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Joint Optimization of Hybrid Energy Storage and Generation Capacity with Renewable Energy
Peng Yang, Arye Nehorai
TL;DR
The paper addresses joint capacity planning for multiple storage and generation technologies in isolated or low-carbon micro-grids with variable renewable availability. It formulates cost-minimizing design and operation and solves the large scenario-based problem with distributed optimization. Numerical examples report 81.78% renewable energy consumption and show that total cost increases rapidly when r_DC decreases below 0.5.
Problem
High-renewable isolated grids need storage, but prior work rarely jointly exploits multiple storage characteristics and renewable-source availabilities when planning hybrid capacity.
Method
The paper jointly optimizes storage, renewable-generator, and diesel-generator capacities and operations while reformulating the design problem as a consensus problem solved in parallel with ADMM.
Results
81.78% of consumed energy is renewable, and total cost increases rapidly when r_DC is below 0.5 and decreases.
Takeaways & Limitations
The framework supports capacity-planning decisions that exploit multiple storage characteristics and renewable availabilities in decentralized high-renewable grids.
Abstract
from arXiv · showhide
In an isolated power grid or a micro-grid with a small carbon footprint, the penetration of renewable energy is usually high. In such power grids, energy storage is important to guarantee an uninterrupted and stable power supply for end users. Different types of energy storage have different characteristics, including their round-trip efficiency, power and energy rating, energy loss over time, and investment and maintenance costs. In addition, the load characteristics and availability of different types of renewable energy sources vary in different geographic regions and at different times of year. Therefore joint capacity optimization for multiple types of energy storage and generation is important when designing this type of power systems. In this paper, we formulate a cost minimization problem for storage and generation planning, considering both the initial investment cost and operational/maintenance cost, and propose a distributed optimization framework to overcome the difficulty brought about by the large size of the optimization problem. The results will help in making decisions on energy storage and generation capacity planning in future decentralized power grids with high renewable penetrations.
R Set of different renewable generators
The notation defines renewable-generator, storage, diesel-generator, demand, shortage, cost, and planning variables used in the optimization model.
- R denotes the set of renewable generator types, while H denotes diesel generator types and S denotes energy storage types.
- Renewable generation and diesel generation are indexed by time period t, with installed maximum capacities represented separately.
- D_t denotes user energy demand, while G_t denotes energy shortage or energy drawn from the main grid during time period t.
- The objective f represents the planning problem, incorporating investment cost, operational/maintenance cost, and amortization through c^inv, c^o/m, and a.
B Group index mapping for boundary conditions
The paper motivates hybrid renewable-storage planning for isolated or low-carbon micro-grids and addresses the computational burden of long, scenario-based design horizons with distributed optimization.
- Renewable availability varies by location, time of day, season, and weather, motivating storage to support uninterrupted supply in high-renewable grids.
- Multiple storage technologies trade off efficiency, capacity and power ratings, energy loss, durability, and investment or maintenance costs.
- Prior work rarely combines the differing characteristics of multiple storage types with the differing availability of multiple renewable sources in one hybrid system.
- The planning objective minimizes storage and generator investment plus operational/maintenance costs by jointly selecting design parameters and operations.
- The design horizon is divided into shorter horizons whose design parameters become consensus parameters across subproblems.
- The consensus formulation is solved in parallel and distributed fashion using ADMM, mitigating dimensionality growth as scenarios increase.
II. RELATED WORK
Related work covers storage and renewable-operation optimization, while this paper emphasizes joint capacity and operation planning for hybrid systems and a distributed solution framework.
- Existing studies optimized energy-storage systems and renewable generation in isolated grids, including wind-diesel storage-rating problems.
- A wind-farm and pumped-storage study optimized day-ahead market bids and facility operations but not generation and storage capacity planning.
- Composite storage research examined high-energy-density and high-power-density storage with converters that distribute demand between storage types.
- Its contribution is to jointly optimize multiple storage and renewable-source capacities and operations, with a distributed framework intended to scale with scenario count.
- The paper models storage using efficiency, power-to-energy ratio, energy-loss ratio, capacity constraints, and charge/discharge-dependent costs.
B. Generator model
The generator model represents diesel and renewable generation with capacity, ramping, availability, and cost constraints, while allowing additional generator types to be incorporated similarly.
- Generators are classified as traditional diesel generators and renewable generators.
- Diesel generation is constrained by installed capacity and ramp-down or ramp-up limits.
- Diesel-generator costs include amortized investment and operational/maintenance components, with quadratic or piece-wise linear operating costs possible.
- Renewable generation from type r is modeled using installed capacity and a random per-unit generation variable that reflects availability.
- Hydro and nuclear generators can also be modeled similarly and included in the planning problem.
C. Load balance constraint
The formulation enforces power balance between generation, storage flows, consumption, and charging while minimizing planning costs. It also constrains shortages, capacities, and operational choices, with a chance-constrained extension for random conditions.
- Generation must equal user demand at every time in the power-grid balance constraint.
- The balance accounts for storage discharge and charging alongside total consumption and generation.For micro-grids, Gt may represent energy drawn from or injected into the main grid; for isolated grids, it may represent shortage or dumped energy.
- The planning objective selects storage and generator capacities while minimizing investment and operational/maintenance costs.
- Shortage or grid-reliance constraints limit unmet demand using a threshold Gth, alongside minimum and maximum capacity constraints.
- Under increasing positive charging and discharging costs, the reformulated model permits nonzero storage charging and discharging at a given time period.
- A chance-constrained version limits the probability of energy shortage exceeding Gth to α and approximates the constraint with sampled deterministic scenarios.The scenario solution satisfies the original chance-constrained problem with probability at least 1 − ϵ when J meets the stated lower bound.
B. Formulation of consensus problem
The large stochastic planning problem is decomposed into scenario subproblems linked by shared design and boundary parameters. Additional consensus constraints replace a potentially suboptimal terminal-energy assumption and yield an equivalent consensus formulation.
- Historical renewable-generation and load realizations make the planning formulation increasingly difficult as their number grows.
- The planning horizon is partitioned into ordered scenario sub-horizons, each with local design variables and a feasible set.
- The prior assumption that storage energy ends at the optimization horizon equal its initial value can make the solution suboptimal.
- The proposed formulation removes that terminal-energy assumption and adds consensus constraints across scenarios.
- Each scenario is connected to global design and boundary parameters through equality constraints xj = ˜zj.
- The consensus solution satisfies the intersection of all scenario feasible sets, z ∈ C = ∩j∈J Cj.
- With scenario approximation, independent identically distributed samples and randomized starting storage energy are required, and the boundary consensus formulation is removed.
C. Distributed Optimization
The consensus problem is solved distributively with ADMM to address the time-complexity growth caused by many scenarios. Local updates run in parallel while global variables enforce agreement across scenario solutions.
- Increasing scenario counts raise the consensus problem’s dimensionality and time complexity.
- The augmented Lagrangian adds a quadratic penalty for differences between local variables xj and corresponding global entries ˜zj.
- The algorithm alternates among x-minimization, z-minimization, and dual-variable updates at each iteration.
- ADMM reformulates the consensus optimization so local minimization problems can be solved in parallel.
- At convergence, the resulting global variable satisfies every scenario subproblem’s constraints and therefore belongs to their feasible-set intersection.
3) Dual-variable update:
Parallelizable ADMM updates make the scenario-based problem scalable, and convergence is guaranteed under the stated algorithm. Numerical examples use real renewable-generation and load data to support planning decisions.
- 3) Dual-variable update:: Parallel x- and dual-variable updates make the optimization scalable as the number of scenarios increases.
- 3) Dual-variable update:: ADMM convergence is guaranteed, with adaptive adjustment of the penalty parameter ρ to balance primal and dual residuals.
- 3) Dual-variable update:: The algorithm stops when both primal and dual residuals fall below a specified threshold.
- 3) Dual-variable update:: Numerical examples evaluate wind and solar generation using online renewable data and hourly ERCOT load data.
- 3) Dual-variable update:: Solar output is calculated from modeled direct-normal radiation, zenith angle, panel tilt, and panel efficiency.
- 3) Dual-variable update:: Wind output uses air density, turbine diameter, and turbine efficiency, with zero output below cut-in or above cut-out wind speed.
- 3) Dual-variable update:: Renewable-generator costs are obtained from cited sources and included in Table I.
2) Load data:
The study constructs normalized load and renewable-generation inputs, then evaluates joint storage and generation planning under diesel-capacity and shortfall constraints. In the case study, renewable generation supplies most consumed energy while total generation generally tracks demand.
- Storage and generation setup: Three prototype storage technologies are used: flywheel storage, Li-ion battery, and pumped storage.Their parameters are taken from and included in Table I.
- Load and renewable data: The simulations use ERCOT hourly load data from 2008–2010 and NSRDB data for renewable generation.The illustrative average hourly load is 1 MW.
- Constraints and optimization: The case study sets rDC = 0.50 and rSD = 0.05, limiting diesel capacity to half the peak load and local-generator shortfall to 5% of demand.The optimization is solved with the CVX toolbox over a three-year design horizon with investment costs amortized across three years.
- Planning results: 18.22% of consumed energy comes from diesel generators, leaving renewable energy at 81.78% of total consumed energy.The planned diesel capacity reaches its upper limit, which is half the peak load.
- Planning results: For most of the time, energy shortage is close to zero, indicating generation including storage discharge is close to demand.Occasional excess generation may be dumped in isolated grids or injected into the main grid in grid-connected micro-grids.
C. Result with different diesel generation capacities
The study examines how diesel capacity limits and geographic renewable availability shape joint planning of renewable generation and energy storage. Lower diesel limits can sharply raise cost without substantially increasing renewable penetration, while local resource characteristics alter planned capacities.
- Diesel generation capacity: Lower allowed diesel capacity requires more renewable generation and energy storage to maintain uninterrupted supply.The maximum diesel capacity ratio is varied while other parameters remain fixed.
- Diesel generation capacity: When rDC falls below 0.5, total cost increases rapidly while the renewable-energy percentage changes little.The analysis varies rDC from 0 to 1 in increments of 0.1 and evaluates three-year planning results.
- Geographic location: Renewable-resource availability significantly affects the planned capacity of different renewable generators across geographic locations.San Francisco’s greater wind availability means relatively little wind capacity is needed, while San Antonio and St. Louis require less solar installation.
- Geographic location: Variability in renewable sources also affects the planned capacity of different energy-storage types.The geographic comparison uses the same hourly load data for all three regions because hourly data was unavailable for every location.
- Interconnected micro-grids: Interconnecting multiple micro-grids could reduce the renewable generation and storage capacity needed to satisfy energy demand, but introduces transmission issues.The paper’s relatively isolated-grid setting motivates considering inter-grid energy transmission in future planning.
- Framework and planning use: The proposed planning framework jointly optimizes multiple storage, renewable-generation, and diesel-generator capacities using a consensus formulation solvable in parallel and distributed form.Numerical examples cover different diesel-capacity limits and geographic regions to illustrate practical planning use.
- Framework and planning use: The study supports capacity-planning decisions for decentralized grids with high renewable penetration.Future work will address operation under stochastic demand and renewable generation and interconnected micro-grid networks.
APPENDIX
The appendix examines charge/discharge pairs and relates their selection to operational cost in the optimization problem.
- The appendix considers two pairs of charge/discharge variables satisfying the stated conditions.
- The operational cost is described as an increasing function of the charging-power variables.
- The charge/discharge variables in the optimal pair must satisfy the resulting cost-related condition.