## 1 Introduction  
Steam systems are a part of almost every major industrial process, in nearly all industrial sectors. Steam generation systems were estimated to account for 38% of global final manufacturing energy use or 44 EJ in 2005 [1], corresponding to 9% of the global final energy consumption. Steam production is still primarily based on the use of fossil fuels, and all the major industrial energy users devote significant proportions of their fossil fuel consumption to steam production [2]. There is thus an urgent demand to develop cost-efficient alternatives for fossil-based steam generation. Among these, thermal energy storage (TES) in combination with power-to-heat (P2H) conversion technologies such as electric boilers or high-temperature heat pumps (HTHPs) may enable a rapid transition towards renewables-based steam production with rather small changes in the infrastructure. Moreover, P2H combined with TES allows active participation of energy-intensive industries in the energy markets, which will be necessary for stable and flexible electricity supply in future decarbonized, renewables-based energy systems. At the same time, the industry can decrease its energy costs by shifting the electricity consumption to low-cost periods, and the security of supply can be increased. Since short payback time and profitability are key criteria for investment decisions in the industry, it is necessary to identify cost-optimal integration scenarios for TES that also consider technical restrictions, such as available conversion technologies and thermodynamic constraints. Cost-optimal integration of TES has been studied in many different settings. Especially within the context of concentrating solar power plants, in combination with distributed energy systems, as well as in combined heat and power (CHP) and tri-generationsystems (combined cooling, heat and power--CCHP), cost optimal storage sizing and optimal operation are often addressed using mathematical programming techniques. For example, for use in combination with a CHP unit, a sensible hot water storage model based on a network-flow model, which is a special case of linear programming model, was introduced [3]. The objective in this case was to optimize energy planning and trading within distributed energy systems, also targeting short-term trades at the spot market and participation at the reserve market providing balancing power. The DESOD (distributed energy system optimal design) tool is based on mixed-integer linear programming for optimal design and operation of distributed energy systems providing heating, cooling and electricity [4]. Within this tool, TES is considered using a capacity model (costs are driven by capacity, capacity is derived from the maximum energy content throughout the optimization period). Capacity models have also been used for the optimization of a tri-generation system including TES [5], within a simple storage model for optimization of a poly-generation district energy system [6], and for optimization including a simple ice storage with loss free heat transfer [7]. In the latter, the storage operates solely at phase change temperature and consists of a mixture of water and ice depending on the state of charge (SOC) of the storage. Optimization performance and results for four different formulations for stratified TES using mixed integer linear programming (MILP) were investigated and compared to the widely used capacity models [8]. The authors showed that for their use-case, an energy system for building application, the capacity model overrates the system's efficiency and underestimates operating costs by 6-7%. Within a design methodology based on linear programming for designing and evaluating distributed energy systems, the authors use ideally mixed hot water tanks as thermal energy storage [9]. The storage thus shows a linear correlation between SOC and the storage temperature. Similarly, discrete temperature layers were introduced in a hot water storage tank model [10].