# Online Learning Algorithms for the Real-Time Set-Point Tracking Problem
## 2 Mathematical Framework of Online Decision Making
However, in practice, better performance can be achieved with specially tuned step sizes for both algorithms. For instance, in the case of OFW, it was always observed that if some warm-start modification step was added to the approach, this could quickly decrease the loss and achieve a very small accumulative loss. Similarly, for the OGD case, some time-varying step sizes could result in better performance; however, this depends on the type of problem and the related parameters. To obtain better insights for the step size tuning, sensitivity analysis is a future extension of this work to construct a warm start modification of OFW and a step-size problem-dependent tuning for the OGD algorithm-based approaches. Figure 12: Comparison of the time consumption of each approach for different numbers of TCLs. \begin{table}
\begin{tabular}{c c} \hline \hline
**N** & **Time Consumption Ratio (OGD/OFW)** \\ \hline
10 & 1.09 \\
100 & 1.12 \\
1000 & 1.21 \\
10,000 & 1.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Quantitative comparison of the time consumption with different numbers of loads

## 7 Conclusions

Evolving technologies make it possible to become more productive while economizing customer energy consumption. This growing suite of technologies includes the possibility to provide market data very fast for demand-side customers, thus, activating them to respond to energy-related signals in a quick fashion. In this study, the problem of real-time set-point tracking was addressed. In the introduced real-time settings, it was difficult to make fast exact predictions with the popular approaches and even utilizing off-shelf solvers and powerful computing systems still could not produce an optimized solution within the required time (a few seconds). Thus, a decision-making framework based on the online optimization was introduced, and the performances of two candidate approaches (with and without projections) were fully analyzed--namely, OGD and OFW. We demonstrated that both approaches successfully decreased the loss over time and achieved sub-linear regret, which is the goal of online optimization. Thus, the error of the learning decreased over time and did not increase linearly. In addition, both approaches require further tuning to achieve an ideal performance, which can be a task for future works. The time consumption of both approaches was investigated. As was clearly observed, both algorithms were very fast; however, in applications with more limited time settings, the OFW was preferable. Conceptualization, A.A., D.P. and M.F. ; methodology, A.A.; software, A.A.; validation, A.A., D.P. and M.F. ; formal analysis, A.A., D.P. and M.F. ; investigation, A.A., D.P. and M.F. ; data curation, A.A.; writing--original draft preparation, A.A.; writing--review and editing, A.A., D.P. and M.F. ; visualization, A.A., D.P. and M.F. All authors have read and agreed to the published version of the manuscript. This work was prepared as a part of the AMPaC Megagrant project supported by Skoltech and The Ministry of Education and Science of Russian Federation, Grant Agreement No 075-10-2021-067, Grant identification code 0000005/07521 QJX0002. Not applicable. The authors declare no conflict of interest. ## Abbreviations

* **Acronyms**
* **EV**: Electric vehicle
* **DR**: Demand response
* **IBR**: Inclining block rates
* **LSE**: Load scheduling learning
* **OGD**: Online gradient descent
* **OFW**: Online Frank-Wolfe
* **TCL**: Thermostatically controllable load
* **Sets & parameters**
* **_a_**: Dimensionless TCL parameter
* **_C_**: Thermal capacitance
* **_i_**: TCL index
* **_K_**: Decision-making set
* **_N_**: Number of loads
* **_pr_**: Rated power
* **_R_**: Thermal resistance
* **_Reg_**: Dynamic regret
* **_s_**: Set-point signal
* **_t_**: Time period

## References

* (1) [PERSON] A survey on smart grid technologies and applications. _Renew. Energy_**2020**, _146_, 2589-2625.