Load Aggregation Effect in Power Grid L. Shalalfeh 1 and E. Jonckheere 1 Abstract—In this paper, the issue of data-driven modeling of a single load in a laboratory set-up is confronted with the same data-driven modeling of the same load, but in a real grid environment. As it is argued here, an aggregation effect of all of the loads in a grid endows a single load with grid- characteristics properties in addition to the usual load-specific properties. Topologically, the hidden feedback structure of a bus model reveals that the resulting digraph is strongly connected, meaning that all loads are intertwined in a single system that cannot be decomposed into islands. I. I NTRODUCTION Load modeling is undoubtedly an important—still active— area of research in the power grid. Indeed, depending on their active and reactive power profiles, loads could be the culprit in voltage collapse or other nuisances in the grid [8], [24]. One issue that has never been made completely clear is what is the difference between, on the one hand, a model of a single isolated load and, on the other hand, a model of a load in a complex grid environment. Berg [4], [5] is adamant that his load models represent the loads in the Scandinavian microgrid environment in which the experiments were done. Hill [14], [20] on the other hand does “curve fitting” of the power disruption resulting from a voltage drop in a single load laboratory environment. A puzzling difference between the two models is that the Berg model involves the frequency in a fractional exponent in a narrow bandwidth around 50 cycles/sec, whereas a lingering issue in the Hill model is its lack of frequency dependence [14, p. 175], [21, p. 24]. A fractional transfer function model of a high voltage transformer was already derived in [15]. The big difference is that the fractional model of the transformer is mandated by matching the frequency response over a very large frequency sweep, exciting parasitic distributed parameter electromag- netic effects present in the wiring of the transformer. Here, we deal with the frequency response over a very narrow frequency band around 50 cycles/sec. Under those circum- stances, it is a bit difficult to see how parasitic modes could be excited, unless another hitherto unknown effect is present. The purpose of this paper is to offer evidence that his other effect is the aggregation of the loads. By “load aggregation,” we mean that, after unraveling the hidden feedbacks in bus model, the resulting interconnected system is made up of one and only one strongly connected component [6], [7]. The latter means that under normal operations loads are not “islanded.” Quite to the contrary, every single load penetrates the whole grid. As an example [12], removing a load at a (1) Dept. of Electrical Engineering, Univ. of Southern California, Los Angeles, CA 90089-2563 {shalalfe,jonckhee}@usc.edu grid point in the Bay Area was observed by a micro-PMU 550 kilometers away in the Los Angeles area. An outline of the paper follows: Sec. II deals with Berg load modeling. Sec. III introduces the basic feedback motif of a single generator, single line, single load bus. This basic feedback motif is repeated to reveal the hidden feedback structure of a more complicated grid in Sec. IV. Sec. V de- scribes the load aggregation effect as the strongly connected property of the hidden feedback structure. Sec. VI describes various contingency scenarios. II. BERG LOAD MODELING AND FRACTIONAL DYNAMICS The Berg model [1], [4], [5], P (V L ,ω)= K p V pv L ω pω , Q(V L ,ω)= K q V qv L ω qω , (1) is usually thought of as a static model, but its frequency dependence gives it some dynamical properties formalized in the describing function technique [2]. The Berg model was derived experimentally from data collected on a Scan- dinavian “micro-grid,” when a generator was deliberately removed from the grid, resulting in transients and eventually a new steady state of the power profile (P,Q) absorbed by the load. Contrary to the Hill model, the Berg model does not attempt to model the transients; it only models the shift in the steady state, but in a manner that goes beyond the Hill model as it involves the frequency—in a rather unorthodox way, with fractional exponents of the frequency. It is straightforward to go from the complex power to the impedance model of the load: Z L = V L V L * I L V L * = V 2 L S * L = V 2 L P (V L ,ω) − jQ(V L ,ω) (2) = 1 K p V pv-2 L ω pω − jK q V qv-2 L ω qω , where boldfaced quantities denote phasors and where S L = V L I L * = P (V L ,ω)+jQ(V L ,ω) is the complex power. Using the experimental p v , p ω , q v , and q ω data derived in [4] for different loads, the impedances are easily obtained as in [24, Table II], in describing function format [2] since they depend on the voltage amplitude. For notational convenience, we switch to the admittance formulation Y L =1/Z L = Lω p − jWω q . Next, we approx- imate Y L (V L ,ω) with a formal circuit-theoretic admittance, except for its amplitude dependency, Y L (V L ,ω) ≈ A(V L ) × (jω) α − B(V L ) × (jω) β , (3) where A, B are real-valued. The construction of the approx- imation is done as follows: First, write j α = a + jb,j β =