Accurate Assessment of Decoupled OLTC Transformers to Optimize the Operation of Low-Voltage Networks

energies Article Accurate Assessment of Decoupled OLTC Transformers to Optimize the Operation of Low-Voltage Networks Álvaro Rodríguez del Nozal 1,* , Esther Romero-Ramos 2and Ángel Luis Trigo-García 2 1Departamento de Ingeniería, Universidad Loyola Andalucía, 41014 Seville, Spain 2 Department of Electrical Engineering, Universidad de Sevilla, 41092 Sevilla, Spain; eromer[email protected] (E.R.-R.); [email protected] (Á.L.T.-G.) *Correspondence: ar[email protected] Received: 7 May 2019; Accepted: 3 June 2019; Published: 6 June 2019   Abstract: Voltage control in active distribution networks must adapt to the unbalanced nature of most of these systems, and this requirement becomes even more apparent at low voltage levels. The use of transformers with on-load tap changers is gaining popularity, and those that allow different tap positions for each of the three phases of the transformer are the most promising. This work tackles the exact approach to the voltage optimization problem of active low-voltage networks when transformers with on-load tap changers are available. A very rigorous approach to the electrical model of all the involved components is used, and common approaches proposed in the literature are avoided. The main aim of the paper is twofold: to demonstrate the importance of being very rigorous in the electrical modeling of all the components to operate in a secure and effective way and to show the greater effectiveness of the decoupled on-load tap changer over the usual on-load tap changer in the voltage regulation problem. A low-voltage benchmark network under different load and distributed generation scenarios is tested with the proposed exact optimal solution to demonstrate its feasibility. Keywords: active distribution networks; voltage control; on-load tap changer transformers; low-voltage grids 1. Introduction Low-voltage (LV) networks are generating increasing interest for a variety of reasons, such as the massive deployment of smart meters [ 1 ], the growing presence of distributed renewable generation (DG) [ 2 ], and important new components such as electric vehicles (EVs) and storage devices (SDs) [ 3 ]. The eruption of all these new actors has completely changed the approach to planning and operating voltage levels in light of two main facts. Firstly, new low-carbon technologies cause new planning and operational problems that were never taken into account when passive consumers were the only clients connected to this voltage level [ 4 – 6 ]. Secondly, a much greater extent of new and detailed information is available from smart meters [ 7 ], generating a very valuable starting point for tackling these new planning and operational challenges. Power quality is one of the main concerns in an LV grid. Classical consumers and the aforementioned new participants demand quality from the distribution utility, with the level of voltage being one of the most important aspects. Many distribution companies have reported voltage complaints because of steady-state undervoltages and/or overvoltages [8–12]. On the one hand, permanent undervoltages/overvoltages are one of the issues that consumers are more concerned about. Both phenomena can lead to issues such as shutdowns, malfunctions, Energies 2019,12, 2173; doi:10.3390/en12112173 www.mdpi.com/journal/energies Energies 2019,12, 2173 2 of 22 overheating, premature failures, and poor efficiency of consumer devices [ 13 ]. On the other hand, the level of supplied voltage is also of great relevance to the distribution company, and not only because of consumer complaints. It is well known that voltage control allows utilities to improve the operation efficiency of their networks by minimizing power losses [ 14 ], taking advantage of conservation voltage reduction [15], delaying new investments [16], and so on. Some of the most common causes of undervoltage and overvoltage are the unsuitable tap setting of secondary distribution transformers, improper voltage regulation setting of substations, transformer/cable overloads (for undervoltages), and three-phase imbalances. As a general rule, the voltage of monitored networks tends to be closer to the upper voltage limits than the lower ones [ 11 , 12 ], which implies that the irruption of massive embedded generation can even worsen the frequency and duration of overvoltage phenomena [17]. The above details highlight that voltage control in low-voltage grids is of great relevance. Volt/var control of distribution networks has usually been confined to the medium-voltage (MV) level by using the on-load tap changer (OLTC) of high-voltage (HV)/MV transformers located at substations, capacitor banks, and/or step voltage regulators that are allocated along distribution feeders. Advanced volt/var control solutions specifically adapted to active distribution systems propose the use of new control sources: utility and customer-owned distributed power electronics (mainly those linked to DG), energy storage systems, power curtailment of DG generation and/or load, etc. All these novel control tools are well developed in the literature but not so much in practice [ 18 , 19 ]. For LV levels, in practice, most utilities with a European-style network design use only the secondary distribution transformers equipped with off-load tap changers to control voltages. Once again, the use of both the traditional and novel MV volt/var control solutions mentioned above has been intensively proposed in the most current literature, but it is not applied in practice [ 20 – 31 ]. The use of OLTC transformers in secondary distribution substations is currently the closest to realizing an LV reality [ 19 , 32 – 35 ]. This paper focuses on this last point: the use of OLTC MV/LV transformers (OLTCST) to control voltage in LV networks. Depending on the technology, two different methods of operating an OLTCST are possible: (1) a synchronized tap change among the three phases or (2) decoupled control [ 32 ]. Most previous works have dealt with uniform and common tap positions for all three phases of the MV/LV-controlled transformer (3P-OLTCST) [ 19 – 21 , 28 , 30 , 31 , 33 – 35 ], but increasingly, more tap position settings among transformer phases are being proposed (1P-OLTCST) [ 24 , 25 , 29 , 32 , 36 ]. Decoupled control is obviously a great deal more attractive for LV networks because of their unbalanced nature. Regardless of whether uniform or non-uniform phase tap movements are employed, there are two main methodologies for fulfilling voltage control: rule-based approaches or analytical solutions. Among the first group, different levels of “intelligence” have been proposed to reach a global solution that is as close as possible to the optimal one [ 21 , 24 , 28 , 32 ]. Analytical approaches are integrated solutions that solve an optimization problem to determine the best control actions that both minimize an objective function and fulfill all the electrical constraints [ 29 , 30 , 36 ]. While rule-based solutions reach a suboptimal solution but are easier to implement and need less network information, analytical approaches are more rigorous from a mathematical point of view and ensure not only an optimal solution but also that all the variables are within their limits. Until recently, the main drawbacks of applying analytical solutions laid in their computational cost (both programming and time consumed) and the necessity of having detailed models of the controlled network and scenarios of demand/generation for each time. Nowadays, the current trend toward active LV networks implies not only that these barriers are starting to disappear but also that the current knowledge of networks and power demand/dispersed generation can generate great value for distribution utilities [37]. On this basis, this paper tackles the optimal voltage control problem of European LV networks by using decoupled OLTC transformers (1P-OLTCST) and implementing an analytical solution. Previous works proposing analytical approaches to this problem have been mostly confined to American MV networks [30,36]. Energies 2019,12, 2173 3 of 22 The authors in [ 30 ] use as voltage controllers 3P-OLTCST and PV inverters to minimize power losses, voltage violations, and lines loading; a three-phase simplified line model is assumed for four-wire three-phase lines as well as approximated voltage drop equations; no electrical model is taken into account for 3P-OLTCTS. The optimization problem proposed in [ 36 ] works with 1P-OLTCTS and static voltage converters, and although three-phase transformer models are incorporated into the mathematical problem, once again a three-phase simplified line model is considered. In addition, ground resistances different from zero are not taken into consideration. An optimal solution specifically designed for European MV and LV grids (both integrated) is published in [ 29 ]. In that approach, the optimal location and size of dSTATCOM, SD, and 3P-OLTCST, as well as the reactive contribution by DG’s inverters are determined in order to manage voltage and minimize investment, operation, and maintenance costs. However, no details are provided in [ 29 ] to clearly deduce if detailed models for transformers, lines, and ground resistances are considered. In contrast, this new proposed work takes note of previous shortcomings and considers them warnings of the importance of neutral grounding considerations [ 38 ] and the need to use a rigorous model for all system components [ 39 ]. Therefore, the main novelties of this paper are the following: • The optimization problem associated with the optimal voltage control of unbalanced LV networks by using transformers with OLTCs as control devices is rigorously defined and solved. No approximations are assumed and detailed models for all the components are considered: – three-phase model of three-phase transformers, incorporating also their ground connection design; –three-phase four-wire lines are neither reduced to a three-phase model nor decoupled, –any earthing system type is taken into account. • The former optimization problem is implemented and solved, comparing the real possibilities of 1P-OLTCST versus 3P-OLTCST. From the knowledge of the authors, no previous work has posed this voltage optimization problem for unbalanced LV networks with such a level of detail, so the results are quite conclusive. The layout of the paper is as follows: Section 2describes the framework in which voltage control is performed, and Section 3tackles the mathematical problem definition by considering the appropriate model for each of the components entering the studied system. Section 4shows the obtained results for a European low-voltage benchmark system to demonstrate both the benefits resulting from decoupled tap control compared with the traditional uniform three-phase tap control and the importance of the model to carry out effective control. Finally, Section 5presents the most relevant conclusions and suggests the next steps for future work. 2. Low-Voltage European Networks Distribution networks with the so-called European design are not limited to Europe; in fact, they are widely used all over the world. Many other countries adopted this arrangement, and the European design is well established, even in urban North American distribution networks [ 40 ]. The general structure of European distribution systems consists of a three-phase medium-voltage system in which three phase distribution transformers are connected to feed a large number of consumers/distributed generators connected to the three-phase low-voltage grid. An LV network can have a radial, ring, or meshed structure, but most are operated radially. Figure 1depicts the schema of a typical radial European low-voltage grid. Distribution or secondary transformers are large three-phase transformers (normally between 100 and 1000 kVA), with 20 kV/400 V being the most typical normalized voltages. One of the most common configurations is delta-wye because this arrangement prevents imbalance on the low-voltage side (due to unbalanced consumption) by transferring loads to the MV level. Nowadays, most of these secondary transformers have off-load tap changers, and OLTC transformers are considered in the rest of the paper, as justified in Section 1. Energies 2019,12, 2173 4 of 22 TP-PV TP-L SP-PV SP-L SP-L Figure 1. Schema of a common European low-voltage (LV) network. Low-voltage lines are three-phase four-wire lines that are usually underground cables in urban areas and overhead lines in rural ones. LV networks are primarily earthed at the star-point of the MV/LV transformer. The earthing at the customer location is usually by the public supply network, and it is most often performed by earthing the neutral along different points of the low-voltage feeder. Single-phase and three-phase consumers connect to the low-voltage network, with the former more prevalent than the latter. With distributed generators, mainly those that are photovoltaic, single-phase or three-phase consumers can also be connected, depending on the nominal power [ 41 ]. Single-phase clients connect one phase to the neutral conductor in an attempt to distribute them among the three phases so that the total three-phase equivalent net load from the secondary side of the transformer is as balanced as possible. In practice, although the same kind of single-phase consumers are equally distributed among all three phases, imbalances occur as a consequence of the different hourly load patterns. These imbalances are one of the main concerns of utility engineers. One of the main conclusions resulting from the former design is that a three-phase model that takes into account all phases and the neutral must be considered, and a single-phase equivalent circuit is not valid for analyzing the whole power system. Even a decoupled single-phase circuit to study each of the three phases independently would lead to erroneous results [42]. 3. Optimization Problem The radial LV system under study comprises N+ 1 nodes (buses are numbered from 0). Without loss of generality, bus 0 applies to the MV side of the distribution transformer at the head of the whole LV grid. The set of three-phase buses, loads, and generators are denoted by B, L, and G, respectively. Note that the number of branches is equal to the number of buses minus one because of the radiality of the system. The optimization problem to be solved involves determining the best value of the tap positions of each phase of a three-phase distribution transformer in order to optimize the operation of the whole system at a specific time h while keeping all quantities within limits. These tap positions are the control variables of the problem. The dependent variables are phase–ground voltages at each node i , UT i= [Uiq]T= [Uia Uib Uib Uin]T , phase current injections at each node i , IT i= [Iiq]T= [Iia Iib Iib Iin]T , phase current flows through series branch ij , IT ij = [Iij,q]T= [Iij,aIij,bIij,cIij,n]T , and current Iig through the ground resistor at bus i ( i , j , k are indexes associated with buses; (ij) denotes a series branch between buses i and j ; p refers to each of the three phases a , b , c ; the neutral phase is denoted by n; and the indexes m,qdenote any of the three phases as well as the neutral). The next subsections detail the objective function and constraints associated with the optimization problem to be solved. Equality and inequality constraints result from modeling the electrical behavior of all the components and from considering all of the operational limits imposed by these components Energies 2019,12, 2173 5 of 22 and the fulfillment of required standards. The phase domain and phase coordinates are considered to set out the problem, with computations being performed in physical units. In what follows, each complex equation that appears should be transformed into two equivalent real equations; this step is omitted throughout the paper. 3.1. Objective Function The minimization of power losses is proposed as the main target objective. The power losses of the system can be formulated from a power balance point of view, so the following objective Function (1) results: min <(Sloss) = min <(UT 0I∗ 0) + N+1 ∑ i=1 ∑ phSG ip +SL ipi, (1) where SG ip refers to the generated specified power at phase p of node i , and SL ip denotes the demanded complex power of a load at phase p of node i . Note the net injected power by shunt elements connected to bus i is Sip =SG ip +SL ip , so the generated power SG ip takes values greater than zero while the demanded load SL ip takes negative values. It is important to underline that the objective Function (1) considers all losses, namely power losses in the transformer, lines, and ground resistors. The minimization of imbalances, voltage drops, or neutral currents are other common objective functions to consider. Power losses have commonly been chosen since they represent a well-defined single key index whose lowest value is usually linked to the best voltage profile. Other possible choices are those defined at a node or branch level, and such options necessitate the definition of a single index for any of them (for example, if the minimization of voltage drops was chosen, a possible voltage index to quantify this objective could be the addition of the deviation of each node voltage from its nominal value). In sum, an objective function that differs from the minimization of power losses could be set out. 3.2. Medium-Voltage Grid Equivalent A three-phase Thévenin equivalent network is considered in the model of an MV grid that is upstream of the distribution transformer, as depicted in Figure 2. A rigid grounded wye three-phase balanced ideal voltage source ET g= [Eg|0Eg|−120 Eg|120 0 ]T in series with a single impedance Zg is assumed. These MV equivalent network parameters correspond with the actual operational medium voltage and the short-circuit impedance, respectively. g + ga Oa g + gb Ob g + gc Oc On Figure 2. Thévenin equivalent for a medium-voltage (MV) grid. 3.3. Distribution Transformer Model A complete and exact model for a three-phase transformer requires of a large number of short-circuit experiments, some of which are difficult to implement, depending on the type of transformer (common-core or shell-type integrated three-phase devices) [ 43 ]. A well-accepted starting assumption is that the Energies 2019,12, 2173 6 of 22 three-phase transformer consists of three independent and identical single-phase transformers [ 43 , 44 ] since it has been demonstrated that the resulting inaccuracies from this simplification are subtle [ 43 , 45 ]. In this context, modeling the three-phase transformer starts by considering a single-phase transformer such as that shown in Figure 3, whose admittance-bus-based electrical model follows Equation (2): "I1 I2#="Yt+Ym−rYt −rYtr2Yt#" U1 U2#, (2) where Yt is the short-circuit or leakage admittance, Ym is the open-circuit or magnetizing admittance—with both referring to the primary side of the transformer—and r is the transformer ratio. r:1 t m 1 1 2 2 + - + - Secondary Primary side side Figure 3. Equivalent circuit for a single-phase transformer. The electrical model associated with three independent and decoupled single-phase transformers follows Equation (3), in which the three single-phase transformers are denoted by a, b, c, and their primary and secondary sides are numbered consecutively from 1 to 6. For further developments, three different transformer ratios are adopted for each of the individual phases and are denoted by ra , rb, and rc.          I1 I2 I3 I4 I5 I6          =          Yta +Yma −raYta 0 0 0 0 −raYta r2 aYta 0 0 0 0 0 0 Ytb +Ymb −rbYtb 0 0 0 0 −rbYtb r2 bYtb 0 0 0 0 0 0 Ytc +Ymc −rcYtc 0 0 0 0 −rcYtc r2 cYtc                   U1 U2 U3 U4 U5 U6          (3) Different three-phase transformers can result from these three independent single-phase transformers, depending on the connections between their terminal pairs, with wye and delta connections the most commonly used. The specific connection for the three-phase transformer allows for the formulation of mathematical expressions that link the voltages and currents of the three single-phase transformers with the voltages and currents of the high- and low-voltage sides of the three-phase transformer. For example, Figure 4shows the ∆− yg configuration, one of the most common schemes in European low-voltage networks. For this configuration, the relations among the former electrical quantities are defined by Equations (4) and (5), as can be easily deduced from Figure 4.          U1 U2 U3 U4 U5 U6          =          1−1 0 0 0 0 0 0 0 0 1 0 0 −1 0 1 −1 0 0 0 0 0 0 0 0 1 0 −1 −1 0 1 0 0 0 0 0 0 0 0 0 1 −1                     UA UB UC Ua Ub Uc Un            =N            UA UB UC Ua Ub Uc Un            (4) Energies 2019,12, 2173 7 of 22            IA IB IC Ia Ib Ic In            =NT          I1 I2 I3 I4 I5 I6          +           0 0 0 0 0 Un Rg           (5) 1 2 1 + - + -2 3 4 3 + - + -4 5 6 5 + - + - 6 a b c n A B C + -n A B C A + - B + C + a + - b + c + a b c n g g Figure 4. ∆−yg three-phase transformer. By substituting (3) into (5) and using (4), the final model (6) results for a three-phase ∆− yg transformer that consists of three single-phase transformers (Yta =Ytb =Ytc =Yt,Yma =Ymb =Ymc =Ym) that are identical in all respects except for their transformer ratios:            IA IB IC Ia Ib Ic In            =               2Yt+2Ym−Yt−Ym−Yt−Ym−raYt0rcYt(ra−rc)Yt −Yt−Ym2Yt+2Ym−Yt−YmraYt−rbYt0(rb−ra)Yt −Yt−Ym−Yt−Ym2Yt+2Ym0rbYt−rcYt(rc−rb)Yt −raYtraYt0r2 aYt0 0 −r2 aYt 0−rbYtrbYt0r2 bYt0−r2 bYt rcYt0−rcYt0 0 r2 cYt−r2 cYt (ra−rc)Yt(rb−ra)Yt(rc−rb)Yt−r2 aYt−r2 bYt−r2 cYt (r2 a+r2 b+r2 c)Yt +1 Rg                          UA UB UC Ua Ub Uc Un            (6) The described procedure allows the final bus admittance matrix to be obtained for any other group connection of the three transformers. Note that to obtain a perfect earth connection of the neutral, i.e., Rg= 0, the last diagonal element of the three-phase bus admittance matrix is not defined; in this case, Un is zero and the last row of (6) is omitted. The neutral current flowing through the wye side of the transformer would be obtained by applying Kirchhoff’s current law to the secondary bus of the transformer, as described in Section 3.6. 3.4. Low-Voltage Lines and Ground Resistances Three-phase four-wire lines are the most common configuration for European low-voltage networks, and short single-phase sections are used to connect single-phase clients to the main three-phase four-wire LV line. The modeling of overhead and underground line segments must Energies 2019,12, 2173 8 of 22 be as precise as possible since this model plays an important role in the final solution, as illustrated later. This high accuracy demands for a complete and reliable database in relation to low-voltage lines: the type and size of every conductor, the physical geometry of both overhead and underground lines, the resistance of the earth, etc. This information allows Carsons’s equations [ 46 ] to be applied in order to obtain the resistance and the self- and mutual inductive reactance of the conductors that make up the line. These series impedances are shown in Figure 5. A 4 × 4 series impedance matrix Zij results from this model (7). The series impedance matrix is completely full for a three-phase four-wire line, with only non-null elements in one phase and neutral positions for two-phase or single-phase lines. Ui=ZijIij +Uj=     Uia Uib Uic Uin      =      Zaa ij Zab ij Zac ij Zan ij Zab ij Zbb ij Zbc ij Zbn ij Zac ij Zbc ij Zcc ij Zcn ij Zan ij Zbn ij Zcn ij Znn ij            Iij,a Iij,b Iij,c Iij,n      +     Uja Ujb Ujc Ujn      (7) ja jb jc jn ia ib ic Zaa ij Zcc Znn Zan Zbc Zcn Zab Zac Zan Uib Uia Uic Uin Uja Ujb Ujc Ujn Iij,a Iij,b Iij,c Iij,n Rig Rjg Zbb ij ij ij ij ij ij ij ij ij in Figure 5. Three-phase four-wire electric line model. It is common practice to neglect capacitances in the case of low-voltage lines because the lengths of the lines are short. So, throughout this paper, this shunt effect is not considered. It is not always possible to apply Carson’s equations because the requisite information is not usually available in utility-based data (distance between conductors, height of conductors, earth resistivity, etc.). Instead, positive sequence impedance Z1 and, more rarely, zero sequence impedance Z0 are the only practicable line data. In this last case, the best-approximated series impedance ZAprox ij is ZAprox ij =       Z10 0 0 0Z10 0 0 0 Z10 000Z0−Z1 3        . (8) This simplified model is obtained by assuming an identical type of conductor for all phases and the neutral and equal sections, symmetrical spacing between phases (including the neutral), and perfect earthing at both extremes of each line segment [46]. Neutral conductors are commonly grounded at multiple points along the network (neutral grounding) for safety reasons. The values of these resistors depend on the type of earth conductor and electrode, as well as the resistivity of the terrain. These characteristics determine the value of the Energies 2019,12, 2173 9 of 22 resistor Rig to consider at each bus i , and the value can ideally move between zero and infinity. If this resistor is a finite value, Ohm’s law is applied (9): Uin =RigIig ∀bus iearthed with a finite resistor, (9) where Iig denotes the current flowing through the ground resistor. Note that Uin has a null result for rigid earthing. If the neutral of bus i is not connected to the ground, that is, Rig =∞ , the ground current is forced to be zero (10): Iig =0∀busiunearthed (10) 3.5. Loads and Renewable Generators Hereinafter, the appropriate model for consumers and renewable generators connected to the low-voltage system is discussed. The optimal power flow analysis discussed throughout this paper deals with the steady state, so static models are considered. Static load models can be classified into the following groups: exponential, polynomial, linear, comprehensive, static induction motor, and power electronic-interfaced models [ 47 ]. Among all these static load models, the constant real and reactive power load model is the most widely used for utilities [ 47 ], as a particular case of the polynomial or ZIP model (constant impedance Z, constant current I, and constant power P model). Any other model could be considered if the appropriate parameters for each load are known. For renewable generation, with a focus on PV technologies, a steady-state power injection model is also adopted. PV generators operate by injecting the maximum available power, which varies depending on the irradiance. Similarly, the reactive power injection can be set depending on the standing grid code. Unity power factor operation is a common practice for PV generators connected to LV networks. Three-phase loads and generators can be connected in wye or delta configurations, as shown in Figure 6. The neutral can be accessible or inaccessible in wye connections, and they are referred to as a three-phase four-wire connection or three-phase three-wire connection, respectively (see Figure 6a,b). The electrical model is given by (11) for a three-phase four-wire load/generator connected to bus i , while (12) is applicable to a three-phase four-wire load/generator.    SL/G ia SL/G ib SL/G ic   =   (Uia −Uin)(IL/G ia )∗ (Uib −Uin)(IL/G ib )∗ (Uic −Uin)(IL/G ic )∗   IL/G ia +IL/G ib +IL/G ic +IL/G in =0                  ∀Three-phase four-wire - load L/ generator G ∈bus i(11)    SL/G ia SL/G ib SL/G ic   =   (Uia −UinL/G)(IL/G ia )∗ (Uib −UinL/G)(IL/G ib )∗ (Uic −UinL/G)(IL/G ic )∗   IL/G ia +IL/G ib +IL/G ic =0                  ∀Three-phase three-wire wyeload L/ generator G ∈ bus i,(12) where L/Grefers to the chosen load (L) or generator (G), depending on the considered component. Equation (13) defines delta-connected three-phase loads or generators. Energies 2019,12, 2173 16 of 22 As in the previous subsection, all these observations strengthen the confirmation of the benefits gained from using decoupled controls and the importance of being rigorous in electrical modeling. In this case, with the earthing configuration, the knowledge of the neutral earthing is needed not only to understand the real state of the system but also to properly solve the voltage optimization problem. Additionally, in Figure 10, the differences between the bus voltages in Cases 2, 3, and 4 with respect to those of Case 1 are depicted using the 1P-OLTCST. Some noteworthy remarks are as follows: (1) phase-neutral voltages in Case 3 are practically equal to those obtained in Case 1: that is, a ground resistor of 40 Ω produces practically the same voltages as those obtained with a ground resistor of 3 Ω ; (2) larger differences occur when a perfectly rigid ground resistance (Case 2) exists, with phase-neutral voltages not as low as those in the case of a non-null ground resistor (Case 1); (3) the final phase-neutral voltages in Case 4 are approximately between the two previous ones. -2 0 2 4 -5 0 5 -2 0 2 1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 I2 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 -6 -4 -2 0 ( ( ( ( ( ( Figure 10. Differences between the phase-neutral and neutral bus voltages in Cases 2, 3, and 4 with respect to Case 1. 4.4. Distributed Generation In the previous sections, some simulations are presented in order to demonstrate the importance of the model when carrying out effective control of the power system and for depicting a realistic load scenario without the presence of renewable generation. If distributed generation is considered, the degree of imbalances is expected to be higher, and the differences shown when considering an accurate or a simplified model of the power lines, together with the improvements introduced by the use of a 1P-OLTCST, can become even greater. In this subsection, distributed PV generation units that can act as single-phase or three-phase generators with regard to their power capability are installed in some buses of the network. A 24-hour time period, during which load and generation profiles are collected every 15 minutes, is considered in this subsection using profiles obtained from [ 49 ]. These profiles are adapted to our problem by scaling them to the maximum power and the power factor detailed in Table 7. The distribution of load/generation at each of the three phases of every bus is also specified. Figure 11 shows the total net apparent power at each hour for the described daily load/generation scenario, so the reader is able to estimate the resulting total imbalances at each time. As expected, the largest imbalances among the three phases are identified as occurring during the hours of highest photovoltaic generation. The final adopted ground configuration is specified in Table 3. Energies 2019,12, 2173 17 of 22 Table 7. Load/distributed generation configuration. Bus R2 R11 R15 R16 R17 R18 I2 C2 Smax [kVA]20 −75 25 −135 17 80 −105 −36 cos ϕ0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 Ph. a (%) 0 30 0 30 100 33.33 30 30 Ph. b (%) 100 40 0 40 0 33.33 40 40 Ph. c (%) 0 30 100 30 0 33.33 30 30 Bus C12 C13 C14 C17 C18 C19 C20 Smax [kVA]−101 −36 35 20 −24 −48 −24 cos ϕ0.95 0.95 0.95 0.95 0.95 0.95 0.95 Ph. a (%) 30 40 33.33 100 30 30 30 Ph. b (%) 40 30 33.33 0 40 40 40 Ph. c (%) 30 30 33.33 0 30 30 30 Figure 12 details the obtained optimal tap values when using a 1P-OLTCST or 3P-OLTCST for the specified daily scenario, while Figure 13 shows the evolution of each phase-neutral voltage on the secondary side of the MV/LV transformer. It is easy to identify the numerous times at which the 1P-OLTCST obtains different tap positions for each of the three phases, and it is particularly outstanding for phase b. In reference to the savings in energy losses, that is, the integral of the objective function reached by the two control strategies, the 1P-OLTCTS results in 3% lower daily losses, while the 3P-OLTCTS saves only 1.05% at the end of the day. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 -100 -80 -60 -40 -20 0 20 40 60 80 Figure 11. Net total apparent power at the MV/LV transformer without considering losses. Energies 2019,12, 2173 18 of 22 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0.95 1 1.05 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0.95 1 1.05 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0.95 1 1.05 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0.95 1 1.05 Figure 12. Optimal tap positions when considering a 1P-OLTCTS or 3P-OLTCTS. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 220 230 240 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 220 230 240 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 220 230 240 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0 1 2 3 Figure 13. Phase-neutral voltage evolution for bus R1 when considering a 1P-OLTCTS or 3P-OLTCTS (dashed lines) for phases a, b, and c and neutral voltage. Figure 14 shows the maximum and minimum phase-neutral voltage recorded throughout the day at every bus. It is noticeable that the 1P-OLTCTS has lower min–max voltage differences than those obtained with the 3P-OLTCTS. This means that daily voltage variations at any bus are narrower when decoupled voltage control is considered. It is also apparent that the 3P-OLTCTS is not able to maintain voltages within limits, as it exceeds the lower limits at almost all buses at least once a day. Energies 2019,12, 2173 19 of 22 1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 I2 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 200 205 210 215 220 225 230 235 240 245 250 Figure 14. Maximum and minimum phase-neutral voltage at every bus for both the 1P-OLTCTS and 3P-OLTCTS (dashed lines). 5. Conclusions and Future Work This work aimed to study optimal voltage control in low-voltage systems by using transformers with on-load tap changers for two options: the classical OLTC that modifies the transformer ratio and a decoupled OLTC that allows different tap positions at each phase of the three-phase transformer. In contrast with previous research, the associated exact optimization problem was detailed and programmed with a special emphasis on the three-phase modeling of all network components as a requirement due to the unbalanced nature of these systems. The proposed tool was tested on a benchmark low-voltage network under different load and renewable generation scenarios. The main conclusions are as follows: • Using simplified models for three-phase four-wire lines or excluding considerations of the neutral earthing configuration led to erroneous or suboptimal solutions, regardless of whether 1P-OLTCST or 3P-OLTCST was used. This conclusion was obtained even for low unbalanced scenarios (32.1%, 22.7%, and 44.2%), so we can infer that even worse results would occur with an increase in the level of renewable penetration. This conclusion is quite relevant: simplified models are attractive because of their simplicity, but their use can lead to insecure operation of the system. • The use of simplified models for three-phase four-wires lines caused an underestimation of the voltage drops in most cases, for either of the two OLTCs considered. The worst identified underestimation reached 10 V. • Regarding the grounding resistance configuration, two main solutions were obtained. It was observed that when the ground resistance increases above zero, the same optimal tap positions are obtained no matter the resistance value (ground resistances of 3 Ω and 40 Ω produce almost the same results, but quite different from those corresponding to Rg= 0 Ω ). Conversely, a different solution is reached when rigid ground connection is taken into consideration. • Decoupled OLTCs are more effective voltage controls than the common OLTC transformers that change the three-phase transformer ratio. It was proven that the 1P-OLTCTS fulfills its primary function of maintaining the voltage within limits most of the time, even in very unbalanced scenarios. This ability is crucial when analyzing networks with high levels of penetration of renewable energy sources. In contrast, the 3P-OLTCTS did not succeed in fulfilling that requirement in many cases. The definition and implementation of the exact optimization problem Energies 2019,12, 2173 20 of 22 carried out throughout this work was essential to come to this conclusion and would have not been possible if approximated models had been assumed. • Besides more effectively maintaining voltage within limits and having lower power losses, the 1P-OLTCTS had lower voltage drops and daily voltage variation than the 3P-OLTCTS. For the tested cases, the best saving in power losses was 85.1% for the 1P-OLTCTS, versus 91.7% for the 3P-OLTCTS; the highest voltage drop was 4.5% for the 1P-OLTCTS, versus 9% for the 3P-OLTCTS; and the maximum daily voltage variation was reduced by up to more than 10 V with 1P-OLTCTS versus 3P-OLTCTS. Future work will examine the coupling of all time frameworks under study in order to reduce the number of movements of tap positions and to maximize the lifetime of the device; it is clear that such tap movements do not lead to profitable savings of power losses and although voltages are within limits, tap movements should be avoided. In addition, the current work focused on the decoupled OLTC as the only means of voltage control in order to understand its real possibilities and because it is a solution that is close to the current reality of utilities. However, additional voltage controls, such as statcoms, inverters linked to renewable generators, or storage systems, would be interesting to study in future analyses. Author Contributions: Conceptualization, Á.R.d.N., E.R.-R., and Á.L.T.-G.; Formal analysis, Á.R.d.N., E.R.-R., and Á.L.T.-G.; Validation, Á.R.d.N., E.R.-R., and Á.L.T.-G.; Writing—original draft, Á.R.d.N., E.R.-R., and Á.L.T.-G. 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