Unified Framework for the Analysis of the Effect of Control Strategies on On-Load Tap-Changer’s Automatic Voltage Controller

With the advent of new loads and generation on the low voltage grid, voltage fluctuation has increased, especially in active distribution grids with a high penetration of distributed resources and a large deployment of electric vehicles. The coordination of different technologies has emerged as the best way for voltage regulation, among others, smart inverters, open soft points or transformers with on-load regulation capability. This paper proposes a novel way to model the control strategies for the automatic voltage controller of On-Load Tap-Changer transformers. The purpose is to standardize and simplify the way these strategies are represented, in order to facilitate (i) their selection by Distribution System Operators, (ii) their future integration with other systems, and (iii) to increase the ability to anticipate On-Load Tap-Changer behavior. The proposal has been validated using real data, obtaining an accuracy of 99.15% in the tap changer positions. A unified framework is also introduced, which allows the proposed functional representation to be combined with the On-Load Tap-Changer controller behavior estimation. A experimental validation has been carried out where more than 150 000 strategies have been simulated, finally determining the one that best fits the objectives. Note to Practitioners—Historically, on-load tap-changing transformers have been used in high-voltage substations. However, with the integration of electric vehicles and the penetration of distributed energy resources, the need to implement this type of solution in low-voltage substations has grown. However, the characteristics and requirements are not the same, for example, given their nature, low voltage grids are more unbalanced and are often regulated to ensure the quality of supply. Therefore, inheriting the control strategies of traditional on-load tap changers may pose a serious risk. This paper proposes a novel unified framework that facilitates the modeling and simulation of almost any control strategy for distribution transformers with on-load tap changers. This will allow the choice of control parameters that minimize voltage deviation from the voltage setpoint and maximize device lifetime. Our proposal can be considered as a solution to the uncertainty of which control parameters to use. It can be performed before commissioning, based on historical data, or a posteriori, based on the collected data.

maximize device lifetime.Our proposal can be considered as a solution to the uncertainty of which control parameters to use.It can be performed before commissioning, based on historical data, or a posteriori, based on the collected data.
Index Terms-Distribution transformer, on-load tap changer, automatic voltage controller, control strategy, automatic voltage regulator.N EW renewable energies (RE) and new consumption technologies (electric vehicles, heat pumps, etc.) lead to voltage fluctuations, making it difficult for Distribution System Operators (DSOs) to comply with standards.In order to maintain the quality of supply, a continuous modernization of smart grids (SG) is necessary.To achieve this, regulators of privately owned monopolies often use rate-of-return regulation.For example, in Spain, the CNMC (National Commission for Markets and Competition) already rewards investment in assets such as On-Load Tap-Changing Transformers (OLTCTs) [1].

NOMENCLATURE
Therefore, strong investment in voltage regulation assets, both local (e.g., photovoltaic (PV) inverters [2] or reactive power injectors [3]) and global (OLTCTs) [4], is expected in the coming years.In general, and analogous to already existing solutions for energy management based on demand response [5], the correct coordination of technologies is expected to play a key role [6].European projects such as Smart Street [7] or U-Control [8] have shown that the optimal control of OLTCs, coordinated with other technologies, is the best solution for voltage control in terms of cost-benefit ratio, although more research is still needed.
Regarding the OLTCTs, the automatic voltage controller (AVC) gives the tap up or tap down orders, varying the voltage of the secondary side, adjusting the level for the final consumers.These controllers are critical to the behavior of the OLTCTs, since, depending on the control strategy (CS) used, they will try to maintain the secondary side voltage close to the pre-defined voltage set point (V t ) in one way or another, generating a specific number of tap changes (or tap operations, TO) and voltage states [9].
The most widespread control strategies are of the hysteresis type, i.e., a tolerance band (also known as dead band) is defined and the time relative to the voltage deviation from the V t is measured [10].The determination of the bandwidth (BW t ) depends on the method employed, and can be of fixed or variable width [11].The way in which voltage deviation from the BW t and delay times (κ t ) before tap changes are related is a critical issue, and is the main question discussed in this paper.Despite being a crucial aspect, there are very few comparative studies of the effect of these deviationtime relationships.The few reports that can be found in the literature propose and evaluate strategies that are biased towards cases of high renewable penetration [12], [13].
In general, the OLTC manufacturers themselves are the ones who define this functional deviation-time relationship, among others, we can find such specifications in the guides of Maschinenfabrik Reinhausen (MR) [14], a-eberle [15] or Fundamentals [16].However, there is a notable absence of consensus and standardization, which is especially evident in the lack of scientific articles in this field.
There are three main methods for relating voltage deviation and delay times: (i) constant (or definite, depending on the source), (ii) linear and (iii) integral (or inverse, depending on the source).The advantages and disadvantages of each of them are listed in Table I.
The main problem is that the method to be used is not clearly defined by the manufacturers, leaving the decision  I and others to be represented, followed by a unified framework that allows this representation to be integrated and the transformer behavior to be simulated.This process has been empirically tested by validating the results on real data (99.15%accuracy in TC positions) and by simulating more than 150,000 CSs, finally determining the one that best fits the number of TO and the voltage deviation from V t .The paper is organized as follows: in Section II, a novel approach to OLTC modeling based on discrete-step control theory is presented.In Section III, the basic concepts for understanding the automatic voltage control algorithms that regulate the OLTC are described and their operation methodology is illustrated with two examples.Section IV introduces a novel functional representation for control strategies.Section V describes the features of our unified experimental framework to perform the distribution transformer (DT) simulations and optimal control problem is postulated.After that, an application example of the proposed representation is explained in Section VI.Section VII validates the framework.The experiment to evaluate the behavior of different strategies and select the best one is presented in Section VIII.Section IX concludes the paper and discusses future research directions.

II. ON-LOAD TAP-CHANGER
First, we present a novel way to model the behavior of a DT with an OLTC device in automatic mode.Given the dynamic nature of the problem, where the evolution of the system (the tap position x(t) = x t ) depends on time (t) and is controlled by an external agent or control (control variable u(t) = u t ), we have relied on the formulation of theoretical models for the optimal control problem with discrete time steps.
The OLTC characteristics will be given by the inter-tap voltage variation coefficient (ρ), the set of positions ( ) of the tap changer (TC) and the duration of each tap change (w).We define T ∈ {0, . . ., n} as the (discrete) time set of evaluation, note that the control variable u has one less time instant than the state variable x, i.e., with t ∈ T = {0, . . ., n −1}.The state function f k (x k , u k ) is directly defined as follows in Equation (1).
Some constraints inherent to the physical problem are now defined.Regarding the state variables, since the TC positions Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Regarding the control variables, there are additional restrictions, since the tap changes must be discrete and one at a time, it follows that u k ∈ {−1, 0, 1} ∀k ∈ T .On the other hand, since each tap change has duration w, it follows that, where T = {0, 1, . . ., n − w − 1}, i.e., only one tap change can occur every w seconds.In addition, the total number of TO is also constrained, Finally, since the state variable is contained in , it follows that if x k = 1, then u k ≥ 0, and if x k = ω, then u k ≤ 0. The contour restriction is of the kind "fixed starting point and free end point".

III. AUTOMATIC VOLTAGE CONTROLLER STRATEGIES
When operating in automatic mode, the main parameters of the OLTC AVC strategy will be the voltage set point (V t ), the tolerance Band-Width (BW t ) and the time delay or response time (κ t ).AVC follows the CS with the aim of keeping the voltage level as close as possible to the given V t by performing the fewest possible TO, i.e., the optimum V t has had to be stipulated beforehand in order to accomplish a certain objective, for example, to avoid consumers' over-and under-voltage excesses [17].
Since this control method is based on the step-by-step principle, these strategies are usually of the hysteresis type, a BW t (allowed deviation range) is introduced to avoid unnecessary tap changes around the target voltage V t .
Typically, a standard controller measures the busbar voltage (V t ) on the LV side of the transformer and this voltage is used for voltage regulation.The voltage control algorithm compares the V t with the V t and decides what action should be taken based on the difference (V t ) = |V t − V t | [10] and on the the sign of the difference determined by the function sign(V t ) defined as follows in Equation (4), Tap changes are decided comparing the waiting time (κ t ) with the out-of-band time counter (θ t ) defined by Equation ( 5), Usually, the BW t is determined by the coefficient db symmetrically around the V t , such that, Different criteria are used to decide the optimal BW t , but the most common is that the total width of the tolerance band (BW + t − BW − t ), must be greater than the range of voltage variation between taps (ρ • V t ).This is so that by overcoming the band on one side and changing the tap, it doesn't come out at the other end.
Let α be the coefficient that determines how wide the band is, Equation ( 6) gives the coefficient of the band that determines the width on each side, so that 2 • db ≥ ρ with db ≥ 0 and V t ≥ 0, in order to maintain voltage variations between taps lower than the 2 Once the band has been determined, instead of considering the voltage deviation from the V t , as in [14], we will frame the analysis on the voltage deviation with respect to the band, i.e., δ t such that, Usually, in order to define the control strategy, two delay times are established: (i) κ S , for small voltage deviations from the BW t (δ S ) and (ii) κ H , for significantly larger voltage deviations (δ H ). Always Given the parameters listed in Table II, variations in the waiting time before changing the tap according to the voltage deviation can follow different relations according to the CS.The main ones are constant, linear and integral.The longer the waiting time (κ S or κ H ), the greater the voltage variations (V t ), the shorter the waiting time, the greater the TO.Fig. 1 shows two types of control strategies (constantleft and integral -right) deduced from the same parameters (listed in Table II).The effect of applying the constant control strategy (left chart of Fig. 1) on a voltage curve and the decisions involved can be seen in Fig. 2. Analogously, the effect of applying an integral control strategy can be seen in Fig. 3.
In Fig. 2 the different constant tolerance bands are well differentiated by hatches.First, the range of acceptability with respect to the constant V (V k = 230 ∀k) is indicated using "•" (i.e., the constant BW k ), where the κ k or waiting time Fig. 2. Effect of the constant strategy applied to a voltage curve.BW ± k with hatch "•", slow action range with hatch "/" (κ S ) and fast action range with hatch "-" (κ H ). The "-" line is the V , the "--" line marks the first limit (δ S ), and the "-•" line the second limit (δ H ). The dots are instants of voltage measurement, the black dots indicate the decision to change the tap (θ k > κ k ), accompanied by an arrow indicating the direction of change (up or down). is infinite.With "/", the second tolerance range containing variations between δ S = 0 and δ H = 3 with the corresponding waiting time κ S = 10 seconds is represented.Finally, with "-", it can be seen that any voltage deviation greater than δ H = 3 has a delay time κ H = 2 seconds.
The white dots are instants of voltage measurement, the black dots symbolize the decision to change the tap, accompanied by an arrow indicating the direction of change (up or down).When the number of instants in a given voltage range is greater than the waiting time (κ k ), the tap change is performed, i.e., when the out-of-the-band counter θ k is greater than κ k .
In the second example (Fig. 3), after the BW k (white range), κ k depends on the voltage deviation, this functional relationship has been represented with a grey gradient, the higher the color intensity, the shorter the waiting time.

IV. CONTROL STRATEGY FUNCTIONAL REPRESENTATION
The aim of this section is presenting a general formulation that accommodates the examples of strategies described in the previous sections and other possible ones.
In order to provide a standardized functional representation, the functional representations of constant, linear and integral strategies must be combined, and the possibility of generating alternative strategies must also be considered.For this purpose, a combination of the Polynomial (linear strategies), Rational, Exponential (integral strategies) and Logistic (constant strategies) functions (PREL) is proposed: The set of admissible control laws is fully determined by the coefficients of the PREL functional that passes through the points (δ S , κ S ) and (δ H , κ H ). In other words, P(δ S ) = κ S and Hereafter, to simplify the notation, we will only show Equation ( 8) for values of δ ≥ δ S .In addition, we refer to (8) as a "PREL" representation.
It is possible to capture the nature of existing strategies by adjusting the coefficients of ( 8) in order to reproduce the intended strategy behavior.In short, obtaining the representation of a particular strategy consists of finding the set of coefficients γ = {α 1 , α 2 , . . ., α 8 } of the functional PREL that passes through the points (δ S , κ S ) and (δ H , κ H ). Once the PREL is defined, for each time instant it is possible to measure δ t , substitute it in P(δ t ) and then know the time steps that may be at that level of deviation δ t before deciding to change the tap position, κ t (P(δ t ) = κ t ).
Equations ( 9) and ( 10) correspond to accurate PREL representation of the strategies illustrated in Fig. 1.The new CS curves are shown in Fig. 4.
As can be seen, the curves in Fig. 1 and Fig. 4 are very similar, i.e., the PREL formalism is able to represent CSs perfectly.In the case of the constant strategy, the jump is eliminated, making it continuous.
Note that in this article we approximate the integral strategies (inverse curves) by an exponential function (for instance, (10)), however, different approximations can be found in the literature, for example, the method proposed by the manufacturer MR [14], consists of an inverse relationship between time and voltage deviation.For more information, see Appendix A.
In general, given (δ S , κ S ) and (δ H , κ H ), determining the coefficients (γ ) of the functional is reduced to a curve fitting problem.Appendix B has been dedicated to clarify these issues in more detail.In summary, except for the case of the constant strategy, the problem is reduced to solving a system of two equations with two unknowns.For constant strategies, some more sensitive parameters, such as the rate of change or the change midpoint, must be taken into account.Fig. 3. Effect of the integral strategy applied to a voltage curve.BW k in white, after that, the grey gradient marks time variation.The higher the color intensity, the shorter the waiting time.The "-" line is the V , the "--" line marks the first limit (BW − k and BW + k ).The dots are instants of voltage measurement, the black dots indicate the decision to change the tap (θ k > κ k ), accompanied by an arrow indicating the direction of change (up or down).

V. UNIFIED FRAMEWORK
In this section, the objective is to provide a unified framework for simulating OLTCT behavior given a CS represented by the PREL functional.This approach ends with the statement of the optimal control problem.
Given the voltage (V k ) at a particular tap position (l k ), the voltage it would have ( V k ) if it were at another tap (x k ) can be estimated by (11), Obviously, when simulating the OLTC operation methodology based on historical data, it has to be taken into account that, when the tap position is changed in each step, all historical values after it will vary.Therefore, the simulation cannot be carried out with a global perspective, as each step is conditioned by the previous steps.
However, since the optimal tap will be determined by the AVC strategy step by step based on the previous step, it is only necessary to calculate the new voltage value for the next one, and only in the next step, for each iteration.That is, we will assume that a Markov process is followed.
Since it is a deterministic system, given a policy {µ γ 0 , µ γ 1 , . . ., µ γ n−1 }, and an initial state x 0 , all future states are determined.Thus, given x 0 and π γ , the state values x 1 , . . ., x n are determined by In addition, we define the cost function of the optimization problem as follows in Equation ( 13), Thus, the total cost given x 0 and a policy π γ is given by An optimal admissible policy π * γ would be one that minimizes the global cost and is generated from a PREL representation with γ coefficients, being the set of admissible policies.Therefore, taking into account the constraints defined in Section II, the following optimization problem unifies the goals of finding an optimal CS with a PREL functional representation, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE III OLTCT PARAMETERS USED FOR THE APPLICATION EXAMPLE TABLE IV AVC PARAMETERS USED FOR THE APPLICATION EXAMPLE
Both the PREL way of representing strategies and the control theory-based model for OLTCs are independently useful.However, in addition, one of the main uses that can be given to them is to integrate them together to study and compare the behavior of the DT following different CSs.Given the time series with the historical voltage values and historical tap position values, the pseudocode to simulate the OLTC control process is described by Algorithm 1.
Step 3: Estimate V k using V k , l k , ρ and x k (11).
Step 4: Add x k and V k to the result lists (X l and V l ).
Step 10: Add u k to U l .end for return X l , U l and V l .

VI. APPLICATION EXAMPLE
We present an example of the application of the unified framework.With this in mind, we consider an OLTC with the parameters listed in Table III.In addition, we assume a strategy with AVC parameters listed in Table IV.
Then, if an integral control strategy is desired, it is sufficient to use the PREL representation, and look for a negative exponential relationship (17).

VII. FRAMEWORK VALIDATION
To validate this methodology, both for strategy representation and the simulation of OLTCs, data from Ormazabal's DT have been used. 1 One time series of 86400 values has been used, i.e., one day of voltage recordings, one voltage measurement every second.We have the output voltage value of the transformer (V (t)), the tap where it was located (l(t)) and the established voltage set point (V (t)).By reverse engineering the voltage curves that would have been produced in the absence of OLTC have been estimated.
By applying the introduced framework on the curve, the behavior of the transformer under the effect of different CSs can be studied.In order to simulate the actual strategy, first, the transformer is modelled with its characteristics, then, the strategy is captured through its PREL representation, finally, the strategy is applied on the transformer model under the voltage curve.The expected result should be the original curves (measured values, i.e., the curve before applying the reverse engineering).
The OLTC parameters are those shown in the PREL application example (Section VI), specifically in Table III.On the other hand, for the AVC, its PREL representation coincides with (9) since the original parameters when the data were collected match those in Table II and with V = 242 (V k = 242 ∀k).The results are shown in Fig. 5.
The actual data is represented with the "x" mark, whereas the simulated data is represented by a "▲".As can be read in the title, of the 86400 instants, the tap position match 99.15%.In addition, the reason for not obtaining 100% has been identified.The main reasons are twofold, first, the time it takes to perform a tap change has been considered constant (w = 2), but in reality it can vary (w = 2, 3, 4 or 5).Secondly, it is due to a rounding error in the operations carried out, sometimes exceeding 1% of voltage deviation, when in reality it was not exceeded.In the zoom-window it can be seen how the simulation needed 3 seconds more (w = 5 instead of w = 2) to make the same decision, to raise one tap.

VIII. EXPERIMENT
On this occasion, a similar process to the previous section (Section VII) will be followed.One day of real data from an OLTCT will also be used and the voltage curves without OLTC will be calculated by reverse engineering.The parameters of the OLTC are analogous to those previously shown in Table III, and the V now is 236V .
The objective is to find the CS that minimizes the difference between the voltage curve ( V ) and the setpoint (V ).The optimization problem will be approached using the brute force technique, generating a grid-search of logical parameters.In addition, the constraint on the TO will be L = 45.The main reason is that a transformer lasts about 30 years, and the OLTC can perform about 500,000 operations, i.e., about 45 per day.Two criteria have been considered to evaluate the results, • The error between obtained V and the corresponding V .
• The number of tap changes (TO).In order to evaluate the error, two scores have been used, the MSE (Mean Squared Error) and the MAPE (Mean Absolute Percentage Error).
The grid-search has been generated by combinations of parameters δ S , δ H , κ S and κ H , specifically, δ S : from 0 to 3 (step = 0.2), δ H : from 3 to 6 (step = 0.2), κ S : from 55 to 70 (step = 1.0) and κ H : from 1 to 16 (step = 1.0).In addition, for each combination of parameters, the three types of CSs (constant, linear and integral) have been estimated.In total, 151 875 CSs have been tested.Figure 6 illustrates the evolution of the best policies in terms of RMSE with respect to the number of TO.
Among the tested policies, some existing and proposed by the manufacturers are included ( [14], [15], [16]).In general, it can be appreciated that as the number of TO increases, the RMSE decreases.Table V lists the characteristics of the best CS for each case.
Note that the integral relationship is the one that gives the best performance.It coincides with the PREL application example shown in Section VI, therefore, the PREL Fig. 6.Relationship between the best R M S E (the minimum) and the number of TO for each type of CS.The worst error encountered (the maximum) is represented by a horizontal bar.representation is exactly the Equation ( 17).This CS can be entered into the OLTC by typing the generator parameters in the control panel:

TABLE V CHARACTERISTICS OF THE BEST RMSE AND MAPE FOR EACH CS TYPE
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.For illustrative purposes, Fig. 8 shows the simulation performed for the case of the best strategy found (integral CS of Table V).From top to bottom, it is possible to see the waiting time (κ), the voltage value ( V ), the δ value and the tap position (x k ) respectively.
The upper chart shows the evolution of κ k according to the δ k deviation, there are areas with no values because δ k is less than δ S , and the allowed time is infinite.
For the voltage graph, the grey curve represents the used time series, i.e., the real input voltage at constant tap (l k = 5 ∀k).On the other hand, the black (darker grey) curve represents the obtained voltage after tap changes decided by AVC through CS application.It can be seen how the black curve is centred around the V (236V , dashed line), so the algorithm is being applied correctly.
The bottom charts represent the evolution of the δ k and the tap position (x k ) respectively.The correct functioning of the OLTC can be appreciated in the simulation.
This experiment highlights the usefulness of the introduced model.Thanks to its flexibility, it has not only been possible to find the CS that best suits the specified needs, but also to compare the performance among multiple control strategies and to predict the OLTC behavior under a particular CS.In addition, it also allows us to estimate the influence of installing an OLTCT in a transformer substation, helping to evaluate its impact and necessity.

IX. CONCLUSION
In this paper, we have introduced a method to represent, in a functional form (PREL), the control strategies of automatic voltage controllers that determine the tap changes of onload tap-changer transformers running in automatic mode.The framework has been validated against real data with a 99.15% coincidence rate.Using the presented unified framework, we demonstrated the applicability of this method via simulations on one real data set, testing both commonly applied and new control strategies.From the simulation result, the effect of more than 150 000 strategies has been compared in an objective and standardized way, highlighting the option that best fits the OLTC AVC statistically.The PREL method has demonstrated the ability to capture (i) constant, (ii) linear and (iii) integral strategies, and has been postulated as a flexible, scalable and standardized way to represent existing and new control strategies.The proposed representation is expected to be applicable to all AVC for DT with OLTC.Future work will include extending the unified framework to simulate coordination with other voltage regulation technologies external to OLTC, or, where appropriate, with Distribution Management Systems.We also intend to apply this flexibility and standardization approach to other situations, including low voltage grids with high penetration of distributed energy resources.Work is underway to analyze the impact of the predefined voltage set point on optimal control strategies and their effect on consumers.Finally, a methodology is being analyzed to solve the proposed optimal control problem using more efficient approaches.

APPENDIX A OTHER APPROACHES TO INTEGRAL STRATEGIES
In this article, we have chosen to model integral strategies by means of an exponential function for several reasons, among others, (i) because of the continuity and differentiability characteristic of this function and (ii) for the capacity to integrate with the logistic function (used for constant strategies).However, in the literature, particularly in some manufacturers' guides, different approaches to these strategies have been reported.
Another usual way of representing the time-deviation relationship of the voltage is given by (18).
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where T F is the time factor (from 1 to 10 secs), the G I is the insensitivity degree (the BW in %) and DV is the voltage deviation (%).Therefore, it is an expression of the following form, This is why some manufacturers refer to this type of strategy as "inverse."This expression can be adopted by PREL in a simple way, if α 1 = α 2 = α 3 = α 4 = α 8 = 0 and α 6 = α 7 = 1, then, Now, instead of using δ value, passing ln(|δ|), it follows that where D = 30 * T F * G I .In addition, we can calculate V with δ and the inverse function of (7).

APPENDIX B PREL COEFFICIENTS
In the following, the steps to obtain the coefficients γ of the PREL functional that passes through the points (δ S , κ S ) and (δ H , κ H ), where κ S > κ H ≥ 0 and 0 ≤ δ S < δ H are described.

A. Constant Strategy
The objective is, starting from (20), to deduce the coefficients γ to generate an expression such as the logistic function, where l 1 and l 2 are the lower and upper asymptotes respectively and x 0 and k are the midpoint of change and rate of change respectively.Therefore, we know that l 1 = κ H and l 2 = κ S .Furthermore, if we assume ϵ as the step size of the δ measurements (the measurement accuracy), then δ H is the minimum δ with P(δ H ) = κ H and δ H − ϵ is the maximum δ with P(δ H − ϵ) = κ S .Hence, the midpoint of the change must be, x 0 = δ H − ϵ 2 .On the other hand, the rate of change, k, must be sufficient so that P(δ H ) = κ H , P(δ H − ϵ) = κ S and any intermediate point, I , between the two points satisfies that P(δ H ) = κ H < P(I ) < κ S = P(δ H − ϵ).
On the other hand, for the upper and lower asymptotes to be on the left and right respectively, the value of k must be negative.Given these two requirements, a sufficient condition is to establish a rate of change k smaller than the slope of the line joining the two points.

B. Linear Strategy
The objective is, starting from (20), to deduce the coefficients γ to generate an expression such as the linear function, where m is the slope of the line and b is the intersection with the y-axis.Therefore, m = κ H −κ S δ H −δ S and b = κ S − κ H −κ S δ H −δ S δ S .So, the following coefficients are given, (25)

C. Integral Strategy
The objective is, starting from (20), to deduce the coefficients γ to generate an expression as follows in Equation ( 26  All the control strategies proposed have been in line with those already implemented in commercial equipment (constant, linear and integral).However, as mentioned throughout the article, it is possible to generate alternative strategies that break this trend.Let us look at an example with a quadratic function.An expression of the form given in (30). (30) Since we want the curve to be convex, we assume c = −1.Given this assumption, as in the previous case, a and b are directly defined by the two points: It should be noted that these alternative strategies will not always be implementable on the equipment of manufacturers.It will all depend on where they are programmed and the flexibility of this equipment, among others, the control box, the cloud, the PLC, the micro, etc.But this is another subject of study, more associated with the implementation.
In conclusion, we have presented the way in which the coefficients of PREL are defined for each of the cases.There are more casuistry and different combinations that could generate other ways of calculating the delay time.We have described just a selection of illustrative cases in summary form.

Fig. 4 .
Fig. 4. PREL representation of the constant (left) and integral (right) control strategies based on parameters listed in TableII.

Fig. 5 .
Fig. 5. Framework validation through simulation based on real data.The upper chart shows the evolution of the voltage, the lower chart shows the tap position of the TC.The "x" shows the actual values versus the simulated values "▲".The central windows represent a zoom of size 35, for better visualization.

Fig. 8 .
Fig. 8. Evolution over time for the best CS.From top to bottom: κ computed from the voltage deviation, voltage fluctuation (grey: original voltage curve at constant tap l k = 5, black: modified voltage after applying CS), voltage deviation from the BW k (δ k ) and the tap position (x k ) of the TC.

TABLE I ADVANTAGES
AND DISADVANTAGES OF THE MAIN CONTROL STRATEGIES ASSOCIATING VOLTAGE DEVIATION AND TIME DELAYin the hands of the DSOs.The lack of standardization and consensus makes this decision difficult.This paper proposes a functional representation that allows the methods listed in Table

TABLE II CONTROL
STRATEGY PARAMETERS Fig. 1.Constant (left) and integral (right) control strategies based on the parameters listed in TableII.