Assuming the availability of an infrastructure for pervasive measurement, communications, and control, a number of studies explore the possibility of using different types of elastic loads (such as EVs) to smooth out fluctuations of renewable generation and track the difference between demand and supply in real-time [21,89,37,40]. These studies employ
a direct load control mechanism to achieve a desired response to power system dynamics from a large population of elastic loads. Hence, the focus of these studies is on providing power system control services as opposed to achieving the objectives of the electric utility and enhancing the end-use performance, the direction pursued in this thesis. In fact, we are not aware of any related work that has exploited the flexibility of elastic loads to stabilize voltage, relieve congestion, prevent bidirectional flows, and minimize curtailment in a distribution system with a high concentration of solar PV generation.
Furthermore, control will not be limited to elastic loads in the smart grid. Smart inverters can also be controlled by the utility to address growing concerns over widespread adoption of PV systems and simultaneously enhance the EV charging service. Since elastic loads and PV systems have opposite impacts on distribution circuits, it is reasonable to extend the optimal control framework to jointly control elastic loads and PV inverters.
This integrated control system enables the grid to safely accommodate higher penetrations of distributed solar generation and electric vehicles, while enhancing the overall reliability and cost-effectiveness of the power system. We study this problem in Chapter5. To the best of our knowledge, no related work exploits the synergy between solar PV generation and EV charging loads in the distribution network to simultaneously achieve the above-mentioned objectives.
2.8 Chapter Summary
Large-scale integration of elastic loads and solar PV systems can negatively impact reliable and economical generation, transmission, and distribution of power if these end-nodes are not controlled properly. This has given impetus to the design of mechanisms to control these active end-nodes. The extensive body of literature that has been developed around the control of elastic loads can be divided into several categories based on the following criteria:
• Time of control: The control algorithm can run in near real-time or several hours in advance of power delivery.
• Information needs: The control algorithm may require the precise model of the distribution network along with load and generation forecasts, or rely on recent measurements of certain network parameters only.
• Control scheme: Control decisions can be computed in a centralized or decentral-ized manner.
• Optimization time horizon: Control objectives can be myopic or defined over a time horizon.
We believe that a control scheme that fully meets the design goals specified in Chapter1 must be decentralized and based on real-time measurements. The infrastructure presented in this chapter supports the implementation of this control scheme. This scheme should enhance reliability and cost-effectiveness of the power system, satisfy user-level and device-level objectives, and mitigate adverse impacts of large-scale adoption of solar PV systems and EVs, including large voltage fluctuations, network congestion, reverse flow, and violation of voltage limits. None of the control schemes surveyed in this chapter can meet our design goals and satisfy these objectives simultaneously. This calls for the design of scalable control schemes, similar to the TCP congestion control scheme originally developed for the Internet, to balance system-level and user-level objectives.
Chapter 3
System Model
Several studies suggest that disruptive load and generation technologies will primarily affect distribution networks [82,100], which are not typically monitored in real-time for cost reasons. Exploiting the availability of pervasive measurement, broadband communi-cation, and decentralized decision making in the smart grid, different control mechanisms can be designed to effectively mitigate the adverse impacts of these technologies on distribution systems.
In this chapter we describe a simplified model of a radial distribution system and time-slotted models for loads, PV systems, EV chargers, and dedicated storage systems that will be used throughout this work, and present the set of assumptions that are common to the next three chapters.
3.1 Distribution System Model
3.1.1 Network Model
Consider a tree graph G = {B, L} that represents the topology of a radial system (as seen in Figure3.1), comprising a set B of buses and a set L of lines that connect these buses just as edges of a graph connect its vertices. We denote the set of buses that each represents the root of a balancing zone by BZ⊂ B, the set of buses located downstream of bus i by Bi, the set of all network elements (lines and transformers) by N , the set of elements on the unique path from the substation to bus i by Ni, and the set of elements that belong to the subtree rooted at bus i by Ni.
To simplify the analysis, we study the radial system on a per-phase basis, ignoring the dependency between phases. We model homes, businesses, and other end-nodes connected to laterals as single-phase constant complex power loads, i.e., their power consumption is voltage-independent. We encode the topology of the network into a matrix M, where Mij is 1 if bus i is supplied by line or transformer j, and is 0 otherwise. Hence, for every bus i, Mij is 1 when j ∈ Ni.
We denote the setpoint associated with a line l by ξl and the setpoint associated with the substation transformer by ξ01. We assume that setpoints are expressed in Watts for distribution lines and transformers, remarking that the electric utility can easily translate line and transformer ratings which are expressed in Amperes and Volt-amperes,
1The equipment setpoint is its ideal operating point below its nameplate rating, defined in Section2.1.2.
A
substation
primary
secondary
Figure 3.1: A schematic diagram of a radial distribution system representing primary and secondary lines, buses, and loads. Dotted circles represent buses that supply a load connected at point A. We refer to them as upstream buses when we talk about that load.
respectively, into setpoints using conservative estimates of the power factor and the operating voltage at corresponding nodes to err on the side of caution.
To study the dynamics of the system, we use a time-slotted model with time slots of equal length τ (typically of the order of seconds). We assume that during a single time slot, the network configuration, the demand of inelastic loads, the solar power generated by each panel, the output of each storage system, and the number of plugged-in EVs and corresponding charge powers do not change. This assumption permits us to study a dynamical system as a sequence of time slots. To simplify the conversion between energy and power units, we use Watt-τ as the unit of energy transmitted, produced, or consumed.
For instance, if an EV is charged at the constant rate of 1 Watt in a 1 minute time slot, it consumes 1 Watt-minute of energy.
3.1.2 Simplified DistFlow Model
Power flow in a radial distribution system can be approximated with single-phase recursive branch flow equations, known as DistFlow equations [14,15,16]. This specific formulation leads to efficient solution methods for computing bus voltages and branch flows, given real
and reactive powers drawn from or injected to every load bus. We present the DistFlow model and a linearized model based on an approximation that ignores power losses.
Consider the distribution system in a time slot t. For each bus i ∈ B, we denote the voltage magnitude at this bus measured on a per unit basis by vi(t) and real and reactive powers drawn from this bus by pi(t) and qi(t), respectively. Let bus 0 be the substation bus and v0 be its voltage magnitude, which is assumed to be known. We use the substation bus voltage as the base value for voltage in a per-unit system; hence, v0 is equal to 1p.u. in this work. We denote the impedance of a line connecting bus i to bus j by zij= rij + jxij, where j is the imaginary unit, and rij and xij are the line resistance and reactance, respectively. We also denote the sending-end apparent power flow from bus i to bus j by Sij(t)= Pij(t) + jQij(t), where Pij(t)and Qij(t)are real and reactive power flows at the sending-end. The DistFlow model can be described with the following equations:
Pij(t) = pj(t) + X
i(t)2 is the square of the current magnitude that is being carried by the line connecting bus i to bus j, meaning that the quadratic terms in the above equations represent line losses. We note that an OPF problem that incorporates the DistFlow model is not convex and, therefore, finding its solution(s) will be of exponential complexity in the number of nodes.
Since losses are typically quite smaller than real and reactive power flow components, an approximation that ignores the higher order loss terms introduces only a small error on the order of 1%. We refer to this approximate power flow model as simplified DistFlow.
This model was originally proposed in [16] and has been used several times to formulate convex optimal control problems for distribution networks, see for example [97,33]. The
simplified DistFlow equations can be written as follows after unfolding recursions: both bus j and bus k. Observe that these equations are linear in the squared voltage magnitudes, and real and reactive power flows.
We remark that these linearized branch flow equations (3.4-3.6) allow us to enforce capacity and voltage limits in optimal control problems without losing computational tractability. We expand on this in Chapter5.
Note that we use an even more simplified power flow model in Chapter 4 for the purpose of controlling EV chargers on a fast timescale. This approximate model relies on the assumption that the power factor is close to unity in the distribution system and losses are negligible. Thus, it completely ignores the reactive power flow and line losses, and describes only the real power flow as in Equation (3.4).