Expert System Integrating Rule-Based Reasoning to Voltage Control in Photovoltaic-Systems-Rich Low Voltage Electric Distribution Networks: A Review and Results of a Case Study

Abstract

Nowadays, in low voltage electric distribution networks, the distribution network operators are encountering a high number of connected small-scale distributed generation units, mainly photovoltaic prosumers. The intermittent nature of the prosumers, together with the degree of uncertainty of the requested and injected powers associated with all end-users from low voltage electric distribution networks, can cause voltage variations that violate the allowable limits. In this context, this paper developed an efficient and resilient expert system integrating rule-based reasoning applied to the on-load tap changer-fitted transformer to improve the efficiency of the voltage control in the PV-rich LV EDNs. An in-depth analysis based on 75 scenarios, resulting from the combinations of three indicators—the penetration degree of the photovoltaic prosumers, the consumption evolution associated with the consumers, and the energy production of the photovoltaic systems—was performed to demonstrate the efficiency of the proposed expert system in a low voltage electric distribution network from a rural area belonging to a Romanian distribution network operator. The success rate of the expert system was 86.7% (65 out of 75 scenarios did not have voltage issues). All voltages were between the allowable limits in 100% of the time slots associated with the analysed period. For the other scenarios (representing 13.3%), voltages were inside the range [−10%, +10%] in at least 95% of the time slots.

1. Introduction

1.1. Motivation

The reduction in centralized power plants’ energy production has been attributed to the emergence of distributed generation in electric distribution networks (EDNs), particularly at the low voltage (LV) level [1]. The increasing number of photovoltaic systems (PV) installed at the level of consumers (called prosumers) connected to low voltage electric distribution networks (LVEDNs) is expected to increase annually due to their various economic and technical advantages, besides the fact that they also can help reduce greenhouse gas emissions. However, distribution network operators (DNOs) are not fully prepared to integrate a very high number of PV prosumers with a variable character due to, on the one hand, the topologies of LVEDNs and, on the other hand, the ageing technical infrastructure [2].

A lack of voltage quality, defined by the number of incidences where voltages violate the admissible limits, has been identified by DNOs as their main concern in hosting more PV in their LVEDNs to accommodate the growing number of PV prosumers [3,4]. Another uncertainty is the reverse power flow between the LVEDN and medium voltage (MV) EDN causing voltage level issues [5,6].

DNOs should develop new strategies to address these issues and increase the penetration degree of PV prosumers. An overview of the strategies to improve voltage quality emphasizes that most of them belong to passive LVEDNs, without the renewable energy sources integrated. These strategies include replacing the no-load tap changing (NLTC)-fitted MV/LV transformers and reducing the cable impedances to decrease the excessive ratio between the resistance and the reactance of the LVEDNs. However, the expensive cost and the low efficiency from a technical point of view in the active LVEDNs (integrating the renewable energy sources) represent the weaknesses of these strategies [1,7]. On the other hand, the high-frequency solar ramping caused by the rapid emergence and evolution of clouds can lead to over-voltages in LVEDNs [2].

Thus, the high intermittency of these sources could be a burden for DNOs in the voltage control of LVEDNs. As long as new strategies are not developed, the voltage violations could appear more and more often when a high number of PV prosumers are integrated, in conditions in which the voltage control band is usually between 0.9 and 1.1 p.u. of the rated voltage. Developing new strategies will require more advanced voltage control technologies, including smart devices/equipment that can process the data and help to identify the optimal solution [8]. The advanced voltage control technologies, integrated especially in the pilot LVEDNs, are represented by automatic voltage regulators [9,10], energy storage systems [11,12], capacitor banks [13,14], or load balancing devices [15,16]. However, before they can be implemented widely and correlated with each other, DNOs should do a cost–benefit analysis. One of the most efficient devices to ensure the resilience of voltage control is represented by the on-load tap changer fitted to the MV/LV distribution transformers [17,18,19,20,21]. These devices have demonstrated their performance in high (HV) and ultra-high (UHV) voltage electric transmission networks. The research carried out so far has not clearly defined the maximum PV hosting capacity of LVEDNs supplied from an EDS with an OLTC-fitted transformer. In these conditions, efficient strategies must be developed to increase as much as possible the limit threshold.

Based on the aspects outlined above and the current context under which innovative energy solutions must be developed and implemented to fight climate change, we need to find a solution where more and more renewable sources are integrated into the existing EDNs. In this context, this paper will focus on developing an efficient voltage control solution to increase the penetration degree of PV prosumers quantified through the hosting capacity of LVEDNs. The proposed solution aims to change the tap position of the OLTC-fitted distribution transformers based on a rule-based expert system (RBES).

1.2. Literature Review

Most research has evaluated the effects of PV prosumers focusing on the different aspects of the voltage profile from LVEDNs; see Figure 1. However, the results have depended on the type of LVEDNs, test (for example, IEEE EDNs) or real associated with the pilot projects of different DNOs, and the working assumptions considered.

Various definitions regarding the penetration degree of the prosumers from a LVEDN have been given in the literature. The first definition refers to the ratio of roof space needed to install the PV panels to the total surface available [20]. Another definition considers the ratio of annual energy from PV systems to the total energy consumption [21]. Some researchers have attempted to define it concerning the transformer capacity [22], whereas others have calculated it as the ratio of installed PV peak capacity to the maximum load of the feeder [1].

Some definitions have even replaced the “maximum load of feeder” with “minimum load of feeder” [23]. These included the ratio between the actual PV output and the active power load. However, the most accessible, clear, and easy-to-understand definition refers to the number of PV prosumers from the end-users connected in the LVEDNs [19,24].

The studies differ through the factors that influence the design and operation of LVEDNs. They refer to the type of network (rural/urban or aerial/underground), the weather conditions, the generation/load profiles, and the capacity and spread of PV prosumers. Their extension to other networks or areas is difficult since the results are influenced by the analysed LVEDNs. This suggests that further studies must include the multiple aspects of voltage control.

Many voltage control solutions have been developed based on the constant voltage (CV) method and the line-drop compensation (LDC) method.

The CV method [25,26] is represented by fixing the tap position for all time slots regardless of the load variation. It is widely used by DNOs through power transformers with NLTC to prevent voltage issues in the LVEDNs. However, an optimal solution is not easy to identify due to the variations in the load and intermittent nature of the PV prosumers.

The LDC method [26,27,28] solves the decreased voltage issues based on the data performed by the voltage and current measuring devices placed on the LV side of the EDS. The current measured is multiplied by the impedance calculated against a virtual load centre (VLC). The voltage of the VLC is computed using the difference against the voltage of the LV bus of the EDS. The OLTC modifies the tap to ensure that the voltage of the VLC remains at the same value as the reference voltage. It also uses a time delay and dead band to prevent sudden voltage changes.

The limits of both methods have been identified in the case of the PV-rich LVEDNs, being unable to respond to the intermittent nature of prosumers as they only consider the load’s influence. The value of the current at the LV side of the EDS decreases as the PV capacity increases. The small voltages can occur in the phase when only consumers are connected.

Thus, identification of new solutions to improve voltage control in PV-rich LVEDNs represents nowadays a challenge in the LVEDNs. The experience of the decision-makers in the voltage control process from the LVEDNs integrated in a rule-based expert system and real-time communication solutions can represent the poles of new voltage control solutions.

Thus, artificial intelligence (AI) techniques can play a vital role in the integration of PV prosumers by providing a more seamless and accurate voltage control in the LVEDNs. However, despite the progress that AI techniques have made in the field of PV integration, expert systems are still not widely used [29,30]. There are few approaches that integrate an expert system to solve the technical issues (including voltage control) of EDNs.

A comprehensive control system has been presented for UHV EDNs by Lee et al. [31]. It uses an expert system that integrates a numerical subsystem containing power flow software to obtain a sensitivity matrix. The tree space of this parameter depends on the variable sensitivity of different power sources. The system has been implemented at the UHV level, where the control capabilities are more numerous than at the low voltage level.

An expert system that can control the voltage level of a 20 kV MVEDN using a supervisory control and data acquisition (SCADA) system has been proposed by Pimpa and Premrudeepreechacharn [32]. This system can reduce the multiple voltage violations in the EDS and detect abnormal conditions at the buses from the EDN. The expert system focused on the control of the shunt reactive compensating devices and transformer tap change. The pursued objective is to identify the most effective control strategy for keeping the bus voltage within limits. The weak point of this system is represented by a lack of distributed generation sources in the distribution network (this is a passive EDN). An expert system using the fuzzy-logic-based remote monitoring system for a low voltage network has been proposed in [33]. The system provides real-time monitoring of the over-voltage and under-voltage conditions at the end-user level. The system is designed based on a Proteus simulator to communicate with devices that measure the voltage in passive LVEDNs, without the prosumers connected. The use of the LDC method to fix the reference voltage represents the weak point of this system. In addition, the system has been tested only in passive LVEDNs, without prosumers.

Mariaraja et al. [34] proposed an expert system for the reconfiguration process of a MVEDN using a hybrid Fuzzy-Flower Pollination Algorithm. The aim is to decrease the power losses and improve the voltage level. This system can perform optimization in abnormal and normal operating conditions. The weak point identified for this system is associated with the use only in the passive MVEDNs.

Perera et al. proposed an optimal design and control method for an energy hub consisting of a battery bank, wind turbines, and solar photovoltaic panels [35]. Their structure can analyse the power flows in the hub based on an expert system. The optimal operating regime is also determined considering the cost of electricity generated by renewable energy sources and the battery bank’s state of charge. The expert system has not been tested in the case of a LVEDNs. The generalization process from an energy hub to a LVEDN can lead to technical issues due to the higher complexity of an active LVEDN which includes more end-users.

Bennett et al. developed a hybrid expert system to handle the complexity of the LV network demand [36]. It combines a set of modules, such as clustering, correlation, and neural network, to forecast and analyse future energy demand. The system cannot be used in the voltage control of active LVEDNs.

Kirgizov et al. presented in [37] a reactive power compensation solution based on the real-time data collected from the distribution networks. The suggested solution, integrated into an expert system, uses fuzzy logic and heuristic algorithms. The priority of the devices/equipment needing the reactive power compensation is determined based on fuzzy logic. The application allows the expert to take the appropriate steps to establish the optimal location of the devices. The system does not integrate the OLTC-fitted distribution transformers.

Chelaru and Grigoras proposed in [38] an expert-system-based decision-making framework to identify the most appropriate replacement transformer fleet solution. The expert system determines the order of the ageing transformers in a replacement ranking according to the different priority categories. The system is used only in replacing the process of the OLTC-fitted MV/LV distribution transformers in the planning stage of active LVDNs.

The analysis of the references highlighted some limitations of the proposed expert-system-based solutions. The majority have been implemented for the UHV and MV levels. Few of them have been tested in active MVEDNs and LVEDNs; most of them have been proposed for passive EDNs. However, they included the control of the reactive power compensation devices to improve voltage control. The expert-system-based voltage control solutions applied to modify the tap position of the OLTC have been proposed only in passive LVEDNs where the LDC method has been used.

1.3. Main Contributions

The studies regarding the decision-making tools integrating expert systems presented in the literature have highlighted various approaches that solve different technical issues from EDNs at various voltage levels. However, very few references refer to voltage control in LVEDNs, which use expert systems, and these with applications in passive LVEDNs.

In this context, the authors propose an efficient and resilient voltage control solution based on an expert system to remove the over-voltage and under-voltage issues in PV-rich active LVEDNs. The main contributions are the following:

• Proposing another principle compared to the LDC method (used in all current solutions) to solve the decreased/increased voltage issues based on the data performed by the smart meters installed at the end-user level. The OLTC modifies the tap to ensure that the voltages at the end-user level remain inside the range [−10%, +10%] at least 95% of the time slots.
• Developing an expert system, including rule-based reasoning, with the main advantages: the “fast-scanning” of the input data, identification of voltage issues that come up, and determination of a solution associated with the tap position of the OLTC that does not violate the voltage constraints in the PV-rich LVEDNs. The voltage constraints are verified based on the deviations between the reference voltage and the voltages recorded in the nodes in each time slot recognizing the excesses of the allowable limits, regardless of the power flow’s direction.
• Designing a data management framework including a real-time query procedure that uploads data from the smart metering system (SMS) and network data systems (NDS) and saves them in two partitions (static and dynamic) from the knowledge database. The design allows a high speed of data processing.
• Performing an in-depth analysis in a real LVEDN belonging to a Romanian DNO based on more scenarios characterized by the three indicators: the penetration degree of the PV prosumers, the consumption evolution associated with the consumers, and the energy production of the PV systems installed to the prosumers. The number of combinations between the possible values of the three indicators led to 75 scenarios that cover all spectrums of the operating regimes of the LVEDNs.

1.4. Paper Organization

The paper includes the following sections: Section 2 presents the solution used to develop the structure of the proposed expert system, including rule-based reasoning; Section 3 corresponds to the case study in which an in-depth scenario-based analysis has been performed in an active LVEDN from a rural area from the east of Romania; and Section 4 presents the discussion and conclusions, also highlighting the limits of the proposed approach and future work.

2. Our Proposed Solution

A rule-based expert system (RBES) is an artificial intelligence component that uses knowledge-based rules to perform an activity [39].

Their design is similar to an advanced computer program that tries to mimic the capabilities of a human expert collecting knowledge sources used to perform an objective in a particular domain [36,39]. Thus, knowledge is taken from the human expert and converted into a production rule set that represents the domain knowledge. The basic structure of a RBES contains the following main components: knowledge base module, inference engine, decision-making module, explanation facilities, and user interface.

The characteristics of the main components (see Figure 2) included in the proposed RBES used in the voltage control from the active LVEDNs are described in the following.

The knowledge database of the proposed expert system has two partitions: the rule base and the database. The rule base stores the rules and facts. The database is divided into two partitions (static and dynamic) and is associated with recording the immutable facts. The static partition of the database contains information on the features of the active LVEDNs regarding:

• Line sections characterized by the input and end nodes, type (aerial/underground), cross-sections of the phase, and neutral conductors;
• MV/LV distribution transformers from the EDSs characterized by rated power, performance standards identified through the commissioning year, tap changer type (NLTC/OLTC), and tap positions;
• Reactive power compensation devices identified through the installed capacity, type (capacitor banks or static reactive (VAR) compensator), and their locations;
• Energy storage systems identified through the installed capacity and their locations;
• End-users characterized by type (single-phase/1-P or three-phase/3-P), the location in the AEDNs (the connection pillars), the connection phase (for 1-P end-users);
• Energy generation systems installed to the prosumers characterized by the installed capacity.

The RBES uses the data communication between the smart meters installed at all end-users and the data concentrator installed in the MV/LV EDS. These data form the dynamic database of the RBES, together with the outcomes of the steady-state calculations, represented by the values of the voltage from the nodes:

• Upper and lower allowable limits of the voltages imposed by the quality power standards;
• Information from the smart metering system associated with the injected/requested powers by the end-users (prosumers/consumers) at the fixed time slots (depending on setting the sampling step of the smart meters), the annual energy production/consumption.

The nodal voltages, power flows, and power losses resulting from the steady-state calculations are performed with a performance algorithm developed by Grigoras and Neagu in [40].

The algorithm is characterized by a fast convergence with a reduced calculation time. Three stages characterize its internal structure:

• The first stage involves online work, uploading the data from the SMS database, which contains the production and consumption profiles of the prosumers and consumers integrated in the LVEDN.
• The second stage corresponds to the fast recognition of the topology associated with the LVEDN using a structure vector-based method. The data are uploaded from the network topology database.
• The third stage focuses on the steady-state calculations using an improved variant of the forward/backward sweep algorithm determining the power/energy losses in four-wire LVEDNs regardless of the operating regimes (balanced and unbalanced).

It uploads the topology of the AEDNs identified through the data recorded in the static database at the request of the inference engine.

The inference engine accesses the data and, using the rule-based motivation, decides whether to modify the tap position of OLTC. The aim is that all voltages are inside the allowable limits fixed by the DNO. An IF-THEN production rule set represents the domain knowledge in the RBES. The data are associated with a fact set about the current situation. The data and rules are then compared with the facts in the database. When the conditions of the rule match a fact, the system fires the rule and its active component is executed. A code that has the following syntax:

“IF conditions, THEN actions.”

belongs to each rule.

The rule-based reasoning method is integrated into the inference engine and used to analyse the data. The knowledge base contains the rules designed to guide the system in performing its task (suitable voltage level according to the power quality performance standard). Thus, the expert system has the goal, and the inference engine attempts to find the evidence to prove it. First, the knowledge base is interrogated to find rules that might lead to the desired solution. Such rules must have the goal in their THEN (action) parts. If such a rule is found and the part of the condition associated with IF matches data in the database, then the rule is fired, and the goal is proved.

The information collected during a time slot s by smart meters (active and reactive powers) installed at the end-user level (prosumers and consumers) is sent to the data concentrator from the EDS level through the communication system. The data belong to an operation regime of the active LVEDN where the tap position of the OLTC, identified by the variable α, corresponds to the previous time slot (s − 1), with s = 1, …, S, with S representing the analysed period.

The RBRS establishes the tap position of the OLTC starting with the time slot s, s = 1, …, S, based on the data obtained from the steady-state calculations using the tap position from the time slot (s − 1), α_(s−1), and the information from the database (static and dynamic). The next step confirms that the data availability to determine the extreme values (minimum and maximum) of the voltage at the level of active LVEDN.

$$\begin{array}{l}{\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\min } =\underset{\begin{array}{l}\mathsf{\gamma }\in \{{\mathsf{\gamma }}_{\mathrm{a}},{\mathsf{\gamma }}_{\mathrm{b}},{\mathsf{\gamma }}_{\mathrm{c}}\}\\ \end{array}}{\min }\left\{{\mathrm{V}}_{\mathrm{e},\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\left\{\mathsf{\gamma }\right\}}\right\}, {\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\max } =\underset{\mathsf{\gamma }\in \{{\mathsf{\gamma }}_{\mathrm{a}},{\mathsf{\gamma }}_{\mathrm{b}},{\mathsf{\gamma }}_{\mathrm{c}}\}}{\max }\left\{{\mathrm{V}}_{\mathrm{e},\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\left\{\mathsf{\gamma }\right\}}\right\} \\ \mathrm{e}=1,\dots ,\mathrm{NE} , \mathrm{s}=1,\dots ,\mathrm{S}, \mathsf{\alpha }=1,\dots ,\mathrm{a}\end{array}$$

(1)

where:

• γ—the index used to identify one of the three phases γ $\in$ {γ_a, γ_b, γ_c};
• NE—the variable referring to the total number of the nodes from the active LVEDN;
• e—the index associated with a certain node from the active LVEDN;
• A—the upper tap of OLTC;
• α—the index corresponding with the tap position in a certain time slot;
• S—the total number of the time slots defined in the analysed period;
• s—the time slot identified in the analysed period S;
• ${\mathrm{V}}_{{\mathrm{e},\mathrm{s},\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\left(\mathsf{\gamma }\right)}$—the voltage on the phase, γ, γ $\in$ {γ_a, γ_b, γ_c} calculated in each node e, e = 1, …, NE, for the time slot s, s = 1, …, S, and the tap position α associated with the time slot s−1;
• ${\mathrm{V}}_{{\mathrm{s},\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\min }$, ${\mathrm{V}}_{{\mathrm{s},\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\max }$—the extreme values of the voltages from the active LVEDN (regardless of the phase) determined from the time slot s, and the tap position α associated with the time slot (s−1).

The voltage control in each time slot s refers to an iterative process having the stop criterion associated with the extreme values (minimum and maximum) of the voltage at the level of active LVEDN. All voltages must be between allowable limits:

$${\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\min } \ge {\mathrm{V}}_{\mathrm{a}}^{\min }, {\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\max } \le {\mathrm{V}}_{\mathrm{a}}^{\max }, \mathrm{s}=1,\dots ,\mathrm{S}, \mathsf{\alpha }=1,\dots ,\mathrm{A}$$

(2)

If the extreme values V^min_s,α(s−1), V^max_s,α(s−1) exceed the allowable limits (V^min_a and V^max_a), then the tap position is modified with a step Δα_s^(k) in each iteration k of the time slot s based on the following rules [24,41]:

$$\mathrm{IF} \left\{({\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\min }<{\mathrm{V}}_{\mathrm{a}}^{\min }) \mathrm{AND} ({\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\max }\le {\mathrm{V}}_{\mathrm{a}}^{\max })\right\} \mathrm{THEN} \Delta {\mathsf{\alpha }}_{\mathrm{s}}^{(\mathrm{k})}={\mathsf{\alpha }}_{(\mathrm{s}-1)}+1$$

(3)

$$\mathrm{IF} \left\{({\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\min }\ge {\mathrm{V}}_{\mathrm{a}}^{\min }) \mathrm{AND} ({\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\max }>{\mathrm{V}}_{\mathrm{a}}^{\max })\right\} \mathrm{THEN} \Delta {\mathsf{\alpha }}_{\mathrm{s}}^{(\mathrm{k})}={\mathsf{\alpha }}_{(\mathrm{s}-1)}-1$$

(4)

$$\mathrm{IF} \left\{\begin{array}{l}({\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\min }<{\mathrm{V}}_{\mathrm{a}}^{\min }) \mathrm{AND} ({\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\max }>{\mathrm{V}}_{\mathrm{a}}^{\max }) \\ \mathrm{AND} (({\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\max }-{\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\min })>\Delta {\mathrm{V}}_{\mathsf{\alpha }})\end{array}\right\} \mathrm{THEN} \Delta {\mathsf{\alpha }}_{\mathrm{s}}^{(\mathrm{k})}={\mathsf{\alpha }}_{(\mathrm{s}-1)}-1$$

(5)

$$\mathrm{IF}\left\{\begin{array}{l} ({\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\min }<{\mathrm{V}}_{\mathrm{a}}^{\min }) \mathrm{AND} ({\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\max }>{\mathrm{V}}_{\mathrm{a}}^{\max }) \\ \mathrm{AND} (\Delta {\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\min }-\Delta {\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\max }>\Delta {\mathrm{V}}_{\mathsf{\alpha }})\end{array}\right\} \mathrm{THEN} \Delta {\mathsf{\alpha }}_{\mathrm{s}}^{(\mathrm{k})}={\mathsf{\alpha }}_{(\mathrm{s}-1)}+1$$

(6)

where:

$$\Delta {\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\min }={\mathrm{V}}_{\mathrm{a}}^{\min }-{\mathrm{V}}_{\mathrm{s},{\mathsf{\beta }}^{(\mathrm{k})}}^{\max } , \mathrm{s}=1,\dots ,\mathrm{S} , \mathsf{\alpha }=1,\dots \mathrm{A}$$

(7)

$$\Delta {\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\max }={\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\max }-{\mathrm{V}}_{\mathrm{a}}^{\max } , \mathrm{s}=1,\dots ,\mathrm{S} , \mathsf{\alpha }=1,\dots \mathrm{A}$$

(8)

ΔV_α represents the voltage changes per tap.

However, the tap position can be modified only if the constraints regarding the dead band (DB) of the OLTC are satisfied [42]:

$$\left(\frac{{\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\max }-{\mathrm{V}}_{\mathrm{r}}}{{\mathrm{V}}_{\mathrm{r}} }\right)>\mathrm{DB}; \left(\frac{{\mathrm{V}}_{\mathrm{r}}-{\mathrm{V}}_{\mathrm{s},{\mathsf{\alpha }}_{(\mathrm{s}-1)}}^{\min }}{{\mathrm{V}}_{\mathrm{r}} }\right)>\mathrm{DB};\mathrm{s}=1,\dots ,\mathrm{S} , \mathsf{\alpha }=1,\dots \mathrm{A}$$

(9)

where V_r is reference voltage.

The tap position can be changed only when the voltage deviation exceeds DB (having the value ΔV_α/2). The use of the dead band can avoid the effects of unnecessary tap-changing operations.

Figure 3 presents the flow chart of the voltage control using rule-based reasoning.

The user can ask the expert system how it reaches a particular conclusion through the explanation facilities, which allows one to understand the reasoning behind the decision. The RBES can provide a credible and effective solution based on a clear explanation of its analysis and guidance. The proposed guidance is systematic, evaluating the effects of the voltage control on the operation of the active LVEDN, avoiding conflicts between the rules. Although these components represent the core of the RBES, additional parts could be integrated. One of these is the external interface that allows the RBES to interact with other programs and data files used commonly by DNOs (SMS, SCADA, Demand Management System).

The main characteristics of the proposed RBES are efficiency and resilience. Efficiency is the ability to meet the demands of end-users regarding voltage quality. Regarding the second characteristic, it covers the capacity of the RBES to respond to harmful events associated with sudden voltage variations due to the intermittent regime of the small-scale renewable sources (prosumers).

3. Results

The proposed RBES has been tested considering an aerial LVEDN supplied by an EDS whose MV bus (20 kV) represents the CCP with the network of a Romanian DNO that carries out its distribution service in the respective area. Figure 4 shows the structure of the test LV EDN.

The EDS contains an MV/LV (20/0.4 kV) distribution transformer, having the following technical features associated with a Tier 2 transformer [43]:

• Rated power, S_r = 250 kVA;
• Load power loss, ΔP_l = 2.35 kW;
• No-load power loss, ΔP_nl = 0.27 kW;
• OLTC with 9 taps (tapping range ± 10%), where the median tap is 5, voltage step 2.5%, and the number of tap operations without maintenance is 700,000 [41].

The data regarding the consumed active and reactive powers of all end-users are available in the smart metering system. The sampling step set for all smart meters to send this information to the data concentrator was 60 min.

The LVEDN is an aerial network with four conductors (three phases and a neutral), having detailed features in

Table 1. The technical features of the test 88-bus LVEDN.

Type of Section	Conductor			Type of Conductor	Length [km]
	Cross-Section of Phase Conductor [mm2]	Number of Phases	Cross-Section of Neutral Conductor [mm2]
1	50	3	50	C *	2.08
2	50	3	50	S **	0.12
3	35	3	35	C *	0.68
4	35	1	35	C *	0.28
5	25	1	25	C *	0.28
6	25	1	16	C *	0.08

* C—classical conductor (aluminium conductor steel-reinforced cable); ** S—stranded conductor.

. In total, 114 end-users are connected at the three phases, and their distribution is presented in Figure 5.

The DNO provided information about the technical parameters of the MV/LV distribution transformer and conductors after a visual inspection carried out by the maintenance team. After that, a comparison with the data from the database was performed. Modifications were implemented to correspond to the reality of the field when recorded data were erroneous.

The total length of the network is 3.52 km, having two types of conductors (classical and stranded). The sections with the classical conductor have 3.4 km, representing 96.6% of the total length). The most common cross-section is 50 mm² (mainly on the trunk of the network, P1–P88), representing 62.5%, followed by 35 mm² and 25 mm² (found on the lateral branch, P4–P39).

Localization of the network is in an area where the solar potential is high [44] (see Figure 6), so it is expected that the number of consumers who want to become PV prosumers will increase, considering the situation in Romania.

According to the data presented in [45], about 50% of European Union citizens are expected to be able to produce their electricity by 2050. They also plan on meeting 45% of the energy requirements of the European Union. This information is strengthened by the last report of the Romanian Energy Regulatory Authority (RERA) published in 2022 [46], highlighting that the number of prosumers increased more than 6 times in 2021, from 2134 to 13,596. On the other hand, the reports of the RERA [47] regarding the evolution of energy consumption from 2020 to 2022 indicated a decrease at the residential consumer level by approximately 10%. This decrease was due to two factors associated with the pandemic crisis and the geopolitical situation (the war in Ukraine and the energy crisis).

These data represented the basis of the assumptions regarding indicators used in the study. Thus, it has been considered that the number of prosumers will gradually increase to 50%. Regarding consumption, their trend will be associated with possible increases or decreases in the range ±10%. Not least, the energy production of the prosumers will depend on solar radiation, which can have a high uncertainty modelled through a confidence interval characterised by statistical parameters (mean and standard deviation).

Considering these assumptions, more scenarios have been considered with the following indicators: penetration degree, PD, PD $\in$ {10%, 20%, 30%, 40, 50%}; consumption evolution, CEC, CEC $\in$ {−10%, 5%, 0%, +5%, +10%}; and energy production of the PV systems from the month that includes the analysed day, EP_PV, EP_PV $\in$ {${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$_—$2 {\times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$, ${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$_, ${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV} }$+ ${2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$}, where ${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$ and ${\mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$ represent the mean and standard deviation of the energy production (EP) associated with a PV system from the geographical area where the LVEDN is located (see Figure 6), having a certain installed capacity for a month from a year (in our case, June). Knowing the values ${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$ and ${\mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$, EP_PV with probability (confidence level) 0.95 belonging to the interval [${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2 {\times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$, ${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$+ ${2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$] has been considered. This means that, in almost all cases, the frequency will not go beyond the specified interval. Even in 5% of cases, the errors will be minor [48].

The study analysed the penetration degrees considering the assumption that the consumers with the highest energy consumption are qualified and selected to become prosumers, maintaining their connection phase.

Table A1. The allocation of the end-users at each pillar associated with the five penetration degrees.

Pillar	0%		10%		20%		30%		40%		50%		Pillar	0%		10%		20%		30%		40%		50%
	C *	P *	C *	P *	C *	P *	C *	P *	C *	P *	C *	P *		C *	P *	C *	P *	C *	P *	C *	P *	C *	P *	C *	P *
P1	1	0	1	0	1	0	1	0	1	0	1	0	P41	3	0	3	0	3	0	3	0	2	1	2	1
P3	2	0	2	0	1	1	1	1	0	2	0	2	P42	2	0	2	0	2	0	2	0	2	0	2	0
P4	1	0	1	0	1	0	1	0	0	1	0	1	P43	1	0	1	0	1	0	1	0	1	0	1	0
P5	1	0	1	0	1	0	1	0	0	1	0	1	P45	2	0	1	1	1	1	1	1	1	1	1	1
P6	1	0	1	0	1	0	1	0	1	0	1	0	P46	1	0	1	0	1	0	1	0	1	0	1	0
P7	2	0	2	0	2	0	2	0	2	0	2	0	P47	5	0	3	2	2	3	2	3	0	5	0	5
P8	2	0	2	0	2	0	1	1	1	1	1	1	P48	1	0	1	0	1	0	1	0	1	0	1	0
P10	1	0	1	0	1	0	1	0	1	0	0	1	P51	1	0	0	1	0	1	0	1	0	1	0	1
P11	2	0	2	0	2	0	2	0	1	1	1	1	P52	2	0	2	0	2	0	2	0	2	0	2	0
P12	1	0	1	0	0	1	0	1	0	1	0	1	P53	1	0	1	0	1	0	1	0	0	1	0	1
P13	2	0	2	0	2	0	2	0	2	0	2	0	P55	1	0	1	0	1	0	1	0	1	0	1	0
P16	2	0	2	0	2	0	1	1	1	1	0	2	P56	1	0	1	0	1	0	1	0	1	0	1	0
P19	2	0	2	0	1	1	1	1	1	1	1	1	P58	1	0	1	0	0	1	0	1	0	1	0	1
P20	2	0	2	0	2	0	2	0	2	0	2	0	P59	3	0	2	1	0	3	0	3	0	3	0	3
P21	2	0	2	0	2	0	2	0	2	0	1	1	P60	2	0	2	0	2	0	2	0	2	0	2	0
P22	2	0	2	0	2	0	2	0	2	0	1	1	P61	2	0	2	0	1	1	0	2	0	2	0	2
P23	2	0	2	0	2	0	2	0	2	0	1	1	P62	2	0	2	0	1	1	1	1	1	1	1	1
P24	1	0	1	0	1	0	1	0	1	0	1	0	P63	1	0	1	0	1	0	0	1	0	1	0	1
P25	1	0	1	0	1	0	1	0	1	0	1	0	P65	1	0	1	0	1	0	1	0	1	0	1	0
P26	4	0	4	0	4	0	3	1	3	1	1	3	P68	1	0	1	0	1	0	1	0	1	0	1	0
P27	3	0	3	0	2	1	2	1	2	1	0	3	P69	1	0	1	0	1	0	0	1	0	1	0	1
P28	3	0	3	0	3	0	3	0	3	0	2	1	P71	4	0	4	0	3	1	3	1	3	1	3	1
P30	3	0	1	2	1	2	1	2	0	3	0	3	P72	1	0	1	0	1	0	1	0	1	0	1	0
P31	1	0	1	0	1	0	1	0	1	0	0	1	P73	1	0	1	0	1	0	1	0	1	0	1	0
P32	2	0	2	0	2	0	2	0	1	1	1	1	P75	2	0	2	0	2	0	2	0	2	0	2	0
P33	2	0	2	0	2	0	1	1	1	1	1	1	P76	1	0	1	0	1	0	1	0	0	1	0	1
P34	1	0	1	0	1	0	1	0	1	0	1	0	P77	1	0	1	0	1	0	0	1	0	1	0	1
P35	1	0	1	0	1	0	1	0	1	0	1	0	P80	2	0	2	0	2	0	2	0	2	0	2	0
P36	2	0	2	0	2	0	2	0	2	0	1	1	P81	1	0	0	1	0	1	0	1	0	1	0	1
P37	2	0	2	0	1	1	0	2	0	2	0	2	P82	2	0	1	1	1	1	1	1	1	1	1	1
P38	3	0	3	0	3	0	2	1	2	1	2	1	P85	1	0	1	0	1	0	1	0	1	0	1	0
P39	1	0	1	0	1	0	1	0	1	0	1	0	P86	2	0	2	0	2	0	2	0	1	1	1	1
P40	2	0	0	2	0	2	0	2	0	2	0	2	P88	1	0	1	0	1	0	1	0	1	0	1	0

* C—consumer; P—prosumer.

from Appendix A presents the allocation of the end-users. They were classified as consumers and prosumers, at each pillar associated with the five penetration degrees, PD $\in$ {0%, 10%, 20%, 30%, 40, and 50%}. Considering the energy consumption uploaded from the SMS, the PV systems were sized (having the installed capacities of 3 and 5 kWp). The sizing process was based on the generation profiles from the geographical location of the LVEDN uploaded from the PVGIS tool [49]. kWp (kilowatts peak) is associated with the peak power of a PV system. PV systems have an installed capacity expressed in kWp, representing the rate at which they generate the energy at peak performance, such as on a sunny day in the afternoon [50,51]. Another assumption refers to prosumers who do not have energy storage systems, representing the current situation in Romania. Figure 7 and Figure 8 present the generation profiles of the two PV systems (3 and 5 kWp), ${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2 {\times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$ (low energy production), ${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$ (average energy production), and ${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$+ ${2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$ (high energy production), determined for June and used in the analysed scenarios.

The number of the scenarios (SC) obtained from the combinations between the three indicators is N_SC = PD$\times$CEC$\times$EP_PV = 5$\times$5$\times$3 = 75 (see

Table 2. The analysed scenarios (SA) with the features of the three indicators.

SC	PD [%]	EPPV [kWh]	CEC [%]	SC	PD [%]	EPPV [kWh]	CEC [%]	SC	PD [%]	EPPV [kWh]	CEC [%]
S1	10	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−10%	S26	10	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	−10%	S51	10	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−10%
S2	10	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−5%	S27	10	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	−5%	S52	10	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−5%
S3	10	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	0%	S28	10	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	0%	S53	10	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	0%
S4	10	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+5%	S29	10	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	+5%	S54	10	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+5%
S5	10	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+10%	S30	10	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	+10%	S55	10	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+10%
S6	20	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−10%	S31	20	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	−10%	S56	20	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−10%
S7	20	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−5%	S32	20	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	−5%	S57	20	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−5%
S8	20	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	0%	S33	20	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	0%	S58	20	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	0%
S9	20	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+5%	S34	20	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	+5%	S59	20	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+5%
S10	20	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+10%	S35	20	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	+10%	S60	20	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+10%
S11	30	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−10%	S36	30	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	−10%	S61	30	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−10%
S12	30	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−5%	S37	30	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	−5%	S62	30	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−5%
S13	30	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	0%	S38	30	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	0%	S63	30	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	0%
S14	30	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+5%	S39	30	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	+5%	S64	30	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+5%
S15	30	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+10%	S40	30	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	+10%	S65	30	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+10%
S16	40	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−10%	S41	40	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	−10%	S66	40	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−10%
S17	40	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−5%	S42	40	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	−5%	S67	40	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−5%
S18	40	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	0%	S43	40	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	0%	S68	40	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	0%
S19	40	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+5%	S44	40	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	+5%	S69	40	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+5%
S20	40	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+10%	S45	40	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	+10%	S70	40	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+10%
S21	50	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−10%	S46	50	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	−10%	S71	50	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−10%
S22	50	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−5%	S47	50	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	−5%	S72	50	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	−5%
S23	50	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	0%	S48	50	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	0%	S73	50	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	0%
S24	50	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+5%	S49	50	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	+5%	S74	50	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+5%
S25	50	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+10%	S50	50	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$	+10%	S75	50	${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}+{ 2 \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$	+10%

), to which is added the base scenario (S0), represented by the current situation where PD = 0, CEC = 0, and EP_PV = 0.

The MV side of the EDS represents in our study the slack node used in the steady-state calculations. However, the final results will refer only to the low-voltage part to highlight the impact of the RBES on the active LVEDN. Figure 9 presents the aggregation of the active powers at the LV level (0.4 kV) of the EDS for the analysed day from June associated with scenario S0. The OLTC operates on tap α_s = 8 regardless of the time slot s, s = 1, …, 24, in this scenario. The high differences between loads of the three phases can be observed, having an unfavourable effect on the energy losses. This effect is due to the pronounced unbalance degree reflected on the additional circulations in the neutral conductor; see Figure 10.

The highest energy losses are in the conductor from the phase γ_b (48.4%), followed by neutral (38.5%), phase γ_c (9.6%), and phase γ_a (3.4%). The percentage values have been reported for the total energy losses (77.03 kWh). The energy losses contain only the component associated with the losses in conductors. However, the LVEDNs with the high unbalance degree represents a common feature of the Romanian distribution system. Thus, the DNO should solve the unbalance issue before integrating prosumers that could cause additional issues, as will be presented in the following.

Regarding the voltage quality, all phase voltages at the pillar level have values inside the allowable range [−10%, +10%] according to the European Standard EN 50160 [52], even if the OLTC has a constant tap position, α = 8. Figure 11 presents the phase voltage variation in the analysed period, where more values are very close to the lower limit value (0.9).

In the next step, operating the LVEDN without using the RBES in the OLTC-based voltage control in all scenarios presented in Table 2 was considered.

Table 3. The maximum value of the phase voltage obtained after performing the steady-state calculations in the analysed scenarios characterized by the features of the three indicators (PD, E_PV, and CEC) without applying RBES.

CEC [%]	PD [%]
	10			20			30			40			50
	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV
	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$
−10	1.053	1.053	1.053	1.053	1.074	1.130	1.059	1.127	1.201	1.081	1.175	1.263	1.128	1.209	1.285
−5	1.053	1.053	1.053	1.053	1.065	1.125	1.057	1.119	1.196	1.075	1.167	1.259	1.125	1.207	1.283
0	1.053	1.053	1.053	1.053	1.064	1.118	1.054	1.114	1.189	1.071	1.162	1.252	1.123	1.205	1.281
5	1.053	1.053	1.053	1.053	1.063	1.110	1.053	1.109	1.182	1.068	1.157	1.245	1.121	1.201	1.279
10	1.053	1.053	1.053	1.053	1.062	1.105	1.053	1.103	1.177	1.066	1.152	1.241	1.119	1.201	1.278

and

Table 4. The total energy losses in the conductors calculated in the analysed scenarios characterized by the features of the three indicators (PD, E_PV, and CEC) without applying RBES.

CEC [%]	PD [%]
	10			20			30			40			50
	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV
	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2 {\times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2 {\times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$
−10	39.4	42.5	48.0	41.8	42.8	48.2	39.4	47.6	70.8	40.0	62.0	107.5	46.1	85.4	157.1
−5	44.4	48.5	54.3	46.6	48.3	51.9	44.2	51.3	73.0	44.1	64.7	108.5	49.9	87.6	157.4
0	50.0	54.9	61.2	51.8	54.4	56.2	49.5	55.2	75.8	48.7	67.6	110.2	54.1	90.0	158.2
5	56.1	61.3	68.5	57.1	60.9	60.9	55.1	59.4	79.0	53.6	70.9	112.1	58.7	92.8	159.5
10	62.2	68.3	76.2	62.9	65.6	67.8	61.5	64.0	82.3	59.3	74.5	114.3	64.0	95.9	161.0

present the results associated with the minimum and maximum phase voltages and total energy losses obtained after performing the steady-state calculations. The tap position of the OLTC was considered constant, α = 8, for all time slices, s = 1, …, S, in all scenarios characterized by the values of PD, EP_PV, and CEC. The following notations have been used for the indicator EP_PV: low energy production—EP^L_PV (EP^L_PV = ${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}-2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$), average energy production—EP^A_PV (EP^A_PV = ${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$), and high energy production—EP^H_PV (EP^H_PV = ${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$+$2{ \times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$).

The analysis of the results presented in Table 3 highlighted that in 40 scenarios (53.3%) from the total number, the phase voltage exceeded the upper allowable value (1.1 p.u.). All values over 1.1 p.u. are highlighted in the table in bold. These situations were recorded in the phases γ_a and γ_c, where the requested power of the consumers was small, and the injected power of the prosumers was high. The increases were between 0.005 p.u. (scenario S30, with PD = 20%, CEC = 10 %, and EP^H_PV) and 0.185 p.u. (scenario with PD = 50%, CEC = −10 %, and EP^H_PV).

Regarding the energy losses, the values are lower for 58 scenarios (77.33% of the total number) than in the base scenario, S0 (ΔW = 77.03 kWh). The higher values, highlighted with bold font in the table, have been recorded for PD = 40%, 50%, EP^H_PV, and all considered possibilities of CEC, CEC $\in$ {−10, −5, 0, 5, 10}. In addition, the energy losses decrease in the columns top-down (from the smaller to higher CEC) and increase in the rows left to right (from the lower to higher PD and EP_PV), aspects usually encountered in the LVEDNs. The minimum energy losses correspond to scenarios S31–S35 with a PD = 30%, all values of CEC, CEC $\in$ {−10, −5, 0, 5, 10}, and EP^L_PV.

A scenarios-based voltage quality matrix can be available to the DNO, indicating possible future values of the maximum phase voltage in the LVEDNs and highlighting the worst scenarios (bold font and larger size); see Figure 12. The rows of the matrix are associated with the CEC possibilities that go top-down (from small to high). The columns correspond to various alternatives for the EP_PV and PG following one another from left to right.

It is clear that if the RBES-based voltage control is not used, the DNO can have issues with the integration in the LVEDN of an increasing number of prosumers. These issues can be more complicated if there are changes in energy consumption at the level of consumers.

Thus, the proposed RBES-based voltage control has been applied only in the scenarios with voltage issues.

Table 5. The maximum value of the phase voltage obtained after performing the steady-state calculations in the analysed scenarios characterized by the features of the three indicators (PD, E_PV, and CEC) with applying RBES.

CEC [%]	PD [%]
	10			20			30			40			50
	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV
	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2 {\times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$
−10	1.053	1.053	1.053	1.053	1.074	1.090	1.059	1.089	1.098	1.081	1.087	1.143	1.097	1.095	1.137
−5	1.053	1.053	1.053	1.053	1.065	1.100	1.057	1.100	1.100	1.075	1.099	1.138	1.100	1.097	1.135
0	1.053	1.053	1.053	1.053	1.064	1.096	1.054	1.095	1.098	1.071	1.098	1.131	1.098	1.099	1.133
5	1.053	1.053	1.053	1.053	1.063	1.091	1.053	1.089	1.098	1.068	1.093	1.123	1.098	1.097	1.131
10	1.053	1.053	1.053	1.053	1.062	1.091	1.053	1.091	1.098	1.066	1.091	1.118	1.096	1.095	1.129

and

Table 6. The total energy losses in the conductors calculated in the analysed scenarios characterized by the features of the three indicators (PD, E_PV, and CEC) with applying RBES.

CEC [%]	PD [%]
	10			20			30			40			50
	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV
	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2 {\times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$
−10	39.4	42.5	48.0	41.8	42.8	48.1	39.4	47.5	77.4	40.0	65.8	127.8	46.6	95.5	191.3
−5	44.4	48.5	54.3	46.6	48.3	51.7	44.2	51.1	78.6	44.1	67.8	128.0	49.9	97.1	190.5
0	50.0	54.9	61.2	51.8	54.4	55.6	49.5	54.6	80.3	48.7	69.1	128.5	53.7	97.9	190.0
5	56.1	61.3	68.5	57.1	60.9	60.2	55.1	58.7	82.6	53.6	72.1	129.9	58.1	100.2	190.6
10	62.2	68.3	76.2	62.9	65.6	64.3	61.5	62.7	84.9	59.3	75.1	130.0	62.8	102.5	190.5

present the data associated with these scenarios in bold font. For the scenarios without voltage issues, the maximum voltage corresponds to the steady state without applying the RBES-based voltage control.

The voltage issues have been removed for 30 of 40 scenarios where the maximum values of the phase voltage exceeded the upper allowable limit. The remaining scenarios with voltage issues, (S56–S60, and S71–S75), identified with bold font and larger size in Table 5 and the scenarios-based voltage quality matrix represented in Figure 13, are associated with PD $\in$ {40%, 50%}, EP^H_PV, and CEC $\in$ {−10%, −5%, 0%, 5%, 10%}.

The energy losses in the conductors for the scenarios (S26–S30) and (S36–S40) are slightly lower between 0.21% (S26 and S36) and 5.16% (S30); see Table 6. For all other scenarios (S41–S45 and S51–S75) the values are higher, especially for S56–S60. In addition, the last scenarios have voltage issues (as previously emphasized) corresponding to the largest energy amount injected by the prosumers in the network.

Regarding the decision variables (tap positions of the OLTC), Figure 14 presents the total number of tap position changes in the studied period and all scenarios. The maximum changes (over 9 changes) have been recorded beginning with PD = 30%, EP^H_PV, and CEC $\in$ {−10%, −5%, 0%, 5%, 10%}, at the hours with the high power injections to keep the voltage below the maximum admissible limit. About half of the scenarios, S1–S25, S31–S35, and S46–S50 (35, representing 47%), are characterized through the same tap position of the OLTC without additional tap changes.

4. Discussion

The proposed expert system, including rule-based reasoning (RBES), applied in 75 scenarios had a success rate of 86.7% (65 out of 75 scenarios did not have the voltage issues anymore). For the remaining scenarios with voltage issues (S56–S60, and S71–S75), the maximum value of the phase voltage was reduced with the values between 0.12 p.u (9.5%) recorded for scenario S56 (PD = 40%, EP^H_PV, and CEC = −10%) and 0.15 p.u. (11.6%) associated with scenario S75 (PD = 50%, EP^H_PV, and CEC = +10%).

Another voltage quality index can be used for these scenarios to quantify the unallowable voltage variations; see

Table 7. The values of UVD, in [%], calculated for the scenarios S56–S60 and S71–S75. (EP^H_PV = ${\mathrm{m}}_{\mathrm{EP}}^{\mathrm{PV}}$ + $2 {\times \mathsf{\sigma }}_{\mathrm{EP}}^{\mathrm{PV}}$).

PD [%]	CEC [%]
	−10	−5	0	5	10
40	2.86/S57	2.48/S56	2.11/S58	1.87/S59	1.73/S60
50	4.98/S71	4.91/S72	4.71/S73	4.60/S74	4.49/S75

. This index calculated with relation (10) shows the percentage of the time slots in which the phase voltage deviations exceed the allowable limits.

$$\mathrm{UVD}=\frac{{\displaystyle \sum _{\mathrm{e}=1}^{\mathrm{NE}}{\displaystyle \sum _{\mathrm{s}=0}^{\mathrm{s}}\left({\displaystyle \sum _{\mathsf{\gamma }\in \{{\mathsf{\gamma }}_{\mathrm{a}},{\mathsf{\gamma }}_{\mathrm{b}},{\mathsf{\gamma }}_{\mathrm{c}}\}}^{}\left(\left|\frac{{\mathrm{V}}_{\mathrm{e},\mathrm{s}}^{\left\{\mathsf{\gamma }\right\}}-{\mathrm{V}}_{\mathrm{r}}}{{\mathrm{V}}_{\mathrm{r}} }\right|\right)}\right)}}}{{\mathrm{N}}_{\mathsf{\gamma }}\cdot \mathrm{NE}\cdot \mathrm{s}}$$

(10)

where the signification of all variables has been indicated in Section 2.

The values of UVD from Table 7 indicate that the phase voltages in the nodes (at the pillar level) of the LVEDN for all 10 scenarios are outside the range [−10%, +10], in a percentage between 1.73% and 4.98%. Higher percentages, between 4.49% and 4.98%, have been recorded for scenarios S71–S75. Thus, in at least 95% of the time slots, the voltages have been inside the range [−10%, +10%], according to the European Power Quality Standard. In these conditions, the DNOs should be very careful and take additional measures to improve the voltage level in the LVEDN, and one of them can be the phase load balancing [15], taking into account the higher unbalance degree emphasized in the base scenario, S0.

Regarding the energy losses, lower values have been recorded for 58 scenarios (77.33% of the total number) than in the base scenario with PD = 0, CEC = 0, and EP_PV = 0 (ΔW = 77.03 kWh). Higher values have been recorded for PD = 40%, 50%, EP^H_PV, and all considered possibilities of CEC, CEC ∈ {−10, −5, 0, 5, 10}. In addition, the energy losses decrease in the columns top-down (from the smaller to higher CEC) and increase in the rows left to right (from the lower to higher PD and EPPV), aspects usually encountered in the LVEDNs. The minimum energy losses correspond to scenarios S31–S35 with a PD = 30%, all values of CEC, CEC ∈ {−10, −5, 0, 5, 10}, and EP^L_PV.

The obtained results have been compared with an efficient method, which proved its effectiveness in voltage control, having the same assumptions as in the proposed RBES to demonstrate the performance of the RBES. The control voltage has been treated as a multi-objective problem where the optimal tap positions of the OLTC in each time slot are determined to minimize the energy losses and voltage deviations in the presence of technical constraints associated with operating the LVEDN [6]. Combinatorial optimization has been used to solve the mathematical model using the Total Enumeration algorithm for rapid convergence. Thus, identification of the optimal tap position α* has been made from the set of admissible solutions that contains all possible combinations. Finally, not all combinations are memorized, except the optimal position that minimizes both objectives and satisfies the technical constraints.

Table 8. The errors between the energy losses calculated with the MM method and RBES, in [%].

CEC [%]	PD [%]
	10			20			30			40			50
	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV
	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2 {\times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$
−10	2.50	2.85	2.96	2.84	3.54	2.87	3.19	3.32	1.60	3.46	2.05	0.00	3.04	1.04	0.00
−5	2.18	2.10	2.10	2.08	2.78	2.99	2.29	3.60	1.56	3.02	2.28	0.00	2.90	1.02	0.00
0	1.89	1.77	1.64	1.75	2.31	2.68	1.91	2.60	1.61	2.14	2.05	0.00	2.18	1.16	0.00
5	1.50	1.78	1.68	1.77	2.19	2.60	1.84	2.50	1.41	2.10	1.77	0.00	2.31	0.94	0.00
10	1.34	1.61	1.47	1.79	1.84	2.32	1.60	2.21	1.38	1.80	1.50	0.00	2.08	0.85	0.00

presents the errors associated with the energy losses between the two methods calculated with the relation:

$$\delta \Delta \mathrm{W}=\left(\frac{\Delta {\mathrm{W}}_{\mathrm{MM}}-{\mathrm{W}}_{\mathrm{RBE}\mathrm{s}}}{\Delta {\mathrm{W}}_{\mathrm{MM}}}\right) \cdot 100 , [\%]$$

(11)

where ΔW_MM represents the energy losses calculated with the multi-objective method (MM), and ΔW_RBED corresponds to the energy losses determined with the proposed RBES.

The analysis of the data highlighted that the energy losses are slightly smaller in the case of RBES compared with the MM method, with differences between 0.85% (PD = 50%, EP^A_PV, and CEC = 10%) and 3.54% (PD = 20%, EP^A_PV, and CEC = −10%). The similar values of the maximum phase voltage have been obtained in 47 scenarios (62.7%) highlighted through null values of the errors; see

Table 9. The errors between maximum values of the phase voltage calculated with the MM method and RBES, in [%].

CEC [%]	PD [%]
	10			20			30			40			50
	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV
	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$
−10	0.08	0.08	0.08	0.08	1.98	0.53	0.62	0.91	0.35	2.62	0.00	0.00	1.35	0.00	0.00
−5	0.00	0.00	0.00	0.00	0.00	1.83	0.00	2.27	0.00	0.85	1.68	0.00	1.85	0.00	0.00
0	0.00	0.00	0.00	0.00	0.00	1.68	0.00	1.57	0.00	0.17	0.17	0.00	1.87	0.00	0.00
5	0.00	0.00	0.00	0.00	0.00	1.23	0.00	1.11	0.00	0.24	0.24	0.00	1.89	0.00	0.00
10	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.44	0.00	0.00	0.00	0.45	0.00	0.00

. However, the maximum values are slightly higher in the other scenarios (28) for the RBES compared with the MM method, between 0.08% (PD = 10%, EP^L_PV, and CEC = −10%) and 2.62% (PD = 40%, EP^L_PV, and CEC = −10%). In addition, the voltage issues have not been solved fully for scenarios S56–S60 and S71–S75, but in at least 95% of the time slots, the voltages were inside the range [−10%, +10%], according to the European Power Quality Standard, and for the rest of the time slots, up to 100% were very close to the limits of the range.

The highest differences have been identified at the level of the optimization variables (the tap positions). A comparison between methods has been made only for the scenarios with voltage issues highlighted with bold font in the tables. The total number of tap position changes is higher in the MM method than in the RBES, which makes RB more efficient; see

Table 10. The additional tap position changes of the OLTC (MM vs. RBES).

CEC [%]	PD [%]
	10			20			30			40			50
	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV
	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$
−10	0	0	0	0	0	4	0	4	4	0	4	0	6	1	0
−5	0	0	0	0	0	4	0	4	2	0	2	0	4	1	0
0	0	0	0	0	0	4	0	4	2	0	4	0	0	2	0
5	0	0	0	0	0	4	0	4	2	0	4	0	2	2	0
10	0	0	0	0	0	4	0	4	2	0	2	0	2	2	0

.

Table A2. The tap position changes of the OLTC (RBES-based method).

CEC [%]	PD [%]
	10			20			30			40			50
	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV
	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$
−10	0	0	0	0	0	4	0	4	10	0	8	14	4	13	14
−5	0	0	0	0	0	4	0	4	12	0	10	14	4	13	14
0	0	0	0	0	0	3	0	3	11	0	7	13	7	11	13
5	0	0	0	0	0	3	0	3	9	0	7	13	5	11	13
10	0	0	0	0	0	7	0	7	13	0	11	17	9	15	17

and

Table A3. The additional tap position changes of the OLTC (MM-based method).

CEC [%]	PD [%]
	10			20			30			40			50
	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV	EPLPV	EPAPV	EPHPV
	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}-2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$	${\mathbf{m}}_{\mathbf{EP}}^{\mathbf{PV}}$$+2{ \times \mathsf{\sigma }}_{\mathbf{EP}}^{\mathbf{PV}}$
−10	0	0	0	0	0	8	0	8	14	0	12	14	10	14	14
−5	0	0	0	0	0	8	0	8	14	0	12	14	8	14	14
0	0	0	0	0	0	7	0	7	13	0	11	13	7	13	13
5	0	0	0	0	0	7	0	7	11	0	11	13	7	13	13
10	0	0	0	0	0	11	0	11	15	0	13	17	11	17	17

from Appendix A present the tap position changes for the two methods.

Table 11. Comparison between the computational times.

No.	Algorithm	ACT [seconds]
1	RBES (Proposed)	0.265
2	MM	0.253

shows the average computational time (ACT) required to get the optimal solution in a time slot. The algorithms were developed using a Matlab implementation and were run on a computer with an Intel Core i7, 3.10 GHz processor, 4 GB RAM, and a 64-bit Windows 10 operating system.

The ACT is very close between the methods (the error is 4.7%). However, due to the other advantages regarding the smaller energy losses and the total number of tap position changes, the RBES method can represent a better alternative for DNOs.

If the sampling step of smart meters to send information to the data concentrator has smaller values (15 or 30 min), then the total number of tap position changes increases. However, the technical constraint regarding the total number of tap changes from a day will not be violated, as shown in [53]. The maximum number considered in this research was 96, according to [41].

5. Conclusions

The voltage quality has been identified by DNOs as one of the main factors that can decrease the hosting capacity to accommodate the growing number of PV prosumers. The reverse power flows from the LVEDN toward the EDS represent the reason for various voltage issues, leading to uncertainties on the voltage level of the nodes. Thus, new methodologies should be developed to solve the voltage issues caused by PV prosumers based on the devices integrated into the LVEDN.

One of the most efficient devices to ensure the resilience of voltage control is represented by the OLTC, which equips the MV/LV distribution transformers. One of the most challenging issues in the operation of the OLTC in the LV EDNs is the determination of the optimal tap position, which leads to a fit 95% of the time between the admissible limits of the phase voltage in all nodes following the performance standards, in conditions when the reverse power flows can occur.

In this paper, an efficient expert system, including rule-based reasoning (RBES), has been developed to solve the voltage issues caused by PV prosumers, having as advantages the “fast scanning” of the input data, identification of voltage issues that come up, and determination of a solution associated with the tap position of an OLTC that does not violate the voltage constraints in the PV-rich network based on the deviations between the reference voltage and the voltages recorded in the nodes in each time slot, recognizing the excesses of the allowable limits, regardless of the power flow’s direction. These advantages offer RBES efficiency and resilience. Efficiency is associated with their ability to meet the demands of the end-users regarding voltage quality, and resilience is characterized by the response to harmful events represented by sudden voltage variations due to the intermittent regime of the small-scale renewable sources (prosumers).

Testing the RBES was conducted considering an aerial LVEDN supplied by an EDS whose MV bus (20 kV) represents the CCP with the network of a Romanian DNO that carries out its distribution service in the respective area. More scenarios (75) have been considered in the case study, characterized by the combinations between the penetration degree, consumption evolution, and energy production of the PV systems from the month that includes the analysed day. The success rate of the RBES was 86.7% (65 out of 75 scenarios did not have the voltage issues anymore). For the remaining 10 associated with the highest values of PD (40% and 50%), high energy production and all considered consumptions the voltage have been mitigated, being close to the upper allowable limit, but at least 95% of the time slots were inside the range [−10%, +10%], according to the European Power Quality Standard.

The comparison with another method based on the multi-objective optimization with two criteria (minimization of the energy losses and voltage deviations in the nodes of the LVEDN) demonstrated the efficiency of the RBES, quantified through smaller energy losses and the total number of tap position changes.

Even under these conditions, the implementation of RBES in the real application integrated into the LVEDN depends on the speed and sampling step of the data transmission from the smart meters installed at the end-user level (in the study case, the sampling step was 60 min according to the setting implemented by the DNO at the level of the SMS) and the performance of the processing module from the data concentrator at the EDS level. Regarding the expected costs to implement the proposed voltage control solution, DNOs should consider an investment cost of 7000 euros and an approximated cost per tap position change of 0.01 euros, according to [54]. In addition, the costs allocated to the maintenance operations should be integrated into the economic analysis, but these are low due to the high reliability of the OLTC (700,000 tap position changes between two maintenance operations [41]). All these costs should be assumed by DNOs beginning with the initial phase of the planning strategies associated with the PV-rich LVEDNs to avoid voltage issues later. However, the voltage control based on the OLTC represents a practical and efficient solution.

The authors are working now on an improved variant of the RBES, which includes an optimization process associated with phase load balancing to remove the voltage issues for higher penetration degrees, as observed in the case study.