The degree of operational freedom of home appliances is increasing with introducing controllable demand-side appliances by using Home Energy Management System with some communication standard. This situation increases the demands on methodology achieving customer utility maximization, which is able to plan operational strategies of a variety of appliances under various exogenous variables such as energy demand, outlet temperature, and power output of Photovoltaic power generator. Energy demand and photovoltaic output have some uncertainty or variation in the parameter. The objective of this study is to analyze a solution behavior of an operational planning problem adapting Mixed Integer Second-Order Cone Programming extended from traditional Mixed Integer Linear Programming model. The objective function of the mathematical optimization involves mean and its variance by extracting naturally Markowitz model. Above proposed model is intended to sophisticate robust operational planning method, and is also evaluated on the EMS simulation platform that we are developing. As a result, it is found that involving variance to objective function affects to computational time meaning computational load rather than robustness of planned schedule. Additionally, using several scenario results in more robust schedule than using plausible one scenario.