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For the given initial finite high-temperature heat reservoir temperature, continuous Hamilton–Jacobi–Bellman equations are established to obtain optimal finite high-temperature heat reservoir temperature for minimum power consumption of multistage Carnot heat pumping system with generalized convective heat transfer law [ q  ∝ (Δ T )   m   ]. Analytical expression of optimal heat reservoir temperature with Newtonian heat transfer law ( m  = 1) is obtained based on generalized optimization results for minimum power consumption. For other heat transfer laws ( m  ≠ 1), numerical solutions for minimum power consumption are provided. Optimization results for multistage Carnot heat pumps are compared with maximum power output solutions of multistage irreversible Carnot heat engines.

Minimum power consumption of multistage irreversible Carnot heat pumps with heat transfer law of q ∝ (Δ T ) m