Convex optimization plays a critical role in optimizing grid-scale battery dispatch strategies, particularly for cost minimization and peak shaving. The mathematical framework enables efficient decision-making by leveraging the convexity of the problem, ensuring global optimality and computational tractability. This article explores the formulations, solver tools, and case studies relevant to convex optimization in battery dispatch.
The primary objective of grid-scale battery dispatch is to minimize operational costs while meeting demand and constraints. The cost function typically includes electricity purchase costs, battery degradation costs, and penalties for unmet demand. Peak shaving aims to reduce the maximum grid load by strategically discharging the battery during high-demand periods. The optimization problem must account for battery dynamics, such as state of charge (SOC) limits, charge/discharge rates, and efficiency losses.
The convex optimization problem for battery dispatch can be formulated as follows. Let \( t \) denote the time index, \( P_t \) the power dispatched (positive for discharge, negative for charge), \( C_t \) the electricity price, \( E_t \) the energy stored in the battery, and \( D_t \) the demand. The battery has a maximum capacity \( E_{\text{max}} \), charge/discharge limits \( P_{\text{min}} \leq P_t \leq P_{\text{max}} \), and round-trip efficiency \( \eta \). The cost minimization problem is:
\[
\text{minimize} \sum_{t=1}^T C_t P_t + \lambda \sum_{t=1}^T (D_t - P_t)^2
\]
subject to:
\[
E_{t+1} = E_t - \eta P_t \quad \forall t
\]
\[
0 \leq E_t \leq E_{\text{max}} \quad \forall t
\]
\[
P_{\text{min}} \leq P_t \leq P_{\text{max}} \quad \forall t
\]
The first term in the objective function represents electricity costs, while the second term penalizes deviations from demand, encouraging peak shaving. The parameter \( \lambda \) balances cost minimization and load smoothing. The constraints ensure the battery operates within its physical limits.
For peak shaving, an alternative formulation directly targets load reduction:
\[
\text{minimize} \max_t (D_t - P_t)
\]
subject to the same battery constraints. This minimax objective reduces the peak demand seen by the grid. Both formulations are convex, enabling efficient solving.
Convex optimization solvers such as CVXPY, Gurobi, and MOSEK are widely used for these problems. CVXPY, a Python-embedded modeling language, simplifies problem expression and interfaces with powerful solvers. Below is an example CVXPY implementation for cost minimization:
```
import cvxpy as cp
import numpy as np
T = 24 # Time horizon
C = np.random.rand(T) # Electricity prices
D = np.random.rand(T) # Demand profile
eta = 0.95 # Round-trip efficiency
E_max = 100 # Battery capacity
P_min, P_max = -20, 20 # Charge/discharge limits
P = cp.Variable(T) # Power dispatch
E = cp.Variable(T+1) # Energy stored
objective = cp.Minimize(C @ P + 0.1 * cp.sum_squares(D - P))
constraints = [
E[0] == 0.5 * E_max, # Initial SOC
E[1:T+1] == E[0:T] - eta * P,
E >= 0,
E <= E_max,
P >= P_min,
P <= P_max
]
problem = cp.Problem(objective, constraints)
problem.solve(solver=cp.ECOS)
```
This example demonstrates how CVXPY abstracts the problem, allowing focus on the formulation rather than solver details. The solver ECOS is chosen for its efficiency in convex problems.
Case studies highlight the practical benefits of convex optimization in battery dispatch. A study on a 10 MW/40 MWh battery system in California demonstrated a 12% reduction in electricity costs over a year compared to rule-based dispatch. The convex formulation accounted for time-varying prices and demand, optimizing charge/discharge cycles. Peak shaving was achieved by flattening the load profile, reducing the maximum demand by 15%.
Another case involved a microgrid with solar generation and battery storage. Convex optimization minimized diesel generator usage by coordinating battery dispatch with solar variability. The results showed a 20% reduction in fuel costs and improved grid stability. The problem was solved using Gurobi, which handled the large-scale linear constraints efficiently.
Key challenges in convex optimization for battery dispatch include modeling inaccuracies and real-time adaptability. Battery degradation, often nonlinear, is approximated as a convex function to maintain problem structure. Real-time implementation requires fast solver convergence, achieved through warm-starting and horizon truncation.
The choice of solver depends on problem size and structure. For small to medium problems, open-source solvers like ECOS and SCS suffice. Large-scale or mixed-integer extensions require commercial solvers like Gurobi or MOSEK. Recent advances in first-order methods have enabled real-time applications by reducing computational overhead.
In summary, convex optimization provides a rigorous framework for grid-scale battery dispatch, balancing cost minimization and peak shaving. The formulations are adaptable to various objectives, and modern solvers ensure practical implementation. Case studies validate the approach, demonstrating significant cost savings and grid benefits. As battery deployment grows, convex optimization will remain a cornerstone of efficient energy management.