*3.4. Approximate Aging Model*

As introduced in Section 3.1, the normalized aging is computed with the hot-spot temperature through the function denoted fAEQ(*θh*) = 2ˆ(*θh*-98)/6. For this nonlinear equation, we apply the same linearization process as for the temperature functions. The fAEQ is linearized by the introduction of additional variables δ*θ*hAEQ *<sup>t</sup>*,c to represent the contribution of each block that should ultimately be equal to the estimated hot-spot temperature at every time step (*θ<sup>h</sup> <sup>t</sup>* ). Finally, the degradation over the simulated period is computed with the PWL approximation summed over the time horizon and should be below 1 to avoid an accelerated aging (12).

$$\begin{cases} \sum\_{\mathbf{c}\in\mathbf{C}} \delta\theta\_{t,\mathbf{c}}^{\mathrm{hAEQ}} = \theta\_{t}^{\mathrm{li}}\\ 0 \le \delta\theta\_{t,\mathbf{c}}^{\mathrm{hAEQ}} \le \theta\_{\mathbf{c}}^{\mathrm{hAEQ}} - \theta\_{\mathbf{c}-1}^{\mathrm{hAEQ}}\\ \mathrm{AEQ} \leftarrow \sum\_{t\in\mathcal{T}} \left(\mathrm{f}^{\mathrm{AEQ}} \Big(\theta\_{0}^{\mathrm{h}}\Big) + \sum\_{\mathbf{c}\in\mathbf{C}} a\_{\mathbf{c}}^{\mathrm{fAEQ}} \times \delta\theta\_{t,\mathbf{c}}^{\mathrm{hAEQ}}\right) \le 1 \end{cases} \tag{12}$$

Figure 8a shows a three-block PWL for the nonlinear function fAEQ as we previously did for temperature-related nonlinear functions f1 and f2. However, the aging function grows exponentially with hot-spot temperatures over 80 ◦C Consequently, an error in the temperature estimation at *θ<sup>h</sup> <sup>t</sup>* > 80 ◦C would incur larger deviations in the AEQ estimation. For instance, Figure 8b shows that only 5% error in the temperature at 120 ◦C leads to 100% error in AEQ.

**Figure 8.** PWL process: (**a**) functions fitting and breakpoints for fAEQ; (**b**) aging function—fAEQ error versus *θ<sup>h</sup>* error.

Since the PWL formulation of the degradation is expressed with the linearized hotspot temperature, the approach could ultimately lead to significant errors in the AEQ in case of lower numbers of PWL blocks. Thus, a set of preliminary test runs over a single day is performed while varying the numbers of PWL segments for the approximation of the temperature on one side and for the estimation of the AEQ on the other. Note that the AEQ computed with the PWL in the DR design is always equal to 1 as the aging is a binding constraint for the considered test day. For every simulation, this value is compared to the AEQ computed with the exact profiles for the transformer temperatures (i.e., via IEC standard [42]) and the temperature profiles approximated in the PWL process. The results displayed in Table 4 show around 33% difference with three PWL blocks for the AEQ, while six segments are still considered for the temperature (1.00 versus 0.67 in terms of AEQ values). Increasing the number of blocks for the AEQ estimation allows us to reduce the

error compared to the reference. However, it is not worth increasing the number of PWL segments for the AEQ without increasing the number of segments for the temperature (i.e., when increasing from 12 to 18 blocks for the AEQ computation). Thus, the test with 12 blocks for both AEQ and temperature shows the best results. Note that due to the overestimation of the PWL for convex function, the AEQ computed with the PWL will be always higher than the one computed with the PWL estimation for *θ<sup>o</sup> <sup>t</sup>* and *θ<sup>h</sup> <sup>t</sup> ,* which is itself always higher than the exact value. It allows us to give an additional safety margin despite the estimation error.


**Table 4.** Aging EQuivalent (AEQ) computation for different numbers of PWL segments.

## *3.5. Multiple Time Sets*

Additional preliminary tests, performed with a given DR design (in terms of power and energy), show numerical oscillations especially for the DR contribution *PDR <sup>t</sup>* (Figure 9a) and the hot-spot temperature (Figure 9b). This is due to the load profile, which is originally at 1 h resolution and resampled at 1 min to fit in the time resolution for the transformer thermal model. Then, the load (i.e., the transformer loading without DR) is assumed constant between two successive 60-min intervals. Then, there is a wide range of DR power profiles that would lead to the same energy balance (from the transformer perspective) and similar temperature values at the end of the intervals. One way to overcome the observed oscillations would be to assume ramping constrains or ramping penalties for the DR flexibility power to supply/absorb a given amount of energy over a single 60-min period. The choice is made here to decouple the time sets (or "time grids") with a 1-min resolution for the transformer model and 1-h resolution (*t <sup>d</sup>* ∈ *<sup>T</sup>D*) for the equations related to the DR operation. This approach is often used in generation expansion planning similar to [58] where investments on the installed capacity are done periodically over decades and the operation/dispatch of the assets is computed on a finer resolution between two investment periods.

**Figure 9.** Simulation with single and multiple time sets: (**a**) DR flexibility power; (**b**) transformer temperatures.

Then, the constraints for the power dispatch (i.e., power balance (7)) and DR operations (6)–(8) are rewritten following the newly introduced time set. Similarly to the DR power, the transformer loading will be defined along the time set *TD*. Finally, the link between the two time sets is ensured by modifying the constraints for the PWL with the contributions in the block (along the set *T*) that should match the operating point of the transformer (along the set *TD*) in (13). Figure 9 displays the smoothed results obtained when moving from a single time set to the optimization of two time sets. Additionally, the computational time is also significantly shortened from 40 s down to only 2 s for simulation over a single day interval. The complexity of the problem is reduced despite similar numbers of variables and constraints: 50,000 variables/23,000 constraints instead of 59,000 variables/26,000 constraints.

$$\begin{cases} \sum\_{c \in \mathcal{C}} \delta p\_{t,c}^{\text{f2}} = P\_{t^d}^{tr} \\ \sum\_{c \in \mathcal{C}} \delta p\_{t,c}^{\text{f1}} = P\_{t^d}^{tr} \end{cases} \forall t, t^d \text{ with } t \in \left[t^d, t^{d+1}\right] \tag{13}$$

#### **4. Results and Discussion**

The reader could refer to Section 4.1 to see the validation runs for the integrated management and design of DR.

Otherwise, the reader could pass directly to the obtained results presented in Section 4.2.

#### *4.1. Validation Runs for the Integrated Management and Design of DR*

Before investigating different reserve margins with the methodology introduced in Section 2, different validation runs are performed for a better understanding of the DR optimization problem and results of its solution. A first simulation is run over a single day interval. The objective is to validate the DR design and management block, which were introduced in Section 3. At first, the DR flexibility is operated under "energy shifting" conditions. Note that the aging constraint (i.e., AEQ < 1) is not considered here. Figure 10 shows the results without DR and with DR for a given value of reserve margin (i.e., with a given amount of surplus load).

**Figure 10.** Validation run with "energy shifting": (**a**) transformer loadings; (**b**) temperatures.

The reader can see in Figure 10 that adding a DR flexibility allows us to keep the hot-spot temperature below its limit. At the same time, the oil temperature always remains below the limit with or without the use of flexibility. To avoid the winding overheating during the evening, the load shifting is applied by lowering the peak load after 18:00 and transferring some of the load to the morning (Figure 10a). Note that at the beginning of the simulation, the loading is decreased to reduce the temperature profiles before the DR flexibility is fully charged (to ensure the "energy conservation" constraints). This ultimately leads to higher temperatures at night, but it is still far from the overheating limits. Finally, the DR flexibility follows a "charge/discharge/charge" pattern, and 64% of the estimated

capacity (592 kWh here) is necessary to shave the peak loading. Note that DR flexibility cannot be fully "discharged" as the virtual state of charge should return back to 50% at the end of the simulated period.

The second simulation is performed given the same optimization inputs but with a DR flexibility operating as a typical "energy shedding" and no AEQ constraints. Figure 11 displays the results with the DR activated only in the evening to shave the peak transformer loading similar to the previous simulation. Then, the loading and temperature profiles (Figure 11b) remain unchanged for the rest of the considered day, i.e., prior to the peak shaving. This load shaving is equal to 377 kWh. This shed energy corresponds to the optimized capacity of the installed DR flexibility in the "energy shedding" mode. This expected capacity is obviously much lower than the one computed in the case of "energy shifting", since there is no need to shift/recover the shed load at other time during the day.

**Figure 11.** Validation run with "energy shedding": (**a**) transformer loadings; (**b**) temperatures.

Final validation runs consist of introducing the aging constraint for the same simulated day in case of DR flexibility operated under "energy shedding". The obtained results show that the amount of curtailed energy is greater than the one observed in the previous runs (Figure 12a) so that the hot-spot temperature remains far below the limit which would otherwise incur an over degradation of the winding insulation (Figure 12b). The oil temperature is reduced as expected. Note that the DR capacity estimated at 637 kWh is almost twice as much as in the case with no aging constraint.

**Figure 12.** Validation run with "energy shedding" and aging: (**a**) transformer loadings; (**b**) temperatures.

## *4.2. Results for Different Reserve Margins*

Having performed the validation runs in Section 4.1, this subsection addresses results considering a full representative year that were obtained with the methodology introduced in Section 2. Figure 13 illustrates typical results while comparing three scenarios: the base case scenario (i.e., base load), the base case after adding a given reserve (75% here), and the last case considering the application of DTR/DR ("energy-shedding mode").

**Figure 13.** One-week profiles in January, comparison of the base case and Dynamic Thermal Rating (DTR)/DR in "energyshedding mode": (**a**) transformer loading; (**b**) hot-spot temperature.

Figure 13 shows that the total load has increased significantly after connecting a constant load (corresponding to reserve 75%) over the whole year. Although the current limit (1.5 p.u.) is not violated, adding that load leads to heavy violations of the hot-spot temperature up to 140 ◦C (Figure 13b). Note that the curves are given for a week in January corresponding to the peak load period. As previously mentioned and observed, the appropriate DR design and management allows adjusting the transformer loading, and the hot-spot temperature remains well below the limit at 120 ◦C to fulfill the aging constraints. Thus, the DR is activated almost every day here, which is not the case for the rest of the year, as it is discussed further.

Then, different reserve margins can be investigated and the yearly profile reconstructed with optimized DTR/DR in every case following the methodology of Section 2. Figure 14 shows the main results obtained with a DR in "energy-shedding" mode. As expected, DR volumes in kW (Figure 14a) and kWh (Figure 14b) tend to increase with greater reserve margins. Note that the optimization problem is not tractable for reserve above 75% due to the length of the studied intervals (and consequent size of matrix constraints with over 3.10<sup>6</sup> variables). Specifically, the green curve represents a full formulation of optimization problem i.e., with aging, power and temperature constraints. The formulations with thermal and power constraints (i.e., without aging constraint) is shown by the black curve. Then, the green curve is always equal or higher than the black one due to the higher DR required to mitigate the aging constraints. Specific attention should be given to the gray curves in Figure 14a,c which represent DR volumes calculated with a conventional DTR considering a design winding temperature (98 ◦C) as a temperature limit. As it was discussed earlier, this assumption is often taken in the papers dealing with DTR; however, the design temperature is not a temperature limit and therefore should not be considered as such. The obtained results prove that if fulfilling the design temperature

as a constraint, more DR in terms of kW and kWh is required, and, as mentioned earlier, fewer reserve margins could be managed without DR at all (i.e., reserves 30% at 98 ◦C limit versus 50% with a 120 ◦C limit). The reader can note that there is no need of DR for reserve margins below 45% (the green and black curves remain flat in Figure 14a,b). This means that the thermal capacity of one distribution transformer alone is sufficient to withstand the connected load (i.e., as we mentioned earlier without need to apply DR). In other words, if any load, corresponding to reserve margins below 45%, would be connected to transformers, the transformer total load will not violate any temperature or aging limits (see Figure 3 for specific values of temperature and aging).

**Figure 14.** Obtained results for different reserve margins and DR in "energy-shedding" mode: (**a**) DR rated power; (**b**) DR power share compared to the added load; (**c**) DR rated energy; (**d**) DR energy share compared to the total energy of load.

One significant result of this paper can be observed in Figure 14d that displays the total curtailed energy compared to the total consumption over the simulated year. Results show that only 1% of total consumption needs to be curtailed to connect up to 75% of the additional load (for the transformer already loaded on 86% in N-1 mode). We remind that results are obtained for a very strict hypothesis: the constant load profile of new consumers, the maximum ambient temperature, and N-1 condition during the whole year. Even if it is necessary to shed almost 50% of nominal power of the transformer (Figure 14a or around 30% of the peak load in Figure 14b), the curtailed energy remains marginal, and the full DR capacity is activated only for a few hours of the year. DR operation is further depicted in the histograms of Figure 15.

**Figure 15.** Power shedding over the year—75% reserve in "energy-shedding" mode.

The maximum DR shedding (the last bar at the right side) is only activated for 2–3 h per year. In total, the DR is required around 6% of the year if aggregating all the hours of activation. Once again, it is necessary to remind that such time of DR application would be required only in the N-1 condition, which is unlikely to happen all year long. Figure 16 displays the duration curves for the hot-spot temperature over the year and for different simulations with 75% of reserve (i.e., added 375 kVA to the 500-kVA transformer already loaded for 215 kVA in N mode and 430 kVA in N-1). In normal operation, the temperature remains well below the limit, and no power shedding is actually required. However, under the N-1 condition, significant overheating above 150 ◦C is observed and can be avoided with appropriate DR design and operation with regard to thermal constraints. If the aging is considered in the DR optimization, the DSO should ensure the transformer operation even at lower temperatures (see the green curve in Figure 16).

Another set of simulations is performed with the DR operating in "energy-shifting" mode. The results displayed in Figure 17 show slightly higher DR capacities if compared to the case with the "energy-shedding" mode. However, it is impossible to consider more than 60% reserve. Indeed, due to the energy conservation constraint, any load shedding during the peak shall be shifted and compensated at other time steps. At the first order, the thermal model of the transformer can be considered as dependent with the integral of the loading. Thus, even for different power profiles, if important amounts of loaded energy are considered, overheating can no longer be avoided.

**Figure 16.** Yearly temperature duration curves: 75% reserve in "energy-shedding" mode leading to less than 1% energy curtailment, as seen in Figure 14d.

**Figure 17.** Results: DR modeled with or without payback effect/energy conservation: **(a)** DR power in kW; **(b)** DR energy in kWh.

The last but not least result: Figure 18 shows that the suggested PWL formulation of the optimization problem allows reducing computation times in comparison to the nonlinear formulation. As it was mentioned earlier, the nonlinearity is caused by f1 and f<sup>2</sup> formulas used for temperature calculations and by fAEQ used for the aging calculation. Moreover, the high computational times and non-systematic convergences of nonlinear formulation happen due to the large size of the optimization problem, which is caused by the high (1-min) time resolution (required by IEC standard [42] for the numerical stability). Thanks to the PWL of f1,f2 and fAEQ, it is possible to reduce computation times while still keeping the required high (1-min) time resolution for load and ambient temperature profiles.

**Figure 18.** Computation times for the nonlinear formulation (the blue line) and for the suggested PWL formulation (the green line). Both formulations were tested for the "energy-shedding" case (SOC*DR* <sup>0</sup> = 100% and SOC*DR <sup>t</sup>*=*<sup>t</sup>* = 0%).

The reader can see that for horizons up to 1 week, both formulations have approximately the same computation time. Thus, it does not matter which formulation is used if the connected load (a reserve margin) leads to transformer overheating less than 1 week per year. However, when a horizon of the optimization problem overpasses 1 week (i.e., for particular reserve margins), the difference between the two formulations becomes even more evident. For instance, the PWL optimization problem at the horizon of 45 days is solved for about 2 h, whereas the same problem in the nonlinear formulation would be solved for more than 1 day. Despite the benefits of PWL over the nonlinear formulation, it is still impossible to solve the optimization problem at the horizon of one year. The optimization problem at the year-wise horizon, even in PWL formulation, becomes intractable. It seems that the high resolution of data required by IEC standard [42] is a main barrier for solving the optimization at such long horizons with nonlinearities f1, f2, and fAEQ of the transformer thermal model. Therefore, other approaches e.g., [59] can be envisaged together with PWL to enlarge the studied horizons (and thus reserve margins), but such study is outside the scope of this paper.

#### **5. Conclusions**

This paper presents the methodology to increase the available reserve of a transformer using Demand Response and Dynamic Thermal Rating. The maximum reserve estimation relies on a linear programming that simultaneously optimizes the DR volume and operation over a given time interval. The mathematical formulation accounts for the thermal limits of the transformer, the maximum power/current, and the aging effects. The most noticeable result shows that relatively small DR volumes (≤1% of total energy consumption) could ensure high reserve margins of transformers. Although DR volumes in kW could reach 30% of peak loads, such high DR volumes will be needed only if the transformer operates in N-1 conditions and for only a few hours per year. In the N mode, no DR is required at all; no thermal stress of the transformer is observed even if high reserve margins are studied. Additionally, those results are obtained despite very strict hypotheses: the constant load profile of a new consumer, historical maximum ambient temperature over the whole month, and normal cyclic limits. Thus, it is very likely that if DSO adopts the methodology to assess the reserve with consideration of exact load profile, then even a large increase of consumption (reserves) can be approved compared with the results obtained in this paper.

The paper also shows that the optimization problem formulated in this paper becomes intractable for the long horizons (i.e., for high reserve margins). This happens due to the inherit feature of the transformer thermal models, which require a high (few minutes) time resolution of data to keep the numerical stability of temperature and aging calculations. Due to using high resolution over the long horizons, the number of variables and constraints of the optimization problem increase substantially. PWL can reduce the computation times drastically in comparison to the nonlinear formulation but still cannot cope with year-wise horizons. Thus, more research is required to allow solving the integrated design and management problem on the long horizons.

As a conclusion, the observed results are very valuable for DSO and consumers since they could be used to establish a variable network access also known as "flexible network connection agreements" [60]. The general idea of such agreements is that the DSO does not provide a firm capacity all the time for certain consumers (or generators). Depending on different incentives (e.g., lower connections costs), the consumer agrees to have a limited access to the distribution network during certain time/events. Such agreements are already used in the United Kingdom for generators, and they are tested in France [60]. For the considered test case, all consumers have access to the distribution network in the N mode, and no transformer overheating occurs. However, if in case of the N-1 mode, consumers could have a limited access during 5–6% of time, as it was earlier illustrated in Figure 14. At the same moment, we remind that apart from Demand Response, the DSO could also use automation [41], load transfer and reconfiguration [61], volt-Var control [62], electric vehicles management [31,63,64], and standby transformers [65] to quickly mitigate the lack of the transformer capacity in the N-1 mode. Thus, the actual time of limited access for consumers could be further reduced. Another legal possibility for implementing those DR operations is the introduction of interruptible contracts [16,25,41]. The interruptible contracts allow DSO to shed some consumer load in exchange for financial payment to consumers. It is believed that interruptible contracts and flexible network connection agreements could be a legal foundation to connect more load to existing transformers while deferring large investments for reinforcements. Moreover, the recent study [33] shows that the operation of existing transformers (with electric vehicles) ensures less CO2 emission against reinforced transformers. This additionally justifies the utilization of the existing transformers instead of their reinforcement.

The results also showed that for DR application, it is more beneficial to apply a DTR based on Hot Spot Temperature (HST) limit (120 ◦C) rather than DTR based on design HST (98 ◦C), which is widely used in other papers on DTR [66–69]. Specifically, the reader can see in Figure 14 that if DTR based on HST limit is used, then DSO needs to apply less DR volumes both in power and energy terms for studied reserve margins. The authors would like to point out that transformers, thanks to using the HST limit (120 ◦C) instead of the design HST (98 ◦C), could ensure a better utilization of capacity rather than other network elements. Transformers, even in normal mode, can exceed a design HST (98 ◦C) for short time (without exceeding the aging), whereas lines are not supposed to exceed their designed operating temperatures during normal operation [70]. From this point of view, DSO can better utilize a transformer capacity in normal mode and therefore have an additional degree of freedom. However, it is also true that the line's DTR could be twice as great as the line's static thermal rating in MegaVolt Ampere (MVA) [70], whereas a maximal MVA rating of transformers would be limited by a current limit of 1.5 p.u. from IEC standard and even lower current limits [71]. The reader could refer to [72] for details on the difference between HST limit and the design HST as well as their impact on transformer capacity. Permission for lines with higher maximum temperatures is under discussion [73,74], but to the author's knowledge, exceeding the designed operating temperature of lines is not yet approved for normal operation in the standards [75,76] (in contrast to transformers standards [42]).

**Author Contributions:** I.D.: Conceptualization, Methodology, Software, Writing—Original Draft, Writing—Review and Editing, Visualization, Investigation, Validation, Formal analysis, Funding acquisition. R.R.-M.: Conceptualization, Methodology, Software, Writing—Original Draft, Writing— Review and Editing, Visualization, Investigation, Validation, Formal analysis. M.-C.A.-H.: Supervision, Funding acquisition. R.C.: Writing—Review and Editing, Validation, Supervision, Funding acquisition. A.P.: Writing—Review and Editing, Validation, Supervision, Funding acquisition. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Conference des Grandes Ecoles, Tomsk Polytechnic University Competitiveness Enhancement Program, IDEX Université Grenoble Alpes, ATER Grenoble INP.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the authors.

**Acknowledgments:** We deeply appreciate Conference des Grandes Ecoles, who first supported our research and laid the ground for its further development. The research is also funded from Tomsk Polytechnic University Competitiveness Enhancement Program grant. The authors thank the IDEX for funding Ildar DAMINOV travel grant. We gratefully appreciate the funding received from Grenoble INP in the framework of ATER position of Ildar DAMINOV. We thank Egor GLADKIKH (a founder of TOPPLAN start-up) for his help with financing the Ph.D. thesis. Authors thank MeteoBlue for provided data on ambient temperature in Grenoble, France.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


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