Advancing Grid Storage Act of 2021 This bill requires the Department of Energy (DOE) to establish certain research, loan, and grant programs about storing energy, such as renewable energy. DOE must carry out, within the Office of Electricity, a program for the research of energy storage systems (systems or strategies that improve the ability to shift the dispatch of energy in time, across multiple timescales). DOE must carry out a loan program for (1) the demonstration and deployment of energy storage systems in a specific project, and (2) programs that finance the demonstration and deployment of multiple energy storage systems through a financial assistance program. Loans may only be used to fund projects and programs relating to the demonstration and deployment of energy storage systems in households, businesses, and communities. DOE must carry out a technical assistance and grant program for (1) disseminating information and providing technical assistance directly to nonprofit or for-profit entities so those entities can identify, evaluate, plan, and design energy storage systems; and (2) making grants to those entities so that they may contract to obtain technical assistance to identify, evaluate, plan, and design energy storage systems.