In this section, we describe a series of two optimization problems that generate our optimal control in every time slot. We discuss how the second problem can be decomposed into a number of decoupled problems and sketch the operation of our decentralized control scheme.
5.3.1 Optimization Problems
We need a multi-step optimization since the three objectives cannot be satisfied at the same time without using an arbitrary scalarization, i.e., a weighted sum of the objectives.
In particular, reducing the use of conventional power is in conflict with the revenue-maximization objective because it can reduce the total supplied power; therefore, a two-step optimization is inevitable. We discuss these two convex optimization problems next.
Revenue-Maximizing Fair Allocation with Minimum Solar Curtailment
The first optimization problem is to maximize the use of solar power, while allocating the available power in a proportionally fair manner among active chargers. Since the first two objective functions of Section5.2.3are not conflicting, it is possible to optimize them at the same time without introducing weight terms. More specifically, for any feasible control, increasing the use of solar power does not negatively impact the optimal power allocation to EV chargers. Hence, the optimal solution to the sum of these two objectives is the solution to any weighted sum of these two objectives6.
Assuming that real and reactive power consumptions of homes and businesses, the setpoint of feeders and transformers, the available solar power at the point of connection of PV systems, and the set of active end-nodes and their parameters are known in the beginning of every time slot, we pose this problem as a nonlinear convex optimization problem.
Problem (5.14) is subject to the power flow equations (Section3.1.2), the real and reactive power injection equations for buses (Section 3.2.5), the distribution systems constraints (Section5.2.2), and the end-node constraints (Section5.2.1). Note that this is a nonlinear convex problem because it maximizes the sum of two concave functions, subject to constraints that define a convex set. We represent the unique proportionally fair power allocation to EV chargers in time slot t by ˜pe(t), and the optimal real and reactive power contributions of PV inverters by ˜pg(t) and ˜qg(t), respectively. Here the upright boldface letters represent vectors.
6Nevertheless, algorithmically weight terms are important because they influence how fast the optimal solution is found.
Problem 1: the global problem Bus Injection Equations (5.5−5.6) Power Flow Equations (5.7−5.9)
Minimizing the Use of Conventional Power
Given the solution to the first problem, the second optimization problem aims at minimiz-ing the power supplied by the grid in a time slot, which can be written as:
Pgrid(t) =X
Since the three first terms in the right hand side of this equation are fixed, maximizing the total power discharged from storage systems minimizes the use of conventional power supplied by the grid. Given real and reactive power consumptions of homes and businesses, the setpoint of feeders and transformers, the solution to the first optimization problem, the available solar power at the point of connection of PVs, the set of active end-nodes, and their parameters, we pose this problem as an LP. Note that Problem (5.15) does not include end-node constraints except for storage systems. This is because the operations of other active end-nodes have been determined already.
As a practical matter, all storage systems located in the same balancing zone must be either charging or discharging in a given time slot; otherwise, control may discharge one storage system and use the energy stored in that system to charge another storage system in the same zone. The energy transfer between storage systems that are within the same zone results in waste of energy due to storage charge and discharge inefficiencies. Hence, we rule out such controls by requiring all storage systems located in the same zone to either charge or discharge in each time slot, thereby maximizing the system efficiency
Problem 2: the global problem Bus Injection Equations (5.5−5.6) Power Flow Equations (5.7−5.9)
implicitly. We denote the set of storage systems that must be charged and the set of storage systems that must be discharged by SC and SD, respectively, which are defined as:
SC = where BC and BD are balancing zones in which every storage system must be charged and discharged, respectively. We define these two sets as:
BC =
This optimization problem can have multiple solutions, each minimizing the use of conventional power from the grid. We represent an optimal control for storage systems in time slot t by ˜ps(t).
Observe that the optimization problem (5.15) is separable because no constraint couples storage systems that belong to two different balancing zones7. Thus, this problem
7We can ignore line and transformer capacity constraints that are outside balancing zones in
Prob-can be decomposed to several problems of the forms (5.20) and (5.21), each of which corresponds to a single balancing zone. Solving each of these problems can be delegated to the controller installed at the edge of the balancing zone as discussed in the next section.
Problem 2-1: the balancing zone problem: j ∈ BC Inputs: pl(t), ql(t), ξ, ˜pg(t), ˜qg(t), ˜pe(t), ps(t), ps(t), I, E , J , S Bus Injection Equations (5.5−5.6) Power Flow Equations (5.7−5.9)
Problem 2-2: the balancing zone problem: j ∈ BD Inputs: pl(t), ql(t), ξ, ˜pg(t), ˜qg(t), ˜pe(t), ps(t), ps(t), I, E , J , S Bus Injection Equations (5.5−5.6) Power Flow Equations (5.7−5.9)
lem (5.15). This is because 1) storage systems are not charged from the grid due to the third objective and 2) storage systems are discharged at the maximum rate due to the choice of the objective function in Problem (5.15) and this does not overload any line or transformer if we ignore the capacity constraints as Problem (5.14) had a feasible solution.
5.3.2 Operation of the Decentralized Control Scheme
The utility may need to control possibly thousands of PV panels, storage systems, and EV chargers using measurements collected from tens of thousands of end-nodes in the distribution network. This calls for the design of an overall architecture that enables scalable, robust, timely, and secure data transfer between measurement and control nodes.
Following the discussion in Section2.3.3, we adopt a decentralized control architecture that consists of a centralized substation controller that coordinates control with a set of controllers corresponding to balancing zones. A communication network connects the substation controller to the balancing zone controllers and to measurement devices installed at homes and businesses. These devices measure residential and commercial demands and the parameters pertaining to the active end-nodes, discussed in Section2.3.2, and send them to upstream controllers in every time slot.
Control actions are computed jointly by the substation controller and balancing zone controllers as follows: Step 1: the substation controller receives real-time measurements from the end-nodes just before the end of every time slot and uses this information to solves the optimization problem (5.14) for the next time slot. Step 2: it then communicates the optimal control setpoints, ˜pg(t), ˜qg(t), and ˜pe(t), to balancing zone controllers. Step 3:
depending on the optimal setpoints computed by the substation controller, the controller of every balancing zone determines whether the storage systems in that zone must charge or discharge in the next time slot and solves either the problem (5.20) or the problem (5.21). Step 4: the controller of each balancing zone sends the control decisions down to the active end-nodes in its subtree. Step 5: each active end-node carries out the control decision in the beginning of the next time slot.