Current research on average consensus for multi-agent networks mainly focuses on the cases with bidirectional interaction. However, the unidirectional information flow usually exists in practical applications due to different communication ranges of individuals. To achieve state-average over directed networks, this brief introduces an extra variable for each agent, called ‘surplus’, to record the shift in the average. Then, a bank of distributed average consensus control protocols is developed by incorporating appointed-time control framework. It shows that for the multi-agent system with strongly-connected network, the presented protocol ensures that the state-average and the surplus-stability behaviour emerge at an appointed convergence time. Besides, the settling time is independent of any other parameters and can be assigned according to the task requirements. Corresponding techniques for convergence analysis involve time projection method, motion planning technique and matrix perturbation theory. Lastly, the theoretical results are examined by a numerical simulation.