  This paper presents a reformulation for the automatic generation control
(AGC) in a decomposed convex relaxation algorithm. It finds an optimal solution
to the AC optimal power flow (ACOPF) problem that is secure against a large set
of contingencies. The original ACOPF problem which represents the system
without contingency constraints, is convexified by applying the second-order
cone relaxation method. The contingencies are filtered to distinguish those
that will be treated with preventive actions from those that will be left for
corrective actions. The selected contingencies for preventive action are
included in the set of security constraints. Benders decomposition is employed
to decompose the convexified Security-Constrained ACOPF problem into a master
problem and several security check sub-problems. Sub-problems are evaluated in
a parallel computing process with enhanced computational efficiency. AGC within
each sub-problem is modeled by a set of proposed valid constraints, so the
procured solution is the physical response of each generation unit during a
contingency. Benders optimality cuts are generated for the sub-problems having
mismatches and the cuts are passed to the master problem to encounter the
security-constraints. The accuracy of the relaxation results is verified using
the presented tightness measure. The effectiveness of the presented valid AGC
constraints and scalability of the proposed algorithm is demonstrated in
several case studies.
