PowerBiMIP: An Open-Source, Efficient Bilevel Mixed-Integer Programming Solver for Power and Energy Systems

PowerBiMIP is an open-source, efficient solver for Bilevel Mixed-Integer Programming (BiMIP), with a special focus on applications in power and energy systems.

Coming soon: Python + Pyomo. We are developing a Python version of PowerBiMIP with support for Pyomo modeling.

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Overview

PowerBiMIP provides a user-friendly framework to formulate complex hierarchical optimization problems.

The toolbox currently supports:

Bilevel Mixed-Integer Programs (BiMIP):
- Support for continuous and integer variables at both levels.
- Optimistic bilevel optimization with coupling constraints support.
- Pessimistic bilevel optimization.
Two-Stage Robust Optimization (TRO):
- The subproblem of the C&CG procedure is a special BiMIP, thus can be solved by PowerBiMIP efficiently.
- PowerBiMIP now offers a solver interface specifically designed for two-stage robust optimization requirements (currently supporting only LP recourse scenarios; MIP recourse is coming soon).
Multiple Solution Methods: Including exact modes (KKT, Strong Duality) and quick modes (Approximately 1-2 orders of magnitude faster than existing global optimal algorithms for BiMIP).
Power and Energy Systems Case Library: We are actively building a library of benchmark cases for classic BiMIP and TRO problems in power and energy systems, which will be continuously updated and expanded.

PowerBiMIP vs. Existing Toolboxes

Toolbox	Programming Platform	Model Support
Toolbox	Programming Platform	Lower-Level Integer Vars.	OBL	PBL
YALMIP	MATLAB		✓
Pyomo/PAO	Python		✓
BilevelJuMP	Julia		✓
GAMS/EMP	GAMS		✓
MibS	C++	✓	✓¹
PowerBiMIP	MATLAB (Python²)	✓	✓	✓

¹ MibS currently only supports cases where connecting variables are pure integer variables.

² The Python version with Pyomo modeling support is under development. The model-support checkmarks above describe the current MATLAB version.

PowerBiMIP vs. MibS: an energy-system benchmark

Time-to-gap comparison of PowerBiMIP exact mode, PowerBiMIP quick mode and MibS on the IEEE one-area RTS-96 vulnerability-analysis benchmark for K = 2, 4, 6, 8, 10 and 12

Figure. Time-to-gap curves for the IEEE one-area RTS-96 vulnerability-analysis case at attack budgets K = 2, 4, 6, 8, 10 and 12. Red: PowerBiMIP exact mode; blue: PowerBiMIP quick mode; black dashed: MibS. The horizontal axis is time in seconds on a logarithmic scale; the vertical axis shows the reported gap (%). See our paper for the experimental setup and gap definitions. Click the figure to view the full-resolution image.

Installation

Prerequisites

Before installing PowerBiMIP, ensure you have the following dependencies installed:

MATLAB: R2018a or newer.
YALMIP: Use an up-to-date version from the YALMIP GitHub repository. COPT users must ensure their installed revision contains the quadratic-objective interface fix described below.
A MILP Solver: At least one MILP solver is required. YALMIP provides interfaces to Gurobi, CPLEX, COPT and MOSEK; choose a backend compatible with your model and available license. For users in China, we recommend COPT as the preferred alternative.

Solver choice for users in China

According to Gurobi's official notice, Gurobi has discontinued its free academic licensing program in mainland China, Hong Kong and Macau and no longer accepts academic license applications from these regions. This notice concerns academic licensing; commercial licensing remains a separate option. We recommend that affected users consider COPT.

Important: update YALMIP before using COPT

Affected YALMIP versions contain a serious COPT interface bug: when the quadratic-objective matrix is nonzero but has at least one all-zero column, the interface can silently drop the entire quadratic objective while still reporting a successful solve. This can occur when some variables enter the objective only linearly, or when conic reformulations introduce auxiliary variables. Results from such quadratic models can therefore be incorrect even if the solver reports success; the issue does not mean that every purely linear MILP is affected.

Update YALMIP to a revision containing the fix in solvers/yalmip2copt.m. The corrected interface uses the scalar condition nnz(interfacedata.Q) instead of any(interfacedata.Q) at both the quadratic-objective assignment and the dimension-extension check after adding cone variables. The current upstream develop branch contains this fix; an older tagged release may not. Updating COPT alone does not fix the YALMIP interface.

After updating, run which yalmip2copt -all in MATLAB to check that an older YALMIP copy is not taking precedence on the path, then restart MATLAB and verify that the active interface contains the corrected checks.

Installation Steps

We recommend using GitHub Desktop or git to install PowerBiMIP. This allows you to easily receive updates.

Clone the Repository:
- Using GitHub Desktop (Recommended): Clone https://github.com/GreatTM/PowerBiMIP.
- Using Git:
```
git clone https://github.com/GreatTM/PowerBiMIP.git
```
Run Installer:
- Open MATLAB.
- Navigate to the matlab/ folder inside the repository (e.g., cd PowerBiMIP/matlab).
- Run the installation script in the MATLAB Command Window:
```
install
```
- This script adds all necessary folders to your MATLAB path and cleans up stale path entries left over from older layouts, so it is safe to re-run after every update.
Verify Installation:
- Run one of the toy examples to ensure everything is configured correctly:
```
run('examples/BiMIP_benchmarks/BiMIP_toy_example1.m');
```

Updating

To update to the latest version, simply Fetch origin in GitHub Desktop or run git pull in your terminal. If an update moves or renames folders (such as the monorepo restructuring), re-run install from the matlab/ folder once to refresh your MATLAB path.

Power System Case Library

examples/Vulnerability_Analysis: RTS-96 transmission-line interdiction benchmark. Use main_Vulnerability_Analysis.m for a compact demo.

Quick Start Example: BiMIP

Let's walk through a simple example. The following problem is defined in examples/BiMIP_benchmarks/BiMIP_toy_example1.m.

Mathematical Formulation:

Upper-Level Problem:

min_{x}  -x - 10*z
s.t.
    x >= 0
    -25*x + 20*z <= 30
    x   + 2*z  <= 10
    2*x - z    <= 15
    2*x + 10*z >= 15

Lower-Level Problem: Where z is determined by the solution of:

min_{y,z}  z + 1000 * sum(y)
s.t.
    -25*x + 20*z <= 30 + y(1)
    x   + 2*z  <= 10 + y(2)
    2*x - z    <= 15 + y(3)
    2*x + 10*z >= 15 - y(4)
    z >= 0
    y >= 0

MATLAB Implementation with PowerBiMIP:

%% 1. Initialization
dbstop if error;
clear; close all; clc; 
yalmip('clear');

%% 2. Variable Definition using YALMIP
% It's good practice to group all variables in a single struct.
model.var_upper.x = intvar(1,1,'full'); % Upper-level integer variable
model.var_lower.z = intvar(1,1,'full'); % Lower-level integer variable
model.var_lower.y = sdpvar(4,1,'full'); % Lower-level continuous variables

%% 3. Model Formulation
% --- Upper-Level Constraints ---
model.cons_upper = [];
model.cons_upper = model.cons_upper + ...
    (model.var_upper.x >= 0);
model.cons_upper = model.cons_upper + ...
    (-25 * model.var_upper.x + 20 * model.var_lower.z <= 30);
model.cons_upper = model.cons_upper + ...
    (model.var_upper.x + 2 * model.var_lower.z <= 10);
model.cons_upper = model.cons_upper + ...
    (2 * model.var_upper.x - model.var_lower.z <= 15);
model.cons_upper = model.cons_upper + ...
    (2 * model.var_upper.x + 10 * model.var_lower.z >= 15);

% --- Lower-Level Constraints ---
model.cons_lower = [];
model.cons_lower = model.cons_lower + ...
    (-25 * model.var_upper.x + 20 * model.var_lower.z <= 30 + model.var_lower.y(1,1) );
model.cons_lower = model.cons_lower + ...
    (model.var_upper.x + 2 * model.var_lower.z <= 10 + model.var_lower.y(2,1) );
model.cons_lower = model.cons_lower + ...
    (2 * model.var_upper.x - model.var_lower.z <= 15 + model.var_lower.y(3,1) );
model.cons_lower = model.cons_lower + ...
    (2 * model.var_upper.x + 10 * model.var_lower.z >= 15 - model.var_lower.y(4,1) );
model.cons_lower = model.cons_lower + ...
    (model.var_lower.z >= 0);
model.cons_lower = model.cons_lower + ...
    (model.var_lower.y >= 0);

% --- Objective Functions ---
% Note: Maximization problems should be converted to minimization problems 
% by negating the objective function.
model.obj_upper = -model.var_upper.x - 10 * model.var_lower.z;
model.obj_lower = model.var_lower.z + 1e3 * sum(model.var_lower.y,'all');

%% 4. Configure and Run the Solver
% Configure PowerBiMIP settings
ops = BiMIPsettings( ...
    'perspective', 'optimistic', ...    % Perspective: 'optimistic' or 'pessimistic'
    'method', 'exact_KKT', ...                % Method: 'exact_KKT', 'exact_strong_duality', or 'quick'
    'solver', 'gurobi', ...             % Specify the underlying MIP solver
    'verbose', 2, ...                   % Verbosity level [0:silent, 1:summary, 2:summary+plots]
    'max_iterations', 10, ...           % Set the maximum number of iterations
    'optimal_gap', 1e-4, ...             % Set the desired optimality gap
    'plot.verbose', 1, ...
    'plot.saveFig', false ...
    );

% Call the main solver function
[Solution, BiMIP_record] = solve_BiMIP(model, ops);

Solver Output:

Welcome to PowerBiMIP V0.1.0 | © 2026 Yemin Wu, Southeast University
Open-source, efficient tools for power and energy system bilevel mixed-integer programming.
GitHub: https://github.com/GreatTM/PowerBiMIP
Docs:   https://docs.powerbimip.com
Bilevel optimization interface
--------------------------------------------------------------------------
User-specified options:
  verbose         = 2
  optimal_gap     = 0.0001
  plot__verbose   = 1
--------------------------------------------------------------------------
Starting disciplined bilevel programming process...
Welcome to PowerBiMIP V0.1.0 | © 2026 Yemin Wu, Southeast University
Open-source, efficient tools for power and energy system bilevel mixed-integer programming.
GitHub: https://github.com/GreatTM/PowerBiMIP
Docs:   https://docs.powerbimip.com
Bilevel optimization interface
--------------------------------------------------------------------------
Starting disciplined bilevel programming process...
Initial model has coupled constraints. Starting reformulation...
  Preprocessing Iteration 1...
  Applying Transformation 2: [Optimistic + Coupled] -> [Optimistic + Uncoupled]
    Using user-defined penalty kappa = 50.
Identified 4 coupled inequalities and 0 coupled equalities to transform.
Preprocessing complete. Model is now uncoupled.
Disciplined bilevel programming process completed.
Problem Statistics:
  Upper-Level Constraints: 1 (1 ineq, 0 eq), 1 non-zeros
  Lower-Level Constraints: 17 (17 ineq, 0 eq), 33 non-zeros
  Variables (Total): 8 continuous, 2 integer (0 binary)
Coefficient Ranges:
  Matrix Coefficients: [1.0e+00, 2.5e+01]
  Objective Coefficients: [1.0e+00, 1.0e+03]
  RHS Values:          [1.0e+01, 3.0e+01]
--------------------------------------------------------------------------
Solving with optimistic perspective...

-----------------------------------------------------------------------------------------------
  Iter |       MP Obj     SP1 Obj     SP2 Obj |          LB          UB         Gap |  Time(s)
-----------------------------------------------------------------------------------------------
     1 |     -42.0000      2.0000    -22.0000 |      -42.00      -22.00      90.91% |      0.4
     2 |     -26.0000      1.0000    -16.0000 |      -26.00      -22.00      18.18% |      0.6
     3 |     -22.0000      2.0000    -22.0000 |      -22.00      -22.00       0.00% |      0.8

Convergence criteria met (gap = 0.00% <= 0.01%).
-----------------------------------------------------------------------------------------------
Solution Summary:
  Objective value: -22.0000       
  Best bound:      -22.0000       
  Gap:             0.00%
  Iterations:      3
  Time elapsed:    0.78 seconds
-----------------------------------------------------------------------------------------------

Quick Start Example: Two-Stage Robust Optimization

PowerBiMIP introduces support for Two-Stage Robust Optimization (TRO). The following problem is defined in examples/BiMIP_benchmarks/TRO_LP_toy_example1.m.

Mathematical Formulation:

This example solves a Robust Facility Location-Transportation Problem:

First-Stage Problem: (Minimize investment + worst-case operation cost)

min_{y,z}  sum_i (f_i*y_i + c_i*z_i) + max_{d in D} min_{x >= 0} sum_{i,j} t_{ij}*x_{ij}
s.t.
    z_i <= 800*y_i,           for all i
    y_i in {0, 1},            for all i
    z_i >= 0, z_i >= 772,     for all i

Second-Stage Problem: (Minimize operation cost given demand d)

min_{x >= 0}  sum_{i,j} t_{ij}*x_{ij}
s.t.
    sum_j x_{ij} <= z_i,      for all i (Capacity)
    sum_i x_{ij} >= d_j,      for all j (Demand)

Uncertainty Set (D): (Budgeted uncertainty on demand d)

d_j = d_nominal_j + 40*g_j
0 <= g_j <= 1
sum(g) <= 1.8
g_0 + g_1 <= 1.2

MATLAB Implementation with PowerBiMIP:

%% 1. Initialization
dbstop if error;
clear; close all; clc; 
yalmip('clear');

%% 2. Problem Parameters
f = [400; 414; 326];        % Fixed costs
c = [18; 25; 20];           % Capacity costs
T = [22, 33, 24;            % Transportation costs
     33, 23, 30;
     20, 25, 27];
d_nominal = [206; 274; 220];% Nominal demand
n_facilities = 3; n_demands = 3; capacity_limit = 800;

%% 3. Variable Definition
% First-stage variables
model.var_1st.y = binvar(n_facilities, 1, 'full'); 
model.var_1st.z = sdpvar(n_facilities, 1, 'full');

% Uncertainty variables
model.var_uncertain = sdpvar(n_demands, 1, 'full');    

% Second-stage variables
model.var_2nd.x = sdpvar(n_facilities, n_demands, 'full'); 

%% 4. Model Formulation

% --- First-Stage Constraints ---
model.cons_1st = [];
model.cons_1st = model.cons_1st + (model.var_1st.z >= 0); 
model.cons_1st = model.cons_1st + (sum(model.var_1st.z) >= 772); 
for i = 1:n_facilities
    model.cons_1st = model.cons_1st + (model.var_1st.z(i) <= capacity_limit * model.var_1st.y(i));
end

% --- Second-Stage Constraints ---
model.cons_2nd = [];
model.cons_2nd = model.cons_2nd + (model.var_2nd.x(:) >= 0);

% Capacity limit (Recourse)
for i = 1:n_facilities
    model.cons_2nd = model.cons_2nd + (sum(model.var_2nd.x(i, :)) <= model.var_1st.z(i));
end

% Demand satisfaction (Recourse + Uncertainty)
for j = 1:n_demands
    d_j = d_nominal(j) + 40 * model.var_uncertain(j); 
    model.cons_2nd = model.cons_2nd + (sum(model.var_2nd.x(:, j)) >= d_j);
end

% --- Uncertainty Set Constraints ---
model.cons_uncertainty = [];
model.cons_uncertainty = model.cons_uncertainty + (model.var_uncertain >= 0);
model.cons_uncertainty = model.cons_uncertainty + (model.var_uncertain <= 1);
model.cons_uncertainty = model.cons_uncertainty + (sum(model.var_uncertain) <= 1.8);
model.cons_uncertainty = model.cons_uncertainty + (model.var_uncertain(1) + model.var_uncertain(2) <= 1.2);

% --- Objective Functions ---
model.obj_1st = f' * model.var_1st.y + c' * model.var_1st.z;
model.obj_2nd = sum(sum(T .* model.var_2nd.x));

%% 5. Configure and Run the Solver
ops = TROsettings( ...
    'mode', 'exact_KKT', ...          % 'exact_KKT' or 'quick'
    'solver', 'gurobi', ...           % Underlying solver
    'verbose', 2, ...                 
    'plot.verbose', 1, ...
    'plot.saveFig', true ...
    );

% --- SOLVE ---
fprintf('Solving Robust Facility Location Problem...\n');
[Solution, Robust_record] = solve_TRO(model, ops);

%% 6. Analyze Results
if ~isempty(Solution.obj_1st)
    fprintf('\nOptimal Objective: %.4f\n', Solution.obj_1st);
    y_opt = value(model.var_1st.y);
    z_opt = value(model.var_1st.z);
    fprintf('Facility Decisions:\n');
    for i = 1:n_facilities
        if y_opt(i) > 0.5
            fprintf('  Facility %d: OPEN (Capacity: %.2f)\n', i, z_opt(i));
        else
            fprintf('  Facility %d: CLOSED\n', i);
        end
    end
end

Solver Output:

Solving Robust Facility Location Problem...
Welcome to PowerBiMIP V0.1.0 | © 2026 Yemin Wu, Southeast University
Open-source, efficient tools for power and energy system bilevel mixed-integer programming.
GitHub: https://github.com/GreatTM/PowerBiMIP
Docs:   https://docs.powerbimip.com
Two-stage robust optimization interface
--------------------------------------------------------------------------
User-specified options:
  verbose         = 2
  plot__verbose   = 1
  plot__saveFig   = true
--------------------------------------------------------------------------
Robust model components extracted successfully.

-----------------------------------------------------------------------------------------------
  Iter |       MP Obj      SP Obj |          LB          UB      Gap(%) |  Time(s)
-----------------------------------------------------------------------------------------------
     1 |   14296.0000  20942.0000 |  14296.0000  35238.0000    59.4302% |    0.190
     2 |   33680.0000  18034.0000 |  33680.0000  33696.0000     0.0475% |    0.391
     3 |   33680.0000  18024.4000 |  33680.0000  33680.0000     0.0000% |    0.577

Converged! Relative gap (0.0000%) <= tolerance (0.0100%).
Figure saved to: results/figures/

-----------------------------------------------------------------------------------------------
Final Results:
  Lower Bound (LB): 33680.000000
  Upper Bound (UB): 33680.000000
  Final Gap:       0.0000%
  Total Iterations: 3
  Total Runtime:    0.705 seconds
-----------------------------------------------------------------------------------------------

Optimal Objective: 15655.6000
Facility Decisions:
  Facility 1: OPEN (Capacity: 255.20)
  Facility 2: CLOSED
  Facility 3: OPEN (Capacity: 516.80)

Documentation

For detailed documentation, including tutorials, advanced examples, and API references, please visit our official website:

https://docs.powerbimip.com

Development Status & Contribution

PowerBiMIP is under active development. As an early-stage project, it may have known or potential bugs, and some features are still being improved. We are committed to long-term maintenance and plan to release a major update annually and minor updates monthly.

We highly welcome feedback and contributions from the community! If you have any feature requests or encounter a bug, please feel free to:

Contact Yemin Wu directly via email: yemin.wu@seu.edu.cn
Open an issue on our GitHub Issues page.

Acknowledgements

This work is performed under the supervision of Prof. Shuai Lu (@Shuai-Lu) and Prof. Wei Gu at Southeast University.

We extend our sincere gratitude to Prof. Bo Zeng from the University of Pittsburgh for his significant support. The algorithms implemented in PowerBiMIP are based on his pioneering research.

Furthermore, the development of the Fast-R&D algorithm (quick mode) was carefully guided by Prof. Bo Zeng.

We also thank Dr. Ruizhi Yu (@rzyu45) for his technical support and guidance on the development of the PowerBiMIP documentation website.

License and Citation

License

Citation

If you use PowerBiMIP in your research, please cite the following paper (in Chinese):

[1] 吴晔敏,陆帅,顾伟,等.PowerBiMIP：面向电力能源系统优化的规范化双层混合整数规划建模与求解器[J/OL].中国电机工程学报,1-18[2026-09-21].https://doi.org/10.13334/j.0258-8013.pcsee.261230.

Name		Name	Last commit message	Last commit date
.github/workflows		.github/workflows
docs		docs
matlab		matlab
.gitignore		.gitignore
.readthedocs.yaml		.readthedocs.yaml
BiMIP_toy_example1_convergence.png		BiMIP_toy_example1_convergence.png
LICENSE		LICENSE
README.md		README.md
README_CN.md		README_CN.md
TRO_LP_toy_example1_convergence.png		TRO_LP_toy_example1_convergence.png
WechatQRcode.jpg		WechatQRcode.jpg

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PowerBiMIP: An Open-Source, Efficient Bilevel Mixed-Integer Programming Solver for Power and Energy Systems

Fork us on Wechat

Overview

PowerBiMIP vs. Existing Toolboxes

PowerBiMIP vs. MibS: an energy-system benchmark

Installation

Prerequisites

Solver choice for users in China

Important: update YALMIP before using COPT

Installation Steps

Updating

Power System Case Library

Quick Start Example: BiMIP

Quick Start Example: Two-Stage Robust Optimization

Documentation

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Acknowledgements

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PowerBiMIP: An Open-Source, Efficient Bilevel Mixed-Integer Programming Solver for Power and Energy Systems

Fork us on Wechat

Overview

PowerBiMIP vs. Existing Toolboxes

PowerBiMIP vs. MibS: an energy-system benchmark

Installation

Prerequisites

Solver choice for users in China

Important: update YALMIP before using COPT

Installation Steps

Updating

Power System Case Library

Quick Start Example: BiMIP

Quick Start Example: Two-Stage Robust Optimization

Documentation

Development Status & Contribution

Acknowledgements

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