.4.Transformation of reliability matrix into Gaussian-Fuzzy influence scores 𝑀𝑅Each value in 𝑀𝑅 (representing support strength between components) is mapped into Gaussian fuzzy space: 𝑅𝑖→𝑗 𝐺𝑎𝑢𝑠𝑠𝑖𝑎𝑛={𝜇𝐿𝑜𝑤(𝑅𝑖𝑗),𝜇𝑀𝑒𝑑(𝑅𝑖𝑗),𝜇𝐻𝑖𝑔ℎ(𝑅𝑖𝑗)} 𝑅𝑖𝑗→{𝜇𝐿𝑜𝑤,𝜇𝑀𝑒𝑑,𝜇𝐻𝑖𝑔ℎ} This transformation forms the foundation of the fuzzy DEMATEL influence structure. Figure4(a) Gaussian fuzzy membership functions for Low, Medium, High influence levels. Figure4(b) Heatmap of the Low-level Gaussian membership 𝜇𝐿𝑜𝑤(𝑅𝑖𝑗), for the reliability interaction matrix 𝑀𝑅. Figure4(c). Heatmap of the Medium-level Gaussian membership 𝜇𝑀𝑒𝑑(𝑅𝑖𝑗). Figure2(d) Heatmap of the High-level Gaussian membership 𝜇𝐻𝑖𝑔ℎ(𝑅𝑖𝑗).