Power Systems Without Fuel Joshua A

1 Power Systems Without Fuel Joshua A. Taylor, Member, IEEE, Sairaj V. Dhople, Member, IEEE, and Duncan S. Callaway, Member, IEEE

Abstract—The finiteness of fossil fuels implies that future for new optimal power flow objectives to replace fuel costs electric power systems may predominantly source energy from and the attendant need for alternative economic mechanisms to fuel-free renewable resources like wind and solar. Evidently, nodal pricing. In some cases, removing fuel-based generators these power systems without fuel will be environmentally benign, sustainable, and subject to milder failure scenarios. Many of can result in significant benefits like the elimination of the these advantages were projected decades ago with the definition unit commitment problem and the viability of transitioning to of the soft energy path, which describes a future where all energy a predominantly DC infrastructure. We address such issues in is provided by numerous small, simple, and diverse renewable a unified fashion by surveying existing discussions, identifying sources. Here we provide a thorough investigation of power challenges, and suggesting new directions where appropriate. systems without fuel from technical and economic standpoints. The paper is organized by timescale and covers issues like the Two key motifs of this paper are that (see Fig. 1) irrelevance of unit commitment in networks without large, fuel- • all timescales are shrinking, and based generators, the dubiousness of nodal pricing without fuel • a few large components will be replaced by numerous costs, and the need for new system-level models and control methods for semiconductor-based energy-conversion interfaces. small components. Index Terms—Optimization, Power electronics, Power system These changes have a number of salient consequences. For operation, Renewable energy, Soft energy path. instance, renewables can be installed and maintained more quickly than fuel-based generators, bulk generation need not I.INTRODUCTION be scheduled days in advance to accommodate stringent startup and shutdown constraints, and the loss of mechanical VENTUALLY, whether from a deliberate shift to renew- inertia implies that transients will be dominated by the dy- ables or the depletion of planetary fossil fuel sources, E namic behaviors of many small, fast power-electronic energy- major portions of power systems will run without fossil fuels. conversion interfaces. We partially structure our discussions There is now extensive literature on this subject, see, e.g., [1], around timescales by first considering long-term planning, then [2], most of which focuses on the advantages and disadvan- steady-state operation, and then transient dynamics. While so tages of renewable energy sources. Here we also consider this doing, we additionally highlight the emergence and disap- scenario, but focus on the combined consequences of using pearance of new and old couplings. For example, renewable renewables and eliminating fuel-based generators; we fully intermittency couples hourly steady-state dispatch decisions define our scope in Section I-B. We believe that it is important to regulation requirements. Similarly, wind and solar power to discuss these issues both for the limiting case of exclusively sources consume negligible quantities of water compared to renewable power production and also under mostly renewable the cooling needs of fuel-based generators, drastically reducing power production, in which case planning and operational the power system’s dependency on the water infrastructure. practices should reflect the large majority of renewables instead of a small minority of fuel-based generators. Much of the salient physics of present-day power systems A. The soft energy path and the status quo are attributable to fuel-based generators, which are most cost- effective at large unit sizes; for example, thermal limits con- The rationale for a fully renewable energy infrastructure strain power production schedules, and synchronous machine has long been well established; we now briefly summarize arXiv:1506.04799v1 [cs.SY] 15 Jun 2015 rotor inertias dominate transient stability. While the addition its beginnings and provide a few modern examples. In 1976, of intermittent and distributed renewables will change the Lovins described the soft energy path as a future in which soci- form of power systems, so will removing the fuel-based etal energy needs are met by numerous small, simple, diverse generators that account for much of its current character. Many technologies that rely on renewable energy sources [3], [4]. of the consequences of removing fuel-based generators are The soft energy path stands in contrast to the hard energy path, well studied, for example the loss of system inertia and the wherein energy is provided by a few large, resource intensive, need to replace generator reserves with storage and demand complex technologies which inevitably induce caustic political response. However, it is our perception that many important dependencies and are prone to expensive physical failures. The issues have not been thoroughly discussed, such as the need case for the soft energy path—which we note, does not include nuclear power—has been restated and reaffirmed in many ways J. A. Taylor is with the Department of Electrical and Computer over the past four decades; for instance, the capability of wind Engineering at the University of Toronto, Toronto, Canada. E-mail: [email protected]. and solar to fulfill all of our energy needs is addressed in [5]. S. V. Dhople is with the Department of Electrical and Computer Engi- Distributed generation [6] and microgrids [7] are two related neering at the University of Minnesota, Minneapolis, MN, USA. E-mail: architectural paradigms where small, renewable technologies [email protected]. D. S. Callaway is with the Energy and Resources Group at the University meet energy needs locally. Another compelling argument for of California at Berkeley, CA, USA. E-mail: [email protected]. the soft energy path is that unused wind and solar energy are 2

Fig. 1. No-fuel power systems will have fundamentally different spatio-temporal characteristics. Time scales governing operations and control will shrink, and energy-conversion interfaces will be distributed in form and function. lost while fossil fuels may be indefinitely “... [left] ... in the or biomass and biofuels, the latter of which may remain useful ground for emergency use only” [3]. for some non-grid connected types of energy consumers like It may well be some time before continental-scale portions long-distance air travel [1], [14], [15]. of power systems are fully on the soft energy path. However, Throughout our discussion, we implicitly assume the ex- there are a number of encouraging small-scale examples, a istence of large stores of flexibility, by which we mean few of which we list below: the ability to match supply and demand on all timescales • Scotland exceeded its target of 31% renewable electricity and maintain stable voltages throughout the power system. production by 2011, and has established targets of 50% The central challenge in replacing conventional fuel-based and 100% by 2015 and 2020, respectively [8]. generators with renewables is finding inexpensive sources of • Following a destructive fire, the Caribbean island of flexibility. Presently, flexibility is achieved via reserves and Bonaire now aims to generate all of its electricity from regulation. Reserves balance supply and demand in the face of renewables [9], about 45% of which will be wind and the unpredicted load variations and contingencies, i.e., component remainder biofuel or biomass. Geographical remoteness failures. Regulation, which includes automatic generation con- makes local renewable production highly pragmatic for trol, automatic voltage regulation, and droop control, balances small island power systems [10]. supply and demand on faster timescales and maintains stable • Following a nuclear disaster in 2011, the Fukushima pre- voltage frequencies and magnitudes throughout the power fecture in Japan established 100% renewable electricity system. The reader is referred to [16], [17] for comprehensive production as their target for 2040 [11]. coverage of these topics. Presently, most of these services are • Portugal, which has no established companies in fossil or provided by fuel-based generators. Replacing these generators nuclear fuel production [12], aims to have 60% renewable with renewables simultaneously increases the need for flexibil- electricity production by 2020 [13]. ity and removes the main source of flexibility. A core obstacle is that most new sources of flexibility are still quite expensive. B. Setup and organization Because we are taking a long-term view in this paper, we neglect the challenge of financing the procurement of In this paper, we attempt to broadly discuss the implications new sources of flexibility, and rather assume that sufficient of fully transitioning electric power systems to a no-fuel future. flexibility can be provided by the following non-fuel-based We implicitly take the virtues of the soft energy path as given mechanisms. and assume almost 100% renewable power production to be a viable future for a large share of power systems. In this regard, • Energy storage devices like batteries, flywheels, and our main objective is to better understand power systems pumped hydro [18]. A generic model of storage is given without fuel as a limiting case to help plan for an energy by equations (2)–(4) in Section III-A2. future that is potentially very different from the present. • Flexible loads enrolled in demand response pro- By renewable, we primarily connote wind, solar, wave, and grams [19], [20]. Several recent studies have shown that hydroelectric power production, but also allow for sustainable flexible load aggregations can provide similar services low-fuel, low-pollution resources like geothermal energy and to (and are well-modeled by) storage [21]–[23]. For this fuel cells. Because we are interested in both all-renewable reason, we will generically use the term storage to refer to and almost all-renewable power production, we allow for oc- devices like batteries and flywheels and to virtual storage casional instances of these resources that contradict our setup. from aggregations of flexible loads. For example, pumped hydro and concentrating solar power • Spilling renewable power by operating below capacity plants often use synchronous generators and can be in the limits. The maximum power output of a wind or solar hundreds of MW, which is much larger than most renewables; power producer depends on prevailing ambient conditions however, both technologies still have very low marginal costs which are inherently random. Renewables are largely and are amenable to direct current configurations. Note that dispatchable up to their maximum power outputs [24], the scope of the paper does not encompass nuclear generation [25]. 3

• Power electronic devices. Converter interfaces for renew- several decades. Wind turbines are typically designed for a ables, storage [26], and direct current systems [27], [28] twenty-year lifetime [34], while PV panel components can be and FACTS devices [29], [30] can provide voltage support expected to last anywhere from five years to two decades [35], and regulation on fast timescales. [36]. PV can be incrementally installed on daily timescales, The remainder of the paper is organized as follows. In Sec- which should be reflected in scheduling and dispatch routines. tion II, we discuss planning and some broader issues like Renewables are subject to different geographical constraints couplings to other infrastructures and DC systems. We move to than fuel-based generators. Typical generation planning con- the steady-state timescale in Section III, where we discuss the straints may include minimal distance from population centers scheduling problems of unit commitment and load-shifting, and proximity to water sources for cooling. On the other real-time dispatch, and the economics of power systems hand, renewables are geographically constrained by weather without fuel. We discuss dynamics and transient stability in characteristics such as in the cases of offshore wind and Section IV. Figure 1 illustrates the paper’s organization within desert solar power plants, which may entail significant new the motif of evolving spatiotemporal characteristics. transmission investment [37], [38]. Similarly, accommodating rooftop PV systems in existing distribution networks calls for an upgrade in distribution-system infrastructure. In the II.LONG-TERMPLANNINGANDBROADERISSUES short-term, this implies that renewables may require more In this section, we discuss planning with renewables and transmission expansion than fuel-based generators. In the long- the broader issues of direct versus alternating current and term, this may incentivize large power consumers to locate couplings with other infrastructures. in energy dense regions. There are few (if any) geographical constraints on non-hydroelectric storage. A. Economies of scale and geographic constraints In economic terms, the above-described smaller unit sizes make planning renewable energy installations considerably This section focuses on basic differences between planning less lumpy than fuel-based generators; rather than infrequently with fuel-based generators and with renewables and storage. building large generators over long intervals, wind turbines and We structure our discussion around the following generic solar panels can be installed incrementally over shorter time optimal planning problem. periods. We summarize the resulting differences in the context " # X of the optimization problem in (1) below. minimize ci(xi, σ) + f(x, y, σ) x,y Eσ • Planning renewables has lower computational complexity i∈C (1) than fuel-based generators. For instance, because wind such that 0 ≤ x ≤ x , x ∈ , i i i Z turbines and solar panels come in finer size increments y ∈ P(x). than fuel-based generators, the sets of possible farm sizes Here, xi represents the decision to install component i ∈ C, are better approximated by continuous intervals. In (1), where C is the set of candidate components, and y is the vector this means increasing the number of sizes a candidate of operational variables, for example, real and reactive power component can take on, xi, and decreasing the sensitivity P flows and voltage. The first constraint on xi represents the of the cost, ci(xi, σ) + f(x, y, σ), and feasible oper- quantity installed, and the second constraint introduces a dis- ating range, P(x), on the installation variable xi. This crete requirement, e.g., that the number of wind turbines must improves the quality of continuous approximations, and be integer valued. The feasible set of operational variables in turn, the performance of integer programming tech- depends on the installed components via P(x). The first term niques like branch-and-bound [39], [40], making optimal in the objective is the sum of installation costs, and the latter, plans easier to obtain. the total operating cost. Because installation can span weeks • Renewables need shorter planning horizons than fuel- to decades, the expectation of the cost is taken over σ, a vector based generators. This leaves less time for predictions of random variables representing uncertain future information at the start of the project to deviate from reality once like load growth and material and fuel costs. completed, for example due to unexpected load growth. Economies of scale reduce fuel-based generator costs as In (1), this means reducing the variance of the random installed capacity increases. On the contrary, wind turbines and variable σ. While this may not affect the expectation of PV inverters are typically available at smaller ratings, often the objective, it decreases the risk by reducing the like- one to two orders of magnitude less than fuel-based generators lihood of the expected cost diverging from the realized in terms of watts per facility. For example, in 2013, the average cost. Consequently, there is less uncertainty from long U.S. coal, gas, and nuclear plant capacities were approximately planning horizons with renewables, decreasing the risk 272 MW, 85 MW, and 104 MW, respectively [31]. On the of wasted investments. other hand, the Vestas V164 has the largest capacity of • In power systems that conduct generation expansion any individual wind turbine at 8 MW [32], and the average through market mechanisms rather than the centralized installed residential-scale PV system was 6.2 kW in 2012 [33]. solution of mixed-integer programs (see, e.g., [41], [42]), While constructing a fuel-based generator takes around five having more numerous, smaller market participants more years, wind turbines typically take a few months, and photo- nearly satisfies the underlying theoretical assumptions voltaic systems and many storage technologies need only be of competitive markets [43]. In particular, renewable interconnected. Fuel-based generators are built to operate for producers will be closer to price-takers, thus enhancing 4

competition and easing market entry for new firms. Today, power electronics offer alternative mechanisms for • On the other hand, power systems that rely on market varying current to voltage ratios that are compatible with signals to incentivize capacity expansion will likely rely both AC and DC [49], negating the first point above. Absent increasingly on capacity payments, i.e., payments for synchronous generators, synchronous and induction motors generators to be available to deliver power in the future, are the only remaining major classes of devices that are and less on coordinated spot markets for energy, which sinusoidally excited. Electric motors make up approximately currently compensate producers at the marginal cost 45% of global electric power consumption, 23% of which are of the highest accepted bid. “Energy only” wholesale synchronous for a total of approximately 10% of the global markets (i.e., markets with only spot energy prices and load, a clear minority [50]. Almost all other loads merely without capacity payments) such as those in Texas and tolerate AC because the unwanted effects of 60Hz oscillations Alberta will probably need to be re-designed to ensure are imperceptible (e.g., lighting) or because they are equipped that capacity expansion continues to meet reliability with adaptors that locally covert to DC (e.g., computing). requirements in the face of potentially reduced energy The major shortcoming of DC relative to AC is at the market revenues. protection level; AC faults are easily cleared because the current crosses zero 120 times per second, but DC current B. Coupling to other infrastructures is constantly nonzero, leading to potentially dangerous arcing when circuits are opened. Viable protection schemes exist, In this section, we address dependencies on other infras- especially for low-voltage settings [51], but they are more tructures. Removing fuel-based generators decouples power expensive than their AC counterparts. In high-voltage settings, systems from other infrastructures because in the envisioned new technologies like multi-level voltage source converters no-fuel future: offer promising solutions [52], [53]. Since conceptual solutions • fuel does not need to be mined and transported via ship, already exist, we regard DC protection as a surmountable ob- rail, road, or pipe; stacle that will become more economical with future research. • significantly less waste (e.g., ash from coal power plants) The features that made AC indispensable for the power needs to be disposed of; and systems of the last century, voltage conversion via transformers • water is not necessary for cooling, as it currently is for and machine synchronization, respectively can be achieved thermal power plants.1 by other means (power electronics) or serve little purpose in These decouplings further reduce planning complexity by power systems without fuel. We list several distinct differences removing considerations such as collocating fuel-based gen- between AC and DC transmission systems below. erators and desalinization plants. First, high-voltage DC transmission is an established tech- Power systems without fuel will however be more coupled nology [27], [28], [54] that is superior to AC in terms of to the transportation and communication infrastructures. Cou- pling power systems to transportation through electric vehicles • only needing one rather than three phases, and trains will increase complexity, but at the benefit of • more efficient wire utilization because the RMS and peak decoupling transportation from the fuel infrastructure. Lever- currents are approximately the same, aging smart meters and employing more real-time control [45] • providing enhanced stability via additional controllability enhances the achievable performance of power systems while and physical decoupling [55]–[57], introducing security and privacy vulnerabilities [46]. • and requiring reduced communication due to amenability to decentralization [58], [59]. C. AC versus DC and the relevance of frequency DC is also viewed as a conceptually superior option for Since the so-called “War of the Currents” of the 1880s, small power systems such as industrial sites [51], [60] and AC has been a core feature of power systems around the microgrids [61], [62] because of world; [47] provides an excellent historical account. Since then, the question of which would be better today has often • improved efficiency due to decreased losses between DC been posed, cf. [48]. We now do so again for power systems sources and loads, without fuel. For cohesion, we contain our entire discussion of • absence of synchronization and reactive power consider- this issue to this section despite certain aspects like stability ations, informing faster timescales discussed elsewhere in the paper. • and improved isolation from faults and failures on the First, recall the two main reasons for using AC instead of main grid. DC. Finally, operating DC systems in steady-state is mathemat- • Transformers only work with AC and until recently were ically simpler than AC systems. This was part of the reason the only practical solution for long-distance transmission. behind Edison’s opposition to AC. The modern pragmatic • Synchronous machines can directly synchronize with AC implication is that DC systems lend themselves to easier power systems. computations, further reducing the operational complexity of 1We remark, however, that in some cases solar thermal generation requires power systems. In particular, convex relaxations of DC power significant amounts of water [44]. flow [63] are exact under more general conditions than convex 5 relaxations of AC power flow [64], [65].2 This means that Unlike fuel-based generators, wind turbines and hydroelec- solving the standard nonconvex optimal power flow is tractable tric generators can go from off to maximum output in minutes in the DC case for a wider range of systems than the AC case. due to the absence of thermal constraints [70]. Solar panels and distributed energy-storage devices that are interconnected Main points. Planning renewables introduces new transmis- through power electronics can engage and disengage almost sion requirements. However, planning renewables is an easier instantaneously [25]. Without fuel-based generators, the most problem than planning fuel-based generators because stringent of these startup and shutdown constraints amounts to • renewables and storage devices come in smaller units, a few minutes, which is essentially real-time with respect to improving the accuracy of continuous approximations to current power system operational practices. Therefore, a major planning problems and hence decreasing the difficulty of perk of eliminating fuel-based generators is the elimination the associated integer programs; and of the unit commitment problem and the concomitant need • they can be installed over shorter time periods, decreas- to procure reserves long in advance, which in turn has the ing uncertainty from long planning horizons. following benefits: The following two considerations suggest that power systems • The computational complexity of power system opera- without fuel can attain superior performance to that of current tions decreases because combinatorial scheduling prob- power systems. lems, i.e., integer programs, do not need to be solved. • Power electronics and the absence of synchronous gen- • The uncertainty resulting from scheduling fuel-based erators make DC a viable paradigm. generators hours to days in advance disappears. Instead, • With the exception of communications, power systems renewables and storage can be brought online with lead without fuel will be less coupled to other infrastructures, times on the order of minutes. This reduces the time for reducing complexity and enhancing robustness. predictions to deviate from reality, hence reducing the risk of inefficient decisions. III.STEADY-STATE OPERATION Without fuel-based generators, reserves must be supplied by In this section, we discuss the effects of removing fuel- storage to ensure power balance in the face of contingencies, based generators on steady-state operations where bulk supply load variations, and renewable intermittency. However, holding is matched to demand. slack storage capacity does not incur costs in the way that fuel-based generator reserves do. • Assuming the storage’s leakage is low (which is realistic A. Scheduling for technologies like batteries and pumped hydro [18]), The two main scheduling problems in power system oper- keeping a storage device on standby is essentially free. ations are On the other hand, fuel-based generators are expensive • unit commitment, the problem of selecting when fuel- to keep on standby due to fuel consumption and other based generators are online and offline, and operating expenses. • load shifting, the problem of moving energy in time via • Fuel-based generators incur opportunity costs because storage. providing reserves limits the base-load power they can Eliminating fuel-based generators and adding renewables and sell in energy markets. Storage incurs no such opportunity storage eliminates unit commitment and increases the system’s cost since it does not provide base-load power. load-shifting capability. Impacts of renewable intermittency 2) Load-shifting: Load-shifting refers to using storage to are more pronounced on real-time dispatch, which we address extract power from the system (typically at times of low net in Section III-B. demand) and inject power later (typically at times of high net 1) Unit commitment: Unit commitment has long been a demand). central part of power system operations [17], [66], [67]. Unit s = αs + ∆ (η max{u , 0} + η min{u , 0}) , (2) commitment is challenging because a fuel-based generator can t+1 t in t ex t take hours to a day to switch between on and off states. Con- where st and ut are respectively the state of charge and power sequently, the attendant scheduling problem requires binary injection/extraction in time period t, α ∈ [0, 1] is the leakage variables to indicate when fuel-based generators are on and over one time period, and ηin ∈ [0, 1] and ηex ∈ [1, ∞) are off, and unit commitment solutions must be obtained hours the injection and extraction losses, and ∆ is the length of to days in advance of real time. The resulting combinatorial each time period. Typically, the state of charge, injection, and problems are often posed as mixed-integer programs [68], extraction are subject to energy and power capacity constraints which, despite the development of powerful heuristics, are of the form NP-hard and hence difficult to solve [39], [40]. Generator 0 ≤ st ≤ C, |ut| ≤ R, (3) reserves—idling generators kept online as insurance against failures—must also be scheduled in advance, and are usually Observe that the above model only contains linear constraints. incorporated into unit commitment routines [17], [69]. If the storage has a power-electronic grid interface and is to be used for voltage support, reactive power can be included 2To avoid confusion amidst the growing presence of DC technologies, via the convex quadratic constraint we suggest that the phrase “DC power flow” no longer be used to refer to 2 2 2 linearizations of AC power flow. ut + qt ≤ R , (4) 6

where qt is the reactive power into or out of the storage regulation, which are discussed in Sections III-B and IV. device. Since this model only contains convex constraints, load Finally, a key difference between load shifting and unit shifting is a fairly tractable problem that can be addressed via commitment is that the load-shifting decisions can be im- dynamic programming [71] or multiperiod optimal power flow plemented in near real time. This is because storage needs [72]; we discuss this further in Section III-B. only a few seconds to change its power output while a fuel- With the storage and power-electronics models outlined based generator may take half of a day to turn on or off. above, we can now dwell on the following generic optimal Consequently, although load-shifting schedules can span more load-shifting problem. than a day, the attendant computations and decisions can be T made just prior to real-time. For instance, future schedules X X minimize fi,t(pi,t) could be continually updated in the fashion of model predictive p,u t=1 i∈G or receding horizon control [75] by solving (5) every five such that ui satisfies (2)–(4), i ∈ S, (5) minutes. p ≤ p − u ≤ p , i ∈ S, i,t i,t i,t i,t Main points. Unit commitment and advance procurement p ∈ P. of reserves are not necessary without fuel-based generators. The objective is the sum of production costs for all generators Load-shifting is critical to accommodating non-dispatchable over the time period of interest. Load-shifting schedules can supply and reduce capacity requirements. Although load- span hours (T ≈ 10) to a few days (T ≈ 100). A subset of shifting schedules can span days, load-shifting actions can be nodes, S, have storage. The second constraint represents the implemented in near real-time. Consequently, there is little (if offset to the net supply and demand resulting from storage any) need for day-ahead scheduling in power systems without injections and extractions. The last constraint requires that all fuel. power quantities, p, be physically feasible; for example, P could represent the set of feasible power injections under lin- B. Dispatch earized power flow and line flow limits. Note that uncertainty does not factor into our discussion of load-shifting, and is Power system operators continually update device settings addressed later when we discuss dispatch with multiperiod such as transformer tap positions, the reactive power outputs optimal power flow in Section III-B. of FACTS devices, and, most importantly, the real power We distinguish between three functions of load-shifting. outputs of power producers. These dispatch decisions are made by solving an optimal power flow, which minimizes • Storage can flatten the net load seen by power producers. instantaneous operating costs subject to the resulting power This leads to more efficient power production because the flow satisfying various network constraints. Optimal power functions f (p ), i ∈ G in (5) are approximately convex i,t i,t flow can be solved at higher physical resolutions than unit and increasing [17]. Storage profits from this practice commitment because the continuous nonconvexities arising by arbitraging price differences over time. Without fuel- from power flow constraints admit accurate linear and con- based generators, system efficiency is far less dependent vex conic approximations [64], [65], [76]. As mentioned in on output levels, and thus the price of power may not Section II-C, DC is a viable basic paradigm of power systems change in time; we discuss this further in Section III-C. without fuel, which would lead to more power transfers and As such, we see little benefit from load-shifting for slightly more tractable optimal power flow computations [63]. efficiency in power systems without fuel. A generic multi-period optimal power flow routine can be • Load shifting can move power to accommodate production capacities. This makes the second constraint in written in the form " T # (5) easier to satisfy, thus enlarging the feasible range X X X min E fi,t(pi,t, σ) − di,t(pi,t, σ) of operation. Much of present-day load-shifting of this p,q,u,v σ (6) type accommodates the startup and shutdown constraints t=1 i∈G i∈L of fuel-based generators. For example, flattening load s.t. {p, q, u, v} ∈ P. makes nuclear power plants more economical by enabling Here, p is real power, q reactive power, v voltage, and u them to continuously operate near their capacity limit represents storage as in (5); T is the total number of time and avoid going offline [73]. In power systems with high periods, e.g., 48 hours would be appropriate if multi-day wind dependence on non-dispatchable resources like wind and events were an issue. The function fi,t(pi,t, σ) is the cost of solar, more load-shifting is needed to match supply and producing pi,t at time t if i is in the set of generators, G; and demand [74]. di,t(pi,t, σ) is the utility of consuming pi,t at time t if i is in the • Load-shifting can be used to reduce peak consumption set of loads, L. The expectation is over σ, a random variable levels, in turn reducing system capacity requirements. In that captures, e.g., uncertainty in demand or renewable output. (5), this means attaining feasibility without changing each The constraint represents all of the system physics, including pi,t, hence reducing component installation costs in (1). power flow and storage constraints. We remark that (6) is a This sort of load-shifting, also known as peak-shaving, is generalization of the load-shifting problem (5), and that load- useful with and without fuel-based generators. shifting schedules can be computed alongside other dispatch Much of the storage necessitated by renewables provides decisions within multi-period optimal power flow. Within the services other than load-shifting such as power balancing and planning problem (1), the optimization problem (6) would be 7 embedded in d(x, y, σ) and P(x), albeit in a simplified form has also seen wide application in power systems [84], for tractability. particularly in unit commitment routines [85]. The dominant portion of the generator cost curves, • Robust optimization is a more tractable, less descriptive fi,t(pi,t, σ), i ∈ G, are the fuel costs. These functions are approach to uncertainty modeling compared to stochas- approximately convex and increasing because generators tend tic programming, which specifies uncertain sets without to burn fuel less efficiently at higher power output levels [17]. probability distributions [86]. Robust optimization has In North America, the fi(pi) are usually based on supply also proven useful in power systems [87], again particu- functions associating prices with output-power levels, which larly with unit commitment [88].3 are declared by generators prior to dispatch. Currently, the We remark that from a computational perspective, replac- load utility curves, di(pi), i ∈ L, are often only present in ing fuel-based generators with renewables entails replacing unit commitment and not optimal power flow because loads intractable discrete constraints with uncertainty. While sig- have little real-time elasticity. This is changing with the advent nificant theoretical and empirical work is needed to quantify of demand response [77], [78]. this tradeoff, this hypothesis appears to be reasonable for the System operators will seek to optimally dispatch power following reasons: systems without fuel, for instance, by setting storage charging • Renewable uncertainty admits more modeling choices and discharging schedules, choosing the set-points of power than discrete unit commitment constraints, which essen- electronic devices, and adjusting the power captured from tially must always be modeled as integer variables. renewables, e.g., by adjusting wind turbine blade pitch and • Models of uncertainty are more amenable to tractable the incidence angle of PV arrays. To accommodate large- (convex) approximations. Most approaches to combina- scale production by renewables, (6) must be modified in the torial optimization retain discrete modeling. following four ways: 2) Dynamic constraints: Flexibility from storage and de- • Uncertainty from forecast errors due to renewable inter- mand response is necessary to accommodate renewable intermittency must be included. mittency. Because both resources have finite energy capacity, • Dynamic constraints from storage, e.g., (2)–(4), and serial dynamic modeling is necessary to describe the evolution of its correlations in renewable intermittency must be included. state of charge. This includes both load-shifting as discussed • The presence of numerous dispatchable resources like in Section III-A and power balancing to match supply and inverters and storage in low-voltage distribution systems demand on sub-minute timescales. Because the basic storage may necessitate decentralization. constraints, (2)–(4), are linear and convex quadratic, optimal • New objectives are needed to replace fuel costs. storage decisions can be obtained alongside standard dispatch We discuss these four issues in the order listed. decisions via multiperiod optimal power flow [72]. A multi- 1) Uncertainty: Renewable intermittency adds significant period optimal power flow routine like (6) is a sequence of uncertainty to dispatch decisions in the form of forecast errors. single-period optimal power flow routines that are coupled Of course, lower levels of uncertainty from loads and contin- period-to-period by storage dynamics like (2)–(4) and other gencies (e.g., line outages) have always been present in power constraints such as ramp limits on power production. Multi- system operations, cf. [79]. As mentioned in Section I-B, period optimal power flow has approximately NT variables wind and solar power plants can be operated such that they and MT constraints, where N and M are the number of are dispatchable up to their maximum power outputs, which variables and constraints in each individual period, and T are determined by random factors like wind speed and solar is the number of periods. Despite the increased number of intensity. variables and constraints, fundamentally, multiperiod optimal The following are a few well-known, contemporary ap- power flow is no more complex than ordinary optimal power proaches to incorporating uncertainty into optimal power flow flow, e.g., a multiperiod version of a second-order cone optimal routines. Each approach would be implemented in the context power flow is still a second-order cone program [76]. A of (6) by suitably modifying σ and its distribution and in the useful implication of this is that stochastic and robust optimal feasible set P. power flow models can be extended to the multiperiod case to accommodate serial correlations in wind speeds and incident • The chance constraint is a tool from stochastic program- solar irradiance. ming [80], [81] that requires the probability of an event A recent alternative to multiperiod optimal power flow is to be above or below a certain level, for example, that risk-limiting dispatch [90], [91], which is based on dynamic the probability a collection of renewables satisfies the net programming rather than convex optimization. Risk-limiting load is greater than 0.99. While generally nonconvex, dispatch offers less modeling versatility than multiperiod op- some chance constraints can be cast as linear [80] and timal power flow, but in return produces policies instead of second-order cone constraints [82], the latter of which trajectories, which better accommodate uncertainty by speci- have been used as a tractable basis for chance-constrained fying a decision for any realization of the system state [92]. optimal power flow [83]. 3) Decentralization: We now briefly discuss decentraliza- • The scenario approach is an alternative approximation to tion. For many years, only generators and other transmission stochastic programming in which scenarios are sampled from probability distributions and used to estimate the 3Recently, some theoretical connections between chance constraints, the feasibility of chance constraints. The scenario approach scenario approach, and robust optimization have been established in [89]. 8 level devices could actively change their set points in response timescales [108]. Power quality affects the performance to commands. Now, components in low-voltage distribution and lifetime of grid-connected devices. We also classify systems like PV inverters and small energy storage devices power quality as non-monetary because the costs of must actively change their set points to accommodate un- voltage fluctuations are difficult to quantify. certainty from distributed renewable generation; note that • Presently, most real-time dispatch routines take the loads this largely overlaps with the paradigm of distributed gen- across the system as a fixed set of parameters. Demand eration [6], [93]. Computational and communication-based response enables loads to modify their aggregate con- limitations will likely prohibit system operators from centrally sumption in real-time in response to the system’s state operating these devices. In this case, decentralized optimal and needs. An aggregation of active loads could submit power flow algorithms will be needed to dispatch devices a bid curve to the system operator associating its range with limited communications. Numerous approaches already of power consumptions with varying utility levels. This exist, including consensus-based approaches [94] and the could be a monetary bid, cf. [109]. alternating direction method of multipliers [95], [96]. While The first three objectives overlap significantly. For example, decentralization must always entail a performance loss relative improved reliability should improve power quality and vice to the centralized case and has long been a mathematically versa while reducing maintenance costs. These objectives are challenging topic, the online computational requirements of also closely tied to transient behaviors; for example, increasing practical decentralized algorithms are generally lower than the real power output from a solar power plant will cause more their centralized counterparts simply because they involve disturbances in that part of the network, in turn impacting breaking a large problem into smaller pieces. We revisit the transient stability of the system. We further discuss this decentralization from the perspective of real-time control on coupling in Section IV. As a general rule, we suggest that the faster timescales in Section IV. objective not introduce complexities beyond the constraints 4) Dispatch criteria: Lastly in this section, we address needed to accurately represent the physics, e.g., an objective the fact that renewables operate at zero marginal cost, i.e., that must be evaluated by Monte Carlo integration would once they are built, the power they produce costs very lit- compromise the tractability of the overall dispatch routine. tle [97]. We discuss the economic implications of this feature in Section III-C. After removing fuel-based generators from Main points. Dispatching power systems without fuel will power systems, only the load utilities remain in the objective differ in four ways from current practices. of multiperiod optimal power flow, (6), which are often not • Forecast errors from renewable intermittency must be present in real-time dispatch routines. Because current power accounted for with uncertainty modeling. system operations are almost entirely centered around min- • Multiperiod optimal power flow must be used to model the imizing fuel consumption, alternative objectives are needed dynamics of storage and serial correlations in renewable for power systems without fuel; a few possibilities are listed inputs. below. Note that we do not consider resistive losses to be a • In some cases, distributed resources will need to be valid objective due to the low marginal cost of the power itself. operated in a decentralized fashion. • Maintenance is the next most significant monetary cost • New objectives are required to replace fuel costs in that explicitly depends on how the system is dispatched. dispatch routines. Maintenance and health monitoring in power systems are well-studied topics, e.g., for pumped hydro storage [98], C. Economics wind turbines [99], [100], PV and storage systems [101]– [103], and transmission lines [104], [105]. There is also In this section, we discuss the economics of supplying a well-developed literature on maintenance theory [106]. power with only renewables. A large and growing body of • Reliability-based objectives minimize the risk of compo- literature addresses the assimilation of renewables into the nent failures and instability [79]. We classify reliability current market framework, see e.g. [97]. The reader is referred as a non-monetary objective because it is difficult to to [110], [111] for thorough expositions of power system ascribe explicit costs to potential failures. A reliability economics. While these techniques are effective when some objective might include nuanced considerations such as of the total power is produced by fuel-based generators, we probabilistic failures, or could be as simple as minimizing believe that there is room for new perspectives when all of the the maximum current-to-capacity ratios across the net- power is from renewables. work [107]. The latter can be written as Presently, most producers in North America have several 2 revenue streams. X xi wi , (7) • A bilateral contract is low risk, long-term mechanism for X i i a producer to sell a fixed quantity of power to a consumer where xi and Xi are the usage and capacity of component at a price agreed upon ahead of time. i, e.g., the apparent power flow through and thermal • Producers can sell power in spot markets at nodal prices capacity limit of a transmission line or power electronic that are the dual variables of optimal power flow with a converter, and wi a positive weighting factor. fuel cost objective. This approach was pioneered in [112] • Power quality refers to keeping the system voltage and is a conceptual application of the Fundamental The- close to its nominal profile on steady-state and transient orems of Welfare Economics [43]. An attractive quality of 9

nodal prices is that they are outputted by optimal power the power system, which currently varies on a minute- flow routines and represent the sensitivity of the objective to-minute basis. This results in generators being paid to incremental changes in power levels. Spot markets according to their efficiency, and, under dynamic pric- provide competitive venues for buying and selling power, ing [122], incentivizes efficient power consumption over which fill the essential roles of adding liquidity and time. Although demand and system capacity will vary in driving down prices of bilateral contracts. time, it is not clear that the marginal cost of operating • Producers are paid for providing flexibility under the power systems without fuel varies on a minute-to-minute names reserves and regulation. Reserve providers are or even hourly basis. often paid through call options in which they receive • It is straightforward to map generation levels to the cost capacity payments for holding idle capacity on standby of fuel consumption, making it relatively easy to audit and energy payments if they are actually called on [113]– bidding by fuel-based generators. The costs of mainte- [115]. Resources are also paid for providing regulation; nance, reliability, and power quality are harder to quantify the form of these payments has been a topic of recent and hence harder to audit. Bidding mechanisms based debate [116], [117]. on these costs may consequently be more susceptible to • System operators or load serving entities sometimes market power and gaming. make long-term capacity payments to power producers • Demand-side bidding alone cannot account for the cost for being present to bid in markets and provide power of supplying power, but rather only what consumers are over periods of years. These payments encourage con- willing to pay, hence removing producer input from the struction of new generation. As discussed at the end of price-setting process. Such an arrangement can lead to Section II-A, capacity payments could play a large role excessively small payments to producers. For example, if in power systems without fuel. Note, however, that their we model such demand-side bidding with supply function implementation would likely require a redefinition of the competition [123], [124], it is trivially a pure strategy notion of capacity to accommodate the random maximum Nash equilibrium for all bids to be zero, thus rendering outputs of renewable power sources. zero payments. Bilateral contracts and spot markets are core features of Due to the above issues, it is our stance that replacing fuel- today’s power systems and should be present in some form if based generators with renewables warrants a rethinking of power systems without fuel are to be competitive. Flexibility is power markets. costly and essential to the reliability of power systems without Main points. The low marginal and high fixed costs of fuel and hence should also be represented in markets. renewable power production makes the current nodal pricing Here, we focus on spot markets because they are at the framework dubious for power systems without fuel. The de- foundations of current power markets but are questionable in velopment of new market mechanisms will rely on accurately their current form for power systems without fuel. The core quantifying the cost of operating power systems without fuel on challenge lies in the fact that power systems without fuel all timescales. Since these costs may be based on a number of have high fixed costs and low marginal costs, much like the complex and hard to audit factors, market power and gaming 4 internet, software, and digital media [118], [119]. We discuss are important considerations. the economics of regulation further in Section IV-D2. Because there are no fuel costs, employing nodal pricing IV. TRANSIENT BEHAVIOR as is would simply result in all prices being equal to zero. Several challenges will need to be addressed from modeling, The fuel cost objective in (6) could hypothetically be replaced analysis, and control perspectives as we embrace the transition with one or a combination of the objectives discussed in from electromechanical energy-conversion interfaces (syn- Section III-B such as maintenance costs, and nodal prices chronous machines) delivering power from thermal sources could be defined as usual. In this case, any new objective in (6) (coal, nuclear, gas) to semiconductor-based energy conversion must be monetary for the dual multipliers to have valid price interfaces (power-electronic converters) delivering power from interpretations; if the objective is not monetary, then the dual renewable resources (wind and solar). The main challenges in multipliers are simply the sensitivities of the optimal objective this context are the following. and not marginal cost-based prices. We must, however, ask • The system dynamics in the envisioned power systems whether this is appropriate for power systems without fuel. without fuel are not dominated by mechanical generator The following are three fundamental issues. rotor inertias. Apart from electrochemical and electrome- • Time-varying nodal prices reflect the instantaneous phys- chanical energy storage devices (batteries and flywheels, ical efficiency and hence marginal cost of operating respectively) power-electronic energy conversion interfaces offer limited inertia innately. 4 The issues stemming from the low marginal cost of renewable energy • Steady-state dispatch decisions now have greater effect on also resemble the nuclear power discourse of the 1950s when it was said electricity would be “too cheap to meter,” and consumers would only need transient stability. For instance, procuring a large amount to pay for fixed costs [120]. Of course, this never came to be, and power of power from a PV power plant also increases the systems without fuel will likewise require rationing mechanisms to moderate magnitude of the disturbances it contributes. demand and control capacity expansion requirements. A basic question is how to increase fixed connection charges so that utilities can recover sales losses • Dynamics of fast-switching power converters present im- due to residential PV and avoid so-called “death spirals” [121]. posing analysis challenges. Quasi-stationary phasor rep- 10

resentations of electrical-network voltages and currents reactive-power-balance conditions for nodes in the electrical that are commonly used in bulk power systems lack the network. Given the lack of numerical-simulation, stability- required modeling fidelity in power systems without fuel. analysis, and controller-design methods for DAE models, it is While one could argue that with the reduced mechanical common to resort to reduced models of the electrical network inertia, a no-fuel power system is likely to be more sensi- when analyzing a collection of synchronous generators. These tive to disturbances, conceivably, there is also more control reduced models are typically composed of nonlinear ODEs flexibility since power-electronic interfaces can act on much and are recovered by eliminating the algebraic power-flow faster timescales compared to synchronous generators. Next, equations. A variety of electrical-network model reduction we offer perspectives on how modeling, analysis, and control methods have been developed for bulk power systems [132]. frameworks will need to be revised in a no-fuel future. To illustrate the limitations of existing approaches and outline future directions, we focus the subsequent discussion (without A. Modeling loss of generality) on Kron reduction. Given an electrical network with a set of pre-specified Problems related to stability in power systems dominated terminals, Kron reduction recovers an electrically equivalent by synchronous generators have received significant attention circuit retaining terminal nodes and eliminating non-terminal over the years. A classification of different stability notions nodes [133], [134]. Kron reduction is accomplished by taking and links to reliability and security is available in the re- the Schur complement of the admittance matrix. In general, port [125]. From the generation perspective, stability issues are encoding network interactions through the admittance matrix typically studied for a networked collection of synchronous is only possible when we rely on quasi-stationary sinusoidal- generators, with nodal dynamics modeled by the ubiquitous steady-state representations for the electrical network vari- swing equations. Adopting a standard one-axis model for ables [135]–[139]. However, such quasistationary models ad- synchronous generators in the power system [126], the swing mittedly lack the fidelity required to capture network transient equations are given by behaviors at faster than AC-cycle time scales. Therefore, dδ with regard to analysis of large networks of interconnected = ω − ω , dt synch inverters, there is a pressing need to develop scalable model- dω 1 reduction methods that can be applied across a wide spectrum = (P − P − D(ω − ω )) , (8) dt M mech elec synch of time scales. Noteworthy efforts in this direction are time- where δ and ω are the rotor angle and speed, ωsynch is the domain Kron reduction methods [140], [141]. synchronous speed, M and D are the inertia constant and damping coefficients, Pmech is the mechanical-input power, C. Feedback Control and Pelec is the electrical power that flows into the network. Dynamic models for power-electronic inverters are (circuit) Frequency across the bulk power system is, to the first topology and application dependent, and difficult to abstract order, backed up by the mechanical inertia on offer from in a convenient analytical form akin to the ubiquitous swing the synchronous generators. In particular, notice from (8) that equations. This is particularly due to the following reasons: variations of frequency to load fluctuations, i.e., Pelec are inversely related to the generator mechanical rotor inertia. As • digital control methods that are typically employed in power-electronics circuits involve state-machine-type or such, power imbalances in synchronous generators directly look-up-table-based algorithms that are difficult to encode translate to variations in electrical frequency. This fundamental as dynamical systems; relationship underlies turbine-governor control (at the generator level) and load frequency control (at the system level). • the utilization of switching power semiconductor devices in power electronic systems implies that switching-time- In a future power system dominated by power-electronics scale behavior is only well modeled by hybrid automata interfaces, mechanical inertias will no longer be direct proxies for which scalable simulation platforms are lacking; for nodal frequency dynamics across the electrical network. The limited inertia on offer in power-electronics inverters • characteristics of renewable power sources and energy storage devices are nonlinear, technology specific, and is that provided by inductive and capacitive filter elements; use proprietary control algorithms that are not readily and therefore without fuel-based generators, electrical physics accessible in the literature. can be expected to dominate at even faster timescales. The role of feedback control in this setting is magnified, and The challenges mentioned above have been tackled with next-generation distributed-control algorithms will play an modeling strategies that include: virtual prototyping [127], important role of maintaining stable voltages and frequencies reachability analysis [128], and sampled data modeling and in a network of interconnected power electronic inverters. control for power-electronics circuits [129]–[131]. In the present generation of power-electronics inverters intended for grid-tied operation, controllers are designed with B. Analysis the implicit assumption of the existence of a stiff voltage In their original form, dynamical models that capture the source. The main role of inverters in this setting is to reg- electromechanical behaviors of synchronous generators in a ulate the delivery of real and reactive power. This is the power network are described by differential algebraic equa- so-called grid-following control paradigm. Recognizing the tions. Algebraic equations are introduced from real- and inevitable transition to a network with a high penetration of 11 power-electronics inverters, there has been resurgent interest 1) Feedback control and energy management: The goal in control methods that seek to regulate terminal voltages of feedback control is to maintain stable AC waveforms and frequencies in the absence of a stiff grid voltage. These (uniform voltages and frequency) across the network in spite of approaches have been referred to as grid-forming controls. variations in sources, loads, and exogenous disturbances. The Efforts in this direction are largely based on the so-called objective of energy management is to provide set-points to the droop control [142]. Inspired by the operation of synchronous real-time inverter controllers such that in steady-state, power generators in bulk power systems, the premise of droop control losses and voltage deviations are minimized and economic is to linearly trade off terminal voltage and frequency versus benefits to end users are maximized. active and reactive power. To leverage the increased control Several approaches can therefore be adopted to steer the capabilities offered in fast-acting power-electronics inverters, operating point of a power system without fuel to an energy recently there has been some interest in the development of management routine’s solution. One solution strategy that has time-domain control methods that directly act on time-domain received significant attention in the literature is to synthesize waveforms and do not assume sinusoidal steady-state oper- controllers that seek saddle points of Lagrangian functions ation as is done in droop control. System-theoretic methods corresponding to the network-optimization problems, and then have outlined conditions under which frequency-synchronized integrating these optimization-based controllers with existing electrical variables can be maintained across the network real-time controllers. The general strategy of synthesizing dy- with droop control [142] and time-domain approaches [143], namical systems to solve convex optimization problems goes [144]. Extending these analytical methods to accommodate back to the seminal work of [146], and in recent years, it has a variety of energy-conversion interfaces as well as fuel- received increased attention in the control systems community based synchronous generators is, albeit challenging, critical in light of several motivating applications [147]–[154]. While to understand the implications of high penetrations of power- many synthesis approaches have been put forth, one that is electronics inverters. conceptually straightforward is to formulate dynamical systems (the states of which are proxies for Lagrange multipliers) D. Coupling between the transient and steady-state timescales which evolve in a gradient-ascent-like fashion towards the The spatiotemporal characteristics of power systems without optimal dual solution [151]. Reference inputs for the physical- fuel are fundamentally different from present bulk power layer system are then derived from these optimization-based systems. In particular, the generation and load are closely col- dynamical states. Since these controllers are formulated to located, largely intermittent, and vary at faster timescales. This dynamically seek the Karush-Kuhn-Tucker (KKT) conditions aspect motivates the development of new approaches to system for optimality, they have been referred to as dynamic KKT management: hierarchical operational practices that enforce a controllers [150], [151]. strict time-scale separation between energy management and 2) Economics: Steady-state dispatch influences the location real-time control are incompatible with the form and function of disturbances in power systems without fuel, which in turn of no-fuel power systems. influences which resources will be tapped for regulation. The Traditionally, network-wide energy management, e.g., op- fast adjustments involved in providing regulation, e.g., rapidly timal power flow and load-shifting, has been designed in- cycling the state of charge of a battery, factor significantly into dependently from feedback control mechanisms which track maintenance costs. As discussed in Sections III-B4 and III-C, set-points from energy management systems, e.g., automatic these maintenance costs may play a significant role in the generation control and droop control. This approach is based steady-state dispatch and economics of power systems without on the separation between steady-state (seconds to minutes) fuel. Hence, not only must energy management and feedback and transient (sub-cycle to seconds) timescales. In power sys- control mechanisms be jointly designed, but economic mech- tems without fuel, adopting this time-scale separation to guide anisms must also take into account costs incurred on transient optimization and control is questionable for the following time scales. reasons: We now comment on who should bear the cost of regulation. • Timescale separation is merely a simplifying approxima- Historically, most disturbances have been the result of natural tion; designing energy management and feedback mech- load variations and component failures. Load-following han- anisms with regard to both timescales will result in better dles the first type of disturbance and should be paid for by the performance [145]. Moreover, increasing the frequency at loads. Failures, however, are the result of machine breakdowns which energy management is executed will also improve and exogenous factors like trees and lightning. Failures are performance. Improved computational capabilities make only problematic on fast timescales because loads require such unified designs viable. uninterrupted power. Consequently, it is the responsibility of • The dispatch of variable wind and solar power sources both the component owner and loads to pay the cost of strongly influences the location and magnitude of dis- failures, cf. [155], [156]. Such costs are often termed negative turbances, in turn influencing the system’s transient be- externalities because they are not clearly attributable to the haviors and regulation needs. Hence, decisions at the actions of a specific market participant. 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