By Bert Zwart
The power grid is one of the most significant engineering achievements in the history of humankind, having kept the lights on for more than a century. Traditionally, the supply of electricity has depended on large, controllable generators that transform fossil fuels into power. These generators employ several methods to balance the supply and demand of electricity while accounting for physical constraints on the network. In the past, electricity demand was relatively easy to predict (even several days in advance) and transport was unidirectional; electricity traveled from large suppliers via high-voltage transmission networks to low-voltage distribution networks, and finally to consumers.
This standard setup has already begun to change and will continue to radically do so over the next several decades. In response to global warming, renewable energy sources—such as wind and solar—are gradually phasing out fossil fuels. This ongoing shift will make the supply of electricity less centralized, less predictable, and more difficult to control; for instance, the addition of rooftop solar panels to the power grid means that a significant portion of energy production will take place locally. The system’s inertia is also decreasing as traditional generators become less common, meaning that authorities must handle unforeseen events (like failures) at a much faster rate.
Researchers have proposed numerous technological innovations to address these emerging issues. Storage devices may take over the customary role of balancing supply and demand, and smart devices—such as refrigerators and air conditioners—can respond to fluctuations in electricity supply. Electric vehicles might charge at a variable rate or even deliver electricity back to the grid. In addition, the evolution of new energy markets that enable peer-to-peer trading is an exciting novelty [10]. All of these advancements come with additional privacy and security challenges as well as associated communication requirements, though developments like 5G and cloud-based services will help to address the latter.
The Need for Mathematics
Incremental changes to the current set of mechanisms that balance supply and demand will not sufficiently or effectively incorporate these developments. Existing algorithms mainly consist of (i) automated control mechanisms that handle imbalances on short time scales of up to several minutes to ensure that frequencies remain at a nominal value; (ii) offline optimization techniques that provide readjustments on the time scale of 15 minutes to several hours; and (iii) energy markets that manage challenges on time scales that range from hours to days. These mechanisms collectively balance the transmission grid—which comprises high-voltage networks that span regions or even countries—wherein the demand by local distribution grids (networks that span neighborhoods of consumers) is fixed. The mechanisms (e.g., online frequency control and offline optimization of energy production) also take place separately and depend on a time scale separation assumption. Such simplifying assumptions, which decompose the balancing problem across space and time, are no longer realistic and waste billions of dollars each year. Finally, many algorithms ignore or linearize the network’s physical constraints — an assumption that becomes more problematic when power grids operate closer to their physical capacities, resulting in similar losses [7].
Future coordination mechanisms—such as online control mechanisms, offline optimization, markets, and combinations thereof—must be able to account for the presence of numerous small suppliers, smart devices, low levels of inertia, and high levels of uncertainty while simultaneously meeting high privacy and security standards. This problem yields formidable scientific challenges in the mathematical areas of distributed and real-time optimization, control, game theory, market design, privacy, security, stochastics, and complex systems.
Many individuals and institutions have recognized the need for a focused effort on these areas in recent years. For example, a 2016 report from the U.S. National Academies of Sciences, Engineering, and Medicine identified relevant challenges in dynamical systems and optimization (particularly the AC optimal power flow problem) [7]. And in 2019, the Isaac Newton Institute for Mathematical Sciences organized a semester-long program on the mathematics of energy systems that addressed corresponding issues in stochastics, machine learning, reliability, and market design.
Designing a New Architecture
If we set out to redesign the entire power grid from scratch, how would we do so? From a mathematical perspective, this is a vast decentralized stochastic dynamic optimization problem in which the behavior of physics, networks, and agents must be jointly modeled, analyzed, and optimized. Two frameworks that partially address this challenge have already arisen in the context of the internet: network utility maximization and layered network architectures [3, 5]. These frameworks explain the successful performance of the Transmission Control Protocol (TCP)—a key distributed algorithm that determines the number of packets that may be submitted for a particular connection—and have helped to develop refined versions of the TCP. Essentially, the frameworks seek to design algorithms that decouple a network’s user layer from its physical layer by constructing an intermediate digital layer; the act of packetizing information enables this decomposition.
Figure 1. A power grid architecture that consists of physical, digital, and user layers. Figure courtesy of Pavol Bauer of Delft University of Technology.
Logistics networks exhibit a similar phenomenon in the operation of the
ocean container: a large vessel that can store a wide variety of products; is movable via sea, road, or rail; and effectively optimizes transport [6]. Certain ideas about the analogous design of a control architecture for a quantum internet are currently materializing [3-5]. Typically, the resulting architectures consist of a “thin” digital layer—known as a bowtie or hourglass architecture—that connects the user to physical layers. Such architectures already persist in many other areas of science [9]. Figure 1 utilizes concepts like energy cells, packets, communities, and hubs to envision this type of layering in the power grid.
What stops us from developing these ideas for the grid? First of all, digitizing energy is much harder than digitizing information. Because energy storage is not easy, it is difficult to establish a separation of time scales — which in turn makes it tougher to decompose the resulting optimization problem into several subproblems. And while the internet has grown organically over time, we would have to redesign the new power grid on top of legacy systems. Finally, evaluating an architecture’s efficiency is not a simple task, namely because no Shannon’s theorem exists for power grids.
These and other questions are all part of active research efforts. For example, current studies are exploring and even deploying the idea of packetizing energy [1]. Furthermore, scaling laws from probability theory and statistical physics—which provide mathematically rigorous ways to perform model reduction techniques—suggest that it might be possible to demonstrate the optimality of certain designs in a suitable asymptotic regime [2]. Concepts from the theory of complex networks and extreme value theory [8] have also successfully contributed to the rigorous study of reliability in the context of renewable energy systems. These developments promise great results for this line of research, and I remain optimistic that we will live to see a future in which the power grid is as efficient as the internet.
Bert Zwart delivered a public lecture about this research at the 2023 SIAM Conference on Computational Science and Engineering, which took place last year in Amsterdam, the Netherlands.
References
[1] Almassalkhi, M., Frolik, J., & Hines, P. (2022). Packetizing the power grid: The rules of the internet can also balance electricity supply and demand. IEEE Spectr., 59(2), 42-47.
[2] Aveklouris, A., Vlasiou, M., & Zwart, B. (2019). A stochastic resource-sharing network for electric vehicle charging. IEEE Trans. Control. Netw. Syst., 6(3), 1050-1061.
[3] Chiang, M., Low, S.H., Calderbank, A.R., & Doyle, J.C. (2007). Layering as optimization decomposition: A mathematical theory of network architectures. Proc. IEEE, 95(1), 255-312.
[4] Gauthier, S., Vardoyan, G., & Wehner, S. (2023). A control architecture for entanglement generation switches in quantum networks. In QuNet ’23: Proceedings of the 1st workshop on quantum networks and distributed quantum computing (pp. 38-44). New York, NY: Association for Computing Machinery.
[5] Kelly, F.P., Maulloo, A.K., & Tan, D.K.H. (1998). Rate control for communication networks: Shadow prices, proportional fairness and stability. J. Oper. Res. Soc., 49(3), 237-252.
[6] Levinson, M. (2016). The box: How the shipping container made the world smaller and the world economy bigger (2nd ed.). Princeton, NJ: Princeton University Press.
[7] National Academies of Sciences, Engineering, and Medicine. (2016). Analytic research foundations for the next-generation electric grid. Washington, D.C.: The National Academies Press.
[8] Nesti, T., Sloothaak, F., & Zwart, B. (2020). Emergence of scale-free blackout sizes in power grids. Phys. Rev. Lett., 125(5), 058301.
[9] Sabrin, K.M., & Dovrolis, C. (2017). The hourglass effect in hierarchical dependency networks. Netw. Sci., 5(4), 490-528.
[10] Sousa, T., Soares, T., Pinson, P., Moret, F., Baroche, T., & Sorin, E. (2019). Peer-to-peer and community-based markets: A comprehensive review. Renew. Sust. Energ. Rev., 104, 367-378.
Bert Zwart is a group leader at Centrum Wiskunde & Informatica and a professor at Eindhoven University of Technology in the Netherlands. His mathematical expertise is in probability and operations research. Zwart is a former recipient of INFORMS’ Erlang Prize; in 2019, he co-organized a semester program on the mathematics of energy networks at the Isaac Newton Institute for Mathematical Sciences.