In the capacity planning of hydro-wind-solar power systems (CPHPS)

it is crucial to use flexible hydropower to complement the variable wind-solar power. Hydropower units must be operated such that they avoid specific restricted operation zones

that is

forbidden zones (FZs)

to avoid the risks associated with hydropower unit vibration. FZs cause limitations in terms of both the hydropower generation and flexible regulation in the hydro-wind-solar power systems. Therefore

it is essential to consider FZs when determining the optimal wind-solar power capacity that can be compensated by the hydropower. This study presents a mathematical model that incorporates the FZ constraints into the CPHPS problem. Firstly

the FZs of the hydropower units are converted into those of the hydropower plants based on set theory. Secondly

a mathematical model was formulated for the CPHPS

which couples the FZ constraints of hydropower plants with other operational constraints (e.g.

power balance constraints

new energy consumption limits

and hydropower generation functions). Thirdly

dynamic programming with successive approximations is employed to solve the proposed model. Lastly

case studies were conducted on the hydro-wind-solar system of the Qingshui River to demonstrate the effectiveness of the proposed model.