Massive renewable generation introduces volatility and uncertainty in the grid, leading to sharp ramps and disturbances in net demand (demand minus renewable generation). Traditionally, these ramps and disturbances have been addressed using fossil-based generators such as coal and diesel, which are costly, inefficient, and not environment friendly.
The grid operator can tackle these challenges by intelligently allocating distributed energy resources (DERs), such as solar photovoltaic, wind turbines, battery storage, and flexible loads. We propose a methodology that leverages aggregate DER models to optimally allocate DERs to meet the ramps and disturbances in net-demand forecasts (net-demand is demand minus renewable generation) using a model predictive control (MPC) framework. We do so while including the dynamical behavior and capacity constraints of the DER aggregation.
The grid operator and DER resource aggregators form the agents in our proposed approach. Aggregators are entities that engage with the consumers to aggregate DERs to deploy them based on the requirements of the grid operator (e.g., EnergyHub, Enel X, and utility companies). The role of the grid operator (e.g., ERCOT) is to coordinate DER aggregations and bulk generation to procure and dispatch the resources required to balance supply and demand. The formulation in
This work is funded in part by NSF grants OAC-1934766 and EPCN-2246658, and by the U.S.-Israel Energy Center managed by the Israel-U.S. Binational Industrial Research and Development (BIRD) Foundation.
this paper focuses on DER aggregations: the precise way in which the control is designed at an individual level to achieve tracking is beyond the scope of this paper (see [11] and the references therein for individual-level control).
The control architectures presently used in the grid focus primarily on short-term optimization [5]. Traditionally, economic dispatch with the participation of large generation plants is performed as a static optimization problem [15]. However, the optimal dispatch of DER aggregations needs to account for the dynamics of these aggregations, thereby requiring an optimal control framework. In such scenarios, the solution to short-term optimization is unlikely to be optimal in the long term. The control actions in the grid also vary across different time scales and information architectures: for example, dayahead dispatch relies on forecasts over the next day, while ancillary services rely on near-real time estimates.
To address these challenges, we introduce an MPC framework that approximates the solution to a long-run optimal control problem by solving a sequence of finite dimensional optimization problems. We require two key ingredients in our framework: (a) models for the dynamical behavior of DER aggregations; and (b) accurate short-term predictions for the net-demand forecast. MPC has strong guarantees for transient operations and ensures constraint satisfaction, while also accounting for long-run optimality [14].
Related Research Finite-horizon convex optimization formulations for the control of DER aggregations are discussed in [6]: a centralized quadratic program is solved to generate different command signals for the different classes of DERs. The paper [2] solves a non-linear AC optimal power flow (OPF) with the participation of an aggregation of DERs, specifically thermostatically controlled loads: the problem is modeled as a Markov decision process (MDP) and is reformulated as a finite-horizon convex program. The cost formulations in this paper are related to [4], [12], which also consider finite horizon convex programs to allocate load aggregations. However, all these approaches focus solely on short-term, finite horizon optimization. Model predictive control is used for economic dispatch in power networks in [9]; however, the formulation does not include DERs.
Moreover, these prior works do not consider the near-real time disturbances entering the grid in their optimal control formulations. We consider a general constrained MPC framework for allocation of DER aggregations: this optimization strategy addresses both transient grid disturbances and accounts for net-demand forecasts over a longer horizon, without violating capacity constraints of the DERs. A key ingredient of our setup is the use of convex costs on the state of charge (SoC) of the DER aggregation with hard constraints based on power and SoC capacity. By regularly updating the forecasts, we can use DERs to provide both ramping and regulation-type services to the grid. This reduces the requirement from bulk generation. The simulations in Section IV illustrate the efficacy of the control design in jointly extracting dispatch and regulationtype services from the DERs.
⌧ : control time horizon for an MPC iteration indexed by t 2 {t 0 , t 0 + 1, . . . , t 0 + ⌧ 1}. t 0 : starting time for an MPC iteration. t s : time shift for t 0 between successive MPC iterations. `(t) : forecast of net-demand at time t. g(t) : power from bulk generation (without renewables). M : number of DER aggregations, indexed by i 2 {1, . . . , M}.
x i (t) : SoC for DER aggregation i (units: GWh). p i (t) : power output of aggregator i (units: GW).
Organization The problem setup is explained in Section II; in particular, the linear dynamical models for the DER aggregations are introduced in Section II-A, while the cost functions associated with the optimal control framework are introduced in Section II-B. The model predictive control architecture is introduced in Section III. Section IV provides results from simulation experiments, illustrating the application of the control design in realistic settings. Conclusions and directions for future research are provided in Section V.
We now introduce models for the control architecture, aggregate DER dynamics, and the optimization objective.
We consider linear models for aggregate DER dynamics, which are well studied literature (see [7], [8], [10]). Following [7], we adopt the generalized battery models for DER aggregations with known constraints on the power and the state of charge. The SoC of the i-th DER aggregagation, i 2 {1, . . . , M}, obeys:
where the state x i (t) 2 R models state of charge, and ↵ i 0 is a leakage parameter. For example, for aggregations of loads such as air conditioners and water heaters, ↵ i corresponds to the thermal time constant. The nominal SoC is defined to be 0 (nominal refers the behavior of the DER aggregation when the DERs are not participating in dispatch or regulation). The (virtual) power supplied at time t by the i-th DER aggregation is p i (t) 2 R: this is the nominal power consumption of the DER aggregation minus the total power consumed by the aggregation. The constant is in units of time. The appendix provides details on the derivation of the capacity limits for an aggregation of grid-responsive loads.
We formulate the resource allocation problem as an infinitehorizon optimal control problem with the objective:
where the state cost c X is given by
and the generation cost c g is given by
with `:= T 1 P T 1 0 `(t), in which `(t) denotes the netdemand forecast. i 0 and g 0 are design parameters.
The net-demand forecast `(t) satisfies
where p (t) = P i p i (t) is the total power supplied by all the DERs, and g(t) is the power generated by traditional resources. The initial condition x i (0) is known for each class.
Solving the infinite horizon optimization problem described by (2) in the previous section is not feasible for the following reasons: (i) we may not have accurate long-term estimates of DER models (e.g., ↵ i ) and constraints (e.g., C i ) and netdemand forecasts (`(t)), but have accurate estimates in the short term; and (ii) the capacity-constrained infinite-horizon optimization problem is not computationally tractable. Model predictive control is a well-established control architecture in this setting to convert the infinite-horizon problem to a finitehorizon problem with regular updated forecast `(t) [14].
Model predictive control is an iterative technique that requires a look-ahead time horizon ⌧ and a time shift t s . For an MPC iteration with starting time t 0 0, the input p(t 0 ) is obtained through the following methodology.
Step (i): First, given a starting state x(t 0 ) 2 R M , we solve the following convex optimization problem for the finite time horizon [t 0 , t 0 + ⌧ ]:
Step (ii): The optimal power (input) for an MPC iteration starting at t 0 is the solution p i (t) for t 2 {t 0 , t 0 +1, . . . , t 0 +t s } and i 2 {1, . . . , M}.
Step (iii): Successive MPC iterations are solved by periodically updating the starting time t 0 at intervals of t s , with 0 < t s ⌧ . That is, for an MPC iteration with starting time t 0 , the starting time for the next MPC iteration is t 0 + t s .
The simulations detailed here include a single class of bulk generation and five classes of DERs, each aggregating millions of DERs. The DER classes are as follows: air conditioners (ACs), electric water heaters (E-WHs), building HVACs (bldgs), refrigerators (RFGs), and electric vehicles with 100 kWh lithium ion batteries (EVs). Tab. I provides details about the type and number of DERs as well as the model parameters for each aggregation. The number of DERs in each aggregation are extrapolated based on the population of California. We compute ↵ values and the capacity limits for the different DER aggregations using data from the following papers: for ACs, E-WHs, and RFGs [13], for bldgs [8], and for EVs [10].
See appendix for the precise formulae. The net load trajectory considered in our experiments is from publicly available California Independent System Operator (CAISO) dataset for September 2023 [3]. Furthermore, we model additive (forecast) disturbances to this net load using the balancing reserves deployed (BRD) in September 2023 by the Bonneville Power Administration (BPA) [1]. BRD is considered a type of regulation signal. Both these datasets have a 5-minute resolution.
Par. Unit DER1 DER2 DER3 DER4 DER5 Type -ACs E-WHs bldgs RFGs EVs N million 10 10 1 10 1 ↵ i -0.98 0.99 0.97 0.96 0.99 C i GWh 8 5 2.3 5 50 ⌘ + i GW 20 4 103 2 3.6 ⌘ i GW 30 50 3 3 3.6 i -1 2 5 5 2 s 300 300 300 300 300
TABLE I PARAMETERS FOR EACH CLASS OF DER AGGREGATION, CORRESPONDING TO THE DISCRETE-TIME LINEAR DYNAMICAL MODEL IN EQUATION (1).
Fig. 1 shows the net load, the output from traditional generation, and the cumulative power supplied by the DERs over a seven day duration following MPC (with net-demand data corresponding to September 1 through September 7, 2023). The finite time horizon for each MPC iteration (⌧ ) is 24 hours, while the time shift between the starting time of successive MPC iterations (t s ) is 30 minutes. This implies that forecast disturbances (taken from BPA's BRD signal) are added to the CAISO's day-ahead net-demand forecast every 30 minutes (in 30 minute rolling windows). Observe that the bulk generation is smooth without any sharp ramps. The total power supplied by the DER aggregations provides most of the balancing required to smooth the ramps as well as the additive disturbances (from BPA's BRD) in the net-demand trajectory.
The SoC trajectories of the different DER aggregations deployed via MPC are shown in Fig. 2, while the corresponding power trajectories are shown in Fig. 3. The DERs charge during times of low net demand (i.e., they consume more energy with respect to nominal), so that they can supply energy (consume less than nominal) when the net demand ramps up. The fact that the DER aggregations are providing regulationtype services can be seen in the fast fluctuations of the power trajectories in Fig. 3. The capacity constraints are satisfied for all the DER classes.
We have shown the efficacy of MPC-based control design for the optimal allocation of DER aggregations to meet netdemand: the MPC design is ideal as it uses the more accurate net-demand forecasts available in the short-term for resource allocation, ensures constraint satisfaction, while still account-