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Currently, the electrical system in Argentina is working at its maximum capacity, decreasing the margin between the installed power and demanded consumption, and drastically reducing the service life of transformer substations due to overload (since the margin for summer peaks is small). The advent of the Smart Grids allows electricity distribution companies to apply data analysis techniques to manage resources more efficiently at different levels (avoiding damages, better contingency management, maintenance planning, etc.). The Smart Grids in Argentina progresses slowly due to the high costs involved. In this context, the estimation of the lifespan reduction of distribution transformers is a key tool to efficiently manage human and material resources, maximizing the lifetime of this equipment. Despite the current state of the smart grids, the electricity distribution companies can implement it using the available data. Thermal models provide guidelines for lifespan estimation, but the adjustment to particular conditions, brands, or material quality is done by adjusting parameters. In this work we propose a method to adjust the parameters of a thermal model using Genetic Algorithms, comparing the estimation values of top-oil temperature with measurements from 315 kVA distribution transformers, located in the province of Tucumán, Argentina. The results show that, despite limited data availability, the adjusted model is suitable to implement a transformer monitoring system.

The increase of energy demand caused by the consistent growth of the electricity consumption over the last few years caused the electricity distribution companies to begin considering the integration of the information and communication technologies with the electrical infrastructure. This situation leads to the implementation of Smart Grids, the next generation of power grids [

The relatively slow progress of the Smart Grids in Argentina is due to different factors, mainly the lack of proper investment by companies, and poor planning in the energy distribution context. This situation results in a progressive decrease in the service quality. The margin between installed power and the consumers demand is reduced year after year and, consequently, the electricity distribution system in Argentina is then left with a very small reserve margin, since the electricity demand approaches in many occasions the installed capacity. Particularly in the hot seasons in Argentina, summer temperatures above 35˚C with over 90% humidity are normal, so there are corresponding consumption peaks related to the use of Air Conditioning devices massively. These peaks produce severe damage or reduction of lifespan for several distribution transformers and led to the Energetic Emergency law [

In this sense, estimating and monitoring the lifespan reduction of distribution transformers is a key for decision making, since it enables companies to take the necessary actions for transformer management on time. Several actions can be taken in order to handle the situation (replacing it with a higher capacity transformer, distributing the electricity consumption to near transformers, expanding the electrical system by installing a new transformer whenever possible, etc.). But efficient resource management requires planning and prevision, so an estimate of remaining lifespan and loss rate is necessary.

However, estimating the remaining lifespan of a power transformer is not a simple task for many reasons. The accumulative effects of the transformer deterioration are not well known, so predict this expected lifespan accurately is a big challenge, even under strictly controlled conditions. Recent works performing transformer state diagnosis based on the analysis of cooling oil [

Over time, different models have been proposed to estimate the remaining lifespan of distribution transformers. Each of them takes into consideration different criteria and it is based on different physical effects to explain the transformer deterioration. The Chendong model [

Thermal models are widely used because of their simplicity of implementation (since they require only a few easy to acquire input variables). Also, the incorporation of new sensors for measuring other variables that allow better estimations of transformer remaining lifespan is limited in general and especially in Argentina since the advent of Smart Grids is relatively recent [

This paper presents a methodology to adapt the thermal model to local conditions, to estimate the lifespan reduction in distribution transformers, based on existing loading guides. The used model takes current consumption and ambient temperature data as the only input variables, and we use information about transformer construction characteristics in the model parameter setting step. We use a Niching Genetic Algorithm to determine the best combination of parameters values in the top-oil temperature model (temperature in the upper layer of the refrigerant oil tank). To validate the model, we were applied the methodology to 315 kVA distribution transformers, located in the province of Tucumán, Argentina, whose history was well known (they were not maintained or repaired in its history). The proposed methodology represents an improvement over existing off-the-shelf international guidelines, which takes advantage of minimal measured data and adapts to the prevailing conditions.

The rest of this paper is organized as follows: Section 2 describes the methodology used to estimate the lifespan of distribution transformers and the procedures for parameter setting, both for the top-oil temperature model and the accumulated loss of life; Section 3 describes the tests performed and the results obtained; Section 4 presents the conclusions.

Although deterioration or aging of the transformer insulation is a function of temperature, time and content of moisture, oxygen and acid, the model presented in loading guides [

The analyzed guides (IEC 60076-7 [

The models described in the loading guides analyzed are designed to explain the hot-spot temperature dynamics, in power transformers. For this purpose, they take into account two effects: the heating caused by the electrical currents flowing through the winding, and the effects of the temperature changes due to the oil flow moving around the hot spot. Then, as an intermediate step for estimating the temperature in the hot-spot, we must first determine the oil temperature on the top of the container, where the flow is at its hottest.

The load factor is calculated dividing the current measurements of each phase I f [A] on rated current I n [A], which is obtained from the rated power value of three-phase distribution transformer S [kVA] as shows in Equation (1). For practical reasons, the nominal voltage U n = 220 [V] and the power factor cos ( φ ) = 0.9 are adopted, so the rated current is just obtained as I n = 1.45 × S . Deviations on cos ( φ ) are severely fined both by governments, regulatory authorities and companies on users, son it being above 0.9 is a realistic assumption. Then, for 315 kVA distribution transformers, the rated current I n is equal to 459.3 [A].

I n = 1000 × S / [ 2 3 × U n × cos cos ( φ ) ] (1)

As mentioned above, the first step in the hot-spot temperature estimation process is calculating the temperature in the cooling oil. The Equation (2) describes the dynamics of the oil temperature change on the upper level of the container (top-oil). This equation will first be used to estimate the top-oil temperature rise Δ θ o , which will then be used to calculate the top-oil temperature by solving the differential equation. This procedure corresponds to step 1 in the block diagram shown in

d θ o d t = 1 k 11 × τ o [ ( 1 + K 2 R 1 + R ) x × Δ θ o r − ( θ o − θ a ) ] (2)

On the other hand, we must calculate the hot-spot temperature rise above the top-oil temperature Δ θ h . To model the thermal behavior of the winding hot-spot, the loading guides propose the Equation (3) and Equation (4). The first one represents the fundamental hot-spot temperature rise, without taking into account the effect of the oil flow. The second equation represents the temperature variability that introduces the oil flow through the hot-spot, which changes much slower. Then, combining both effects, according to the Equation (5) we can calculate the hot-spot temperature rise. The solving of these equations corresponds to step 3 in the block diagram shown in

d Δ θ h 1 / d t = 1 / ( k 22 × τ w ) × [ k 21 × K y × ( Δ θ h r ) − Δ θ h 1 ] (3)

d Δ θ h 2 / d t = ( k 22 / τ o ) × [ ( k 21 − 1 ) × K y × ( Δ θ h r ) − Δ θ h 2 ] (4)

Δ θ h = Δ θ h 1 − Δ θ h 2 (5)

Finally, the calculation of the hot-spot temperature (step 4 in the block diagram of

θ h = θ o + Δ θ h (6)

Unfortunately, these differential equations cannot be solved analytically or exact solution in terms of simple mathematical functions. However, they can be solved using well known numerical methods. In this paper, we have selected the Runge-Kutta method using the GNU Octave Ordinary Differential Equations package [

Once the hot-spot temperature has been calculated over a time period, we can proceed to calculate the insulation degradation rate of the transformer.

As mentioned earlier, the first step of this procedure is to estimate the hot-spot temperature, which corresponds to the dotted lines box in

V = exp [ B / ( 110 + 273 ) − B / ( θ h + 273 ) ] (7)

Integrating the relative aging rate V over a certain time period we can obtain the total lifespan L occurred in that period. We use the Equation (8) when we have discrete values, where V n is the relative aging rate at time n, t n is the time duration of n and N is the number of intervals considered.

L = ∫ t 1 t 2 V d t → L ≈ ∑ n = 1 N V n × t n (8)

The models used in the loading guides include certain parameters whose values are established based on many tests performed on the transformers under controlled conditions. Since these parameters depend on the manufacturing characteristics among other things, it is possible that the proposed values may not be suitable in all cases. For these reasons, an adjustment process of these parameters is necessary, in order to adapt the thermal model to different kinds and brands of transformers and its particular conditions. If real temperature measurements are available, the model parameters can be adjusted in such a way that the estimates obtained using the model are as close as possible to the real values. In this context, the problem is referenced as a parameter estimation problem, which belongs to the class of so-called inverse problems [

The model for the top-oil temperature estimation described in Section 2 was adjusted from measurements provided by the local electricity distribution company, Empresa de Distribución Eléctrica de Tucumán (E.D.E.T.s.a.). For this work, data were collected from two 315 kVA distribution transformers located in residential areas with similar consumption profiles. One of these datasets was used in the parameter adjustment process, and the other one in the validation stage, where the top-oil temperature was calculated and compared with real values to determine the error obtained when using the adjusted parameters. Besides the electrical current measurement, the transformers have a temperature sensor located at the top of the oil tank. The available measurements correspond to the period between 13-11-2016 and 25-02-2017, and the records were sampled every 15 minutes. This time period corresponds to the hot season, which facilitates the model adjustment in conditions of higher ambient temperature. About the ambient temperature, the data comes from a meteorological station from the same company, located less than 1.5 kilometers away from the transformers.

To evaluate the performance of the adjusted models, the errors are analyzed by comparing the real temperature values ( Y i ) with temperature estimations produced by the thermal model ( T i ). To do this, we use different metrics commonly used in the literature, such as: Root Mean Squared Error or RMSE (Equation (9)), whose value is interpreted easily because it is expressed in the same unit that the variable to be estimated; Percentage Root Mean Squared Error or RMSE% (Equation (10)), which expresses the RMSE as percentage; Mean Bias Error or MBE (Equation (11)), which allow us to know if there is an underestimation or overestimation, analyzing its sign; Pearson’s Correlation Coefficient R (Equation (12)), which helps to determine the extent in which the model follows the general trend of the data.

RMSE ( T , Y ) = ∑ i = 1 n ( Y i − T i ) 2 n (9)

RMSE% ( T , Y ) = 1 Y max ∑ i = 1 n ( Y i − T i ) 2 n (10)

MBE ( T , Y ) = ∑ i = 1 n Y i − T i n (11)

R ( T , Y ) = n ∑ Y i T i − ∑ Y i ∑ T i n ∑ Y i 2 − ( ∑ Y i ) 2 n ∑ T i 2 − ( ∑ T i ) 2 (12)

Genetic Algorithm (GA) is a stochastic search technique that allows us to find solutions to a wide variety of optimization problems [

To use a GA and find solutions to the parameter adjustment problem, each individual is coded by a vector with real values, where each element represents a particular parameter that we want to find. From Equation (2) we can see that the parameters to be optimized are: k 11 , x , R , Δ θ o r and τ o . However, to prevent the occurrence of different values representing the same solution we set the value of k 11 = 1 as indicated in Loading Guides, so that the combination k 11 × τ o is unique. This is known as redundancy in the context of GA and heuristics, and it should be avoided since it complicates the search unnecessarily. On the other hand, to avoid inconveniences due to a possible wrong calibration or bad location of the temperature sensors (it might not be located exactly in the zone of higher oil temperature), an additional parameter is included by adding a constant C offset to the Equation (2). This extra parameter is used only in the solution search process in order to automatically correct a measurement offset detected in some cases. Nevertheless, we set C offset = 0 once found the best parameter values.

To delimit the search space, we define lower and upper bounds for each parameter. Then, each gene of the individuals will correspond to a single parameter to optimize, which are bounded by the limits described in

Parameter | Parameter value | |||
---|---|---|---|---|

Lower Bound | Upper Bound | IEC Guide | Optimization | |

x | 0.70 | 1.10 | 0.80 | 0.94 |

R | 3.30 | 6.50 | 5.00 | 5.31 |

Δ θ o r | 40.0 | 60.0 | 55.0 | 40.6 |

τ o | 90 | 210 | 180 | 106 |

C offset | −0.20 | 0.20 | 0.00 | 0.00 |

respect the real meaning given to each parameter of the model, that is, to make them consistent with the physical meaning they have.

As discussed above, given the nature of the problem, the combination of parameters of a differential equation that generates a particular solution might not be unique. That is, there could be more than one combination of parameters that produce similar responses. For this reason, it is necessary to search all existing parameter sets to analyze and select the most adequate to ensure that the model generalizes adequately. In this perspective, there are GA architectures specially designed for situations where more than one solution is required. These algorithms, called Niching, allow us to find and preserve all points or zones of the search space corresponding to the local optimums. This means that the Niching algorithms will search for as many different numerical combinations as they can find, of the model parameters that allow obtaining estimates with similar error level.

Among the existing variants of Niching algorithms we use Deterministic Crowding [

・ Initialization: a population is created, consisting of n randomly generated individuals so that each gene will have random values within the range defined by the lower and upper limits for each of them.

・ Evaluation: each individual is evaluated using the fitness function defined for the problem, described in Section 2.2.4.

・ Crossover: all the individuals of the population are grouped in pairs, to recombine them and generate new individuals. We decided to apply more than one operator to achieve a good exploration of the search space as is usual in this architecture. A Uniform Crossover scheme is implemented to the half of the population so, for each gene, one child copies their value from one randomly chosen parent (the other child will take the gene value from the other parent). The other half of the population is generated using Cube Crossover, in which children genes are randomly generated within the range determined by the parent genes values.

・ Replacement: each parent is grouped with their most similar child, and then they are compared to determine which one survives the next generation. The Euclidean distance metric is used to determine which child is the most similar to each parent, and the fitness values associated with each individual are used to decide which is best and therefore allowed to continue into the next generation (see the flow diagram in

To set the GA parameters, some considerations were taken into account. On the one hand, the population size should be large enough to be able to adequately cover the entire search space. In this paper, this was achieved with a population of 100 individuals. On the other hand, to set the number of generations, we established as a termination condition that individuals must generate estimations that exceed a certain threshold of precision. The RMSE is used as the error metric, and the threshold is set to 5 [˚C] for estimations of the top-oil temperature. Thus, the number of generations reached allows us to find solutions that generate estimations with a good accuracy, but at the same time maintain the diversity of the solutions, and do not continue after the majority of the population is made up of solutions that provide good estimations.

As in nature, the selection of individuals in GA is made according to a criterion:

the more suitable they are to solve the problem, the more possibilities they have to survive and reproduce. The fitness function allows us to evaluate each individual and weighs and compare them, in order to determine which solution is more adequate. Thus, the best solutions will be those that enable a better fit of the temperature estimates curve with the measured temperatures curve. Then, the fitness function (which must produce a real numeric value proportional to how good the solution is) is defined in Equation (13), where Y ( S ) are the estimated values of the top-oil temperature using the parameters determined by the solution S, and T represents the real measurements of top-oil temperature. Good solutions will have a low RMSE value and R value close to 1, so the fitness value will be high. The constant term in the fitness function equation is added only to ensure we do not obtain negative values.

fitness ( S ) = 50 − RMSE ( T , Y ( S ) ) R ( T , Y ( S ) ) (13)

The same methodology used to set the parameters of the top-oil temperature equation can be used to adjust the parameters of the hot-spot temperature equations. It would require real temperature measurements of the hot spot taken over a time period to compare them with the results of the adjusted model. However, hot-spot temperature measurements are difficult to get, since the transformers should include a fiber optics sensor that is usually installed in the factory, but this is not a common situation. Furthermore, determining the exact location of the hot spot is not a simple task, since it depends on the particular characteristics of each transformer. Additionally, installing a sensor inside a transformer that does not include this factory feature generally involves dismantling it and taking it out of service for an extended time period. For these reasons, at the moment we do not have measurements of hot-spot temperature. So, we adopt the parameter values proposed by the loading guides for obtaining the first approximation of the results.

The Loading Guides propose a method for estimating the rate of aging based on the hot-spot temperature. The expected lifespan of 20 years is considered as the reference, which is the time it takes to degrade the insulation paper of the winding in normal operating conditions (100% power consumption). From this, the Equation (7) is proposed, and the parameter B = 15,000 is set up for the thermally upgraded paper. This equation is used to determine how fast a transformer reduces its lifespan due to the effect of its winding temperature being at its maximum or exceeded for a certain period of time. However, this reference may be different for distribution transformers with particular characteristics, depending on the type of insulation paper and operating conditions (for example if

Parameter | Value | Description |
---|---|---|

y | 1.6 | Thermal exponent |

k 21 | 1.0 | Thermal constant |

k 22 | 2.0 | Thermal constant |

τ w | 4.0 | Winding time constant |

Δ θ h r | 35.0 | Hot-spot to top-oil gradient at rated current |

it operates in locations with high ambient temperature, direct solar radiation, high humidity, among others).

The hot-spot temperature measurements from the transformer installation time to the moment where it suffers some damage due to overload (i.e. over its whole lifetime) are required to find a better value for the parameter B in the Equation (7). These measurements are unusual and difficult to obtain in Argentina since the installation of new remote measuring Devices on distribution transformers is in progress, and it is relatively recent. Moreover, repair history and traceability is usually not well documented. For this reason, we propose an alternative method to do this adjustment using only the available measurements of the ambient temperature and the electric load at the transformer substation. First, the data time series are filled and completed, considering the period starting from the installation time of the transformer. The data filling procedure is performed by replicating the available data, adjusting the magnitude of the measurements according to the values of annual increments of electricity consumption recorded throughout the province [

After completing the data set, we estimate the hot-spot temperature and the accumulated loss of life. For the last step, it is necessary to find the value of the parameter B so that the PVU calculation is equal to 100% at the moment when we know that the transformer suffered severe damage due to overloads. For this adjustment, we have information on two 315 kVA distribution transformers, which we know that they continuously operated until they were damaged due to overloads (without moving it or repair it). We also assume that there were no events related to failures that could accelerate the aging process (bumps, faults in the contact terminals, short-time over-voltages, among others) as a hypothesis of the problem. One of these cases is used to obtain a proper value of the parameter B, and the other one is used to validate the results achieved with this value. The procedure described above could not be applied to more cases due to the limited information available about the distribution transformers currently installed. Only two cases can be analyzed given the current circumstances, hoping to have more cases as the remote measuring network grows and better care in keeping the history of movements and repairs.

To use the GA described in Section 2.2.3, we used data from a particular Distribution Transformer in order to obtain a set of parameter combinations of the top-oil temperature, and then the model was applied using each combination on a different dataset, from the other transformer, to determine the most appropriate.

axis diagram, and the different temperature curves generated by the model when using each parameter combination. On the one hand, the parallel axis diagram in

Regarding the adjustment of the parameter for the calculation of the relative aging rate, two historical measurements datasets were used. The first one

Parameters | Dataset | RMSE | RMSE% | MBE | R |
---|---|---|---|---|---|

IEC Guide | Training | 7.10 | 14.51 | 6.41 | 0.94 |

Validation | 11.66 | 25.34 | 10.67 | 0.89 | |

Optimized | Training | 2.80 | 5.72 | −1.33 | 0.96 |

Validation | 4.23 | 9.20 | 2.55 | 0.94 |

corresponds to a transformer installed and started working on 13-07-2001, and it was replaced by another transformer due to overload damage on 13-02-2016 (5328 operating days). The second case corresponds to a transformer put into operation on 02-08-2002, and damaged by overload on 23-03-2014 (4431 operating days). The measurements of these transformer substations are only available in the period between 01-01-2013 and 31-12-2016 (4 years of data). There is no record of movements or repairs conducted on any of the transformers, supported by old employees of the company that support the claim that they were never moved or repaired. The previous historical data was completed by replicating the available measurements in the missing periods, taking into account the annual consumption registered in the whole province, reported for residential customers. These annual increments were taken from statistical reports [

Finally, the thermal and the lifespan models adjusted in this paper are applied to the first case dataset to exemplify the use and implementation of a real time monitoring system.

these curves, which would help during the decision making process when the transformers are at their top demand in the year.

Regarding the parameter adjustment of the top-oil temperature model, the optimized values are very similar to the values proposed by the Loading Guides, except for oil time constant τ 0 , which is slightly lower than expected. This difference may be due to the lack of adequate heat dissipation in the transformers so that it heats faster than normal, or may be attributable to the influence of the climate on heat dissipation or the particular characteristics of the transformer

used in the performed tests. These results indicate that the models proposed in the Loading Guides, once adjusted, are adequate for estimating the lifespan of the transformers located in the province of Tucumán. In this sense, the Genetic Algorithms allowed us to obtain good solutions. Some other restrictions might be added (e.g. based on the physics of the problem) that make the problem more complex, what might require adjustments or changes in the algorithm. If adding other restrictions to the parameter adjustment problem was necessary (possibly based on the physics of the problem), other algorithms could be used.

As for the adjustment of the aging rate equation, the results indicate that the lifespan is smaller than the values indicated in the Loading Guides. This result is a product of the particular insulation paper of these transformers degrades faster than expected due to ambient factors (high humidity, solar radiation influence, among others). However, we were unable to confirm this since, in this stage, the adjustment lacks enough data to produce solid results. To improve the accuracy of the prediction we need more data (transformers damaged by overload and with a traceable history). Nevertheless, the model and methodology presented in this paper constitute a good approximation that can be refined and improved as new data becomes available. The model adjustment and validation must be carried on continuously, due to the results are more precise when we have a greater number of measurements. It is important to notice that different brands and construction year seriously affect the transformer lifespan (transformers used to be built more solid and with a larger margin). So, even when more data become available, a clustering process should be performed in order to adjust a model to a group of transformers with similar strength.

From the results presented, we can conclude that the models and methodologies proposed allows for a correct implementation of a transformer monitoring system, helping to plan actions on the transformers near their end of life. As future work, we propose the inclusion of additional variables with greater impact on the internal temperature of the transformer, such as solar radiation [

This work was partially supported by grant UTN3870 for the research project “Intelligent Data Analysis Applied to Energy Management and Optimization”. We also wish to extend thanks to Empresa de Distribución Eléctrica de Tucumán S.A. (E.D.E.T.s.a.) for providing the data necessary to elaborate this paper.

The authors declare no conflicts of interest regarding the publication of this paper.

Jimenez, V.A., Will, A., Gotay, J. and Rodríguez, S. (2018) Adjustment of Model Parameters to Estimate Distribution Transformers Remaining Lifespan. Smart Grid and Renewable Energy, 9, 151-170. https://doi.org/10.4236/sgre.2018.99010