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...nt (d∗, d̂∗) ∈ D′. Let (d′, d̂′) ∈ D be a feasible point of OLC, then dj =( dj + dj ) /2, d̂j = d̂′j − dj + d′j specify a feasible point (d, d̂) ∈ relint D, where relint denotes the relative interior =-=[31]-=-. Moreover the only constraint of OLC is affine. Hence there is zero duality gap between OLC and its dual, and a dual optimal ν∗ is attained since g∗ > −∞ [31, Sec. 5.2.3]. By Appendix B-1), ∑ j∈N Φ ′...

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...appliances, such as refrigerators, water or space heaters, ventilation systems, and air conditioners, as well as plug-in electric vehicles to help manage energy imbalance. For further references, see =-=[12]-=-. Simulations in all these studies have consistently shown significant improvement in performance and reduction in the need for spinning reserves. The benefit of this approach can thus be substantial ...

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...feasibility of fast timescale load control as a supplement to the turbine-governor mechanism to resynchronize frequency and rebalance power. We use a much more realistic simulation model developed in =-=[24]-=-, [25] to validate our simple analytic model. The detailed simulations can be found in [22, Sec. VII]. We summarize the key conclusions from those simulations as follows. 1) In a power network with lo...

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...j ] . The choice of cost functions is based on physical characteristics of loads and user comfort levels. Examples functions can be found for air conditioners in [29] and plug-in electric vehicles in =-=[30]-=-. See, e.g., [5], [27], [28] for other cost functions that satisfy Condition 1. B. Main results The objective function of the dual problem of OLC is∑ j∈N Φj(ν) := ∑ j∈N min dj≤dj≤dj ,d̂j ( cj(dj)− νdj...

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...ed to mature in the last decade or so. The idea however dates back to the late 1970s. Schweppe et al. advocate its deployment to “assist or even replace turbine-governed systems and spinning reserve” =-=[6]-=-. They also propose to use spot prices to incentivize the users to adapt their consumption to the true cost of generation at the time of consumption. Remarkably it was emphasized back then that such f...

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...st functions is based on physical characteristics of loads and user comfort levels. Examples functions can be found for air conditioners in [29] and plug-in electric vehicles in [30]. See, e.g., [5], =-=[27]-=-, [28] for other cost functions that satisfy Condition 1. B. Main results The objective function of the dual problem of OLC is∑ j∈N Φj(ν) := ∑ j∈N min dj≤dj≤dj ,d̂j ( cj(dj)− νdj + 1 2Dj d̂2j − νd̂j +...

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...h frequency adaptive loads will “allow the system to accept more readily a stochastically fluctuating energy source, such as wind or solar generation” [6]. This point is echoed recently in, e.g., [7]–=-=[13]-=-, that argue for “grid-friendly” appliances, such as refrigerators, water or space heaters, ventilation systems, and air conditioners, as well as plug-in electric vehicles to help manage energy imbala...

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...raic equation that represents power balance at bus j:2 0 = Dj∆ωj + ∆dj −∆Pmj + ∆P outj −∆P inj (2) 2There may be load buses with large inertia that can be modeled by swing dynamics (1) as proposed in =-=[23]-=-. We will treat them as generator buses mathematically. 4sgeneratorsj jD jd jkP j ijP for all s.t.i i j for all s.t.k j k buss 0 m j j jP M mjP Fig. 1. Schematic of a g...

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...continuously differentiable on[ dj , dj ] . The choice of cost functions is based on physical characteristics of loads and user comfort levels. Examples functions can be found for air conditioners in =-=[29]-=- and plug-in electric vehicles in [30]. See, e.g., [5], [27], [28] for other cost functions that satisfy Condition 1. B. Main results The objective function of the dual problem of OLC is∑ j∈N Φj(ν) :=...

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...k 4. The standard technique of using a Lyapunov function that is quadratic in both the primal and the dual variables was first proposed by Arrow et al. [34], and has been revisited recently, e.g., in =-=[35]-=-, [36]. We apply a variation of this technique to our problem with the following features. First, because of the algebraic equation (24) in the system, our Lyapunov function is not a function of all t...

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...utomatically reducing their consumption (e.g, the heating element of a clothes dryer was turned off while the tumble continued) when the frequency of the household dropped below a threshold (59.95Hz) =-=[14]-=-. Field trials are also carried out in other countries around the 2globe, e.g., the U.K. Market Transformation Program [15]. Even though loads do not yet provide second-by-second or minute-by-minute c...

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... dispatch) in the U.K. market. Long Island Power Authority (LIPA) developed LIPA Edge that provides 24.9 MW of demand reduction and 75 MW of spinning reserve by 23,400 loads for peak power management =-=[17]-=-. While there are many simulation studies and field trials of frequency adaptive load control as discussed above, there is not much analytic study that relates the behavior of the loads and the equili...

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...he standard technique of using a Lyapunov function that is quadratic in both the primal and the dual variables was first proposed by Arrow et al. [34], and has been revisited recently, e.g., in [35], =-=[36]-=-. We apply a variation of this technique to our problem with the following features. First, because of the algebraic equation (24) in the system, our Lyapunov function is not a function of all the pri...

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...y adaptive loads can rebalance power and resynchronize frequency, just as the droop control of the generators currently does. They fit well with the emerging layered control architecture advocated in =-=[18]-=-. C. Our prior work and structure of paper In our previous papers [19]–[21] we consider a power network that is tightly coupled electrically and can be modeled as a single generator connected to a gro...

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...he (|N | − 1) × |E| reduced incidence matrix obtained from C by removing any one of its rows, and h̃∗ is obtained from h∗ by removing the corresponding row. Note that C̃ has a full row rank of |N |−1 =-=[33]-=-. If (N , E) is a tree, then |E| = |N |− 1, so C̃ is square and invertible and Z∗P is a singleton. If (N , E) is a (connected) mesh, then |E| > |N |−1, so C̃ has a nontrivial null space and there are ...

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... such frequency adaptive loads will “allow the system to accept more readily a stochastically fluctuating energy source, such as wind or solar generation” [6]. This point is echoed recently in, e.g., =-=[7]-=-–[13], that argue for “grid-friendly” appliances, such as refrigerators, water or space heaters, ventilation systems, and air conditioners, as well as plug-in electric vehicles to help manage energy i...

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...., the U.K. Market Transformation Program [15]. Even though loads do not yet provide second-by-second or minute-by-minute continuous regulation service in any major electricity markets, the survey in =-=[16]-=- finds that they already provide 50% of the 2,400 MW contingency reserve in ERCOT (Electric Reliability Council of Texas) and 30% of dispatched reserve energy (in between continuous reserve and econom...

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...r flows to their scheduled values. Economic dispatch operates at a timescale of several minutes or up and schedules the output levels of generators that are online and the inter-area power flows. See =-=[5]-=- for a recent hierarchical model of these three mechanisms and its stability analysis. This paper focuses on load participation in the primary frequency control. The needs and technologies for ubiquit...

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...ission’s Small Grant Program through Grant 57360A/11-16. A preliminary version of this work has appeared in the Proceedings of the 3rd IEEE International Conference on Smart Grid Communications, 2012 =-=[1]-=-. C. Zhao and S. Low are with the Department of Electrical Engineering, California Institute of Technology, Pasadena, CA, 91125 USA (e-mail: czhao@caltech.edu; slow@caltech.edu). U. Topcu is with the ...

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...ment in performance and reduction in the need for spinning reserves. The benefit of this approach can thus be substantial as the total capacity of grid-friendly appliances in the U.S. is estimated in =-=[8]-=- to be about 18% of the peak demand, comparable to the required operating reserve, currently at 13% of the peak demand. The feasibility of this approach is confirmed by experiments reported in [10] th...

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... is almost the same as the frequency at the generator bus even during transient. 2) Different buses, particularly those that are in different coherent groups [24] and far apart in electrical distance =-=[26]-=-, may have different local frequencies for a duration similar to the time for them to converge to a new equilibrium, as opposed to resynchronizing almost instantaneously to a common system frequency w...

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... as the droop control of the generators currently does. They fit well with the emerging layered control architecture advocated in [18]. C. Our prior work and structure of paper In our previous papers =-=[19]-=-–[21] we consider a power network that is tightly coupled electrically and can be modeled as a single generator connected to a group of loads. A disturbance in generation causes the (single) frequency...

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...he droop control of the generators currently does. They fit well with the emerging layered control architecture advocated in [18]. C. Our prior work and structure of paper In our previous papers [19]–=-=[21]-=- we consider a power network that is tightly coupled electrically and can be modeled as a single generator connected to a group of loads. A disturbance in generation causes the (single) frequency to d...

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... in [8] to be about 18% of the peak demand, comparable to the required operating reserve, currently at 13% of the peak demand. The feasibility of this approach is confirmed by experiments reported in =-=[10]-=- that measured the correlation between the frequency at a 230kV transmission substation and the frequencies at the 120V wall outlets at various places in a city in Montana. They show that local freque...

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...ued) when the frequency of the household dropped below a threshold (59.95Hz) [14]. Field trials are also carried out in other countries around the 2globe, e.g., the U.K. Market Transformation Program =-=[15]-=-. Even though loads do not yet provide second-by-second or minute-by-minute continuous regulation service in any major electricity markets, the survey in [16] finds that they already provide 50% of th...

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...tabilizer model, AC (nonlinear) power flows, and non-zero line resistances. The detail of the simulation model including parameter values can be found in the data files of the toolbox. It is shown in =-=[22]-=- that our analytic model is a good approximation of the simulation model. In the test system there are 35 load buses serving different types of loads, including constant active current loads, constant...

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...ctions is based on physical characteristics of loads and user comfort levels. Examples functions can be found for air conditioners in [29] and plug-in electric vehicles in [30]. See, e.g., [5], [27], =-=[28]-=- for other cost functions that satisfy Condition 1. B. Main results The objective function of the dual problem of OLC is∑ j∈N Φj(ν) := ∑ j∈N min dj≤dj≤dj ,d̂j ( cj(dj)− νdj + 1 2Dj d̂2j − νd̂j + νPmj ...

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...s cj(dj) = d2j/(2α), with identical α = 100 pu for all the loads. The loads are controlled every 250 ms, which is a relatively conservative estimate of the rate of load control in an existing testbed =-=[37]-=-. We look at the impact of OLC on both the steady state and the transient response of the system, in terms of both frequency and voltage. We present the results with a widely used generation-side stab...