Physics-Based Data-Driven rSOC Modeling for Power System Operation: An Interpretability-Enhanced Hybrid Neural Network Agent (Supplementary Information)

1 Abstract-- This supplementary material provides critical additional information to assist researchers in understanding the relevant content within the manuscript. Section I presents the “status boundary” characterization of the rSOC under the SOEC mode based on Special Ordered Sets of type 2 (SOS2) constraints with the Big-M method. Section II details the linearization process for Equation [14] in the manuscript. Section III provides key model parameters in the Case Study. Index Terms-- Reversible solid oxide cell, neural network, operation region, power system operation. I. “STATUS BOUNDARY” CHARACTERIZATION As noted in the manuscript, the operation of a rSOC in SOEC mode can be further subdivided into endothermic electrolysis and exothermic electrolysis. The operation regions for these two scenarios are separated by a “status boundary,” defined as follows: ,min ,max min max , ( ,) ,( ) ec ec ec cell cell cell ECED ec cell ec cell P PP PT T T T TP T  ≤≤  Ω=  ≤≤ ≤   (S1) ,min ,max min max , ( ,) , () ec ec ec cell cell cell ECEX ec cell ec cell P PP PT T TT TTP  ≤≤  Ω=  ≤≤ ≤   (S2) 2 1 23 )(() ec ec ec cell cell cell TP k P kP k+= + (S3) where () ec cell TP is the fitting function for the “status boundary”, 1 k , 2 k and 3 k are the fitting parameters. To represent the distribution of operation points in ECED Ω and ECED Ω under the SOEC mode, we employ the Special Ordered Sets of type 2 (SOS2) constraints with the Big-M This work was supported in part by the National Natural Science Funds for Distinguished Young Scholar (52325703), the National Natural Science Foundation of China (52407085), the Natural Science Foundation of Jiangsu Province (BK20241322), and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX24_0392) (Corresponding Author: Guangsheng Pan.) Zhongfan Gu, Guangsheng Pan, Wei Gu, Qinran Hu, and Linwei Sang are with the School of Electrical Engineering, Southeast University, Nanjing, Jiangsu 210096, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Hanyang Liu is with State Key Laboratory of Technology and Equipment for Defense against Power System, Operational Risks State Grid Electric Power Research Institute, Nanjing, China (e-mail: 6040538[email protected]om). method. Equations (S4)-(S6) utilize SOS2 constraints to model the “status boundary”, while (S7)-(S12) further characterize ECED Ω and ECED Ω through the Big-M method. 00 ˆ , () pp NN ec cell i i i i ii P pT Tp ωω = = = = ∑∑ (S4) 0 1, 0, p N ii i i ωω = = ≥∀ ∑ (S5) { } 0 is an SOS2 set p N ii ω = (S6) ˆ(1 ) status TT M δ ≤+− (S7) ˆstatus TT M δ ≥− (S8) ˆ ˆ (1 ) (1 ) ec status ec ec status cell cell cell H MH H M δδ −− ≤ ≤ +− (S9) // ec ec status ec ec ec status cell cell cell P LHV M H P LHV M ηδ ηδ − ≤≤ + (S10) ˆ ˆ (1 ) (1 ) ec status ec ec status cell cell cell Q MQ Q M δδ −− ≤ ≤ +− (S11) status ec status cell MQ M δδ − ≤≤ (S12) where i ω is the weight variable for segment point i , i p is the cell power at segment point i , p N is the number of segment points, ˆ T is the segmented linear interpolation approximation of () ec cell TP , status δ is the binary selection variable representing the operation region (1 indicates operation in ECEX Ω , 0 indicates operation in ECED Ω ), M is a sufficiently large number, ˆec cell H and ˆ ec cell Q are hydrogen and heat under the DNN agent. Here, 0.74 ec η = because the rSOC operates in ECED Ω . II. LINEARIZATION PROCESS FOR EQUATION [14] Equation (14) in the manuscript presents the external characteristics of the rSOC, which involves the product of binary variables and continuous variables. This can be addressed using the linearization method, as described in (S13) -(S16). ( ) ( ) ( ) rSOC ec fc cell cell cell stack rSOC ec fc cell cell cell stack rSOC ec fc cell cell cell stack P P P NN H H H NN Q Q Q NN =−+  = −  = +      (S13) Physics-Based Data-Driven rSOC Modeling for Power System Operation: An InterpretabilityEnhanced Hybrid Neural Network Agent (Supplementary Information) Zhongfan Gu, Graduate Student Member, IEEE, Guangsheng Pan, Member, IEEE, Wei Gu, Senior Member, IEEE, Qinran Hu, Senior Member, IEEE, Linwei Sang, Member, IEEE, Hanyang Liu 2 { } { } ,min ,max ,max ,min ,, (1 ) (1 ), , cell cell cell cell cell cell cell cell P P P ec fc PP PPP ec fc ϑ ϑϑ ϑ ϑ ϑϑ ϑ ϑ ϑϑ ϑ ε εϑ εε ϑ ≤≤ ∈  − −≤≤+ −  ∈     (S14) { } { } ,min ,max ,max ,min ,, (1 ) (1 ), , cell cell cell cell cell cell cell cell H H H ec fc HH HHH ec fc ϑ ϑϑ ϑ ϑ ϑϑ ϑϑ ϑϑ ϑ ε εϑ εε ϑ ≤≤ ∈  − −≤ ≤ + −  ∈     (S15) { } { } ,min ,max ,max ,min ,, (1 ) (1 ), , cell cell cell cell cell cell cell cell Q Q Q ec fc QQ QQQ ec fc ϑ ϑϑ ϑ ϑ ϑϑ ϑϑ ϑϑ ϑ ε εϑ εε ϑ ≤≤ ∈  − −≤ ≤ + −  ∈     (S16) where ec cell P  , fc cell P  , ec cell H  , fc cell H  , ec cell Q  and fc cell Q  are auxiliary variables; ,mincell P ϑ and ,maxcell P ϑ represent the minimum and maximum values of cell power under two modes; ,mincell H ϑ and ,maxcell H ϑ correspond to the minimum and maximum values of hydrogen; and ,mincell Q ϑ and ,maxcell Q ϑ denote the minimum and maximum values of heat. III. KEY MODEL PARAMETERS Fig. S1 presents the modified IEEE 33-node system structure diagram used in the Case Study. Nodes 6, 18, 24, and 32 are equipped with 3 MW wind turbines (WTs) and 0.5 MWh batteries (BTs, maximum charge/discharge duration of 2 hours). The rSOC system deployed at node 10 comprises 20 stacks, each consisting of 60 cells connected in series. It is accompanied by 1.5 tons of hydrogen storage (HS, maximum charge/discharge time of 8 hours) and 60 MWh of thermal storage (TS, maximum charge/discharge time of 6 hours). The energy balance cycle for BTs and TS is intraday (24 hours), while HS operates on a weekly cycle (168 hours). The capacity factor of WTs and power load are shown in Fig. S2. Additionally, Tables SⅠ and SⅡ provide key model parameters and physical parameters of the rSOC module. Fig. S1. The modified IEEE 33-bus system structure. Fig. S2. The capacity factor of WTs and the power load. TABLE SⅠ KEY PARAMETERS OF CASE STUDY [1], [2] Parameters Value Parameters Value stack N 20 LHV 33.3 kWh/kg cell N 60 tm C 10 kWh/K min T 1023 K ,min ec cell P 0.0717 kW max T 1123 K ,min ec cell P 1.6 kW avr T 1073 K ,min fc cell P 0.035 kW amb T 298 K ,max fc cell P 0.4 kW TABLE SⅡ PHYSICAL PARAMETERS OF THE RSOC MODULE [1], [2] Parameters Value Fuel electrode support layer (Ni-YSZ) 250 μm Electrolyte layer (YSZ) 8 μm Oxygen electrode layer (LSCF-CGO) 55 μm Effective area 414 cm2 SOEC Fuel Electrode Feed 10/90 %vol H2/H2O SOFC Fuel Electrode Feed 96/4 %vol H2/H2O rSOC Oxygen Electrode Feed Dry Air SOEC Oxygen/Fuel Electrode Flow Ratio 1 SOFC Oxygen/Fuel Electrode Flow Ratio 13 SOEC Current Density 0-2 A/cm2 SOFC Current Density 0-1.2 A/cm2 SOEC Steam Utilization 90 % H2O SOFC Hydrogen Utilization 85 % H2 rSOC Pressure 1 atm IV. REFERENCES [1] F. R. Bianchi et al., “Modelling and optimal management of renewable energy communities using reversible solid oxide cells,” Appl. Energy, vol. 334, p. 120657, Mar. 2023. [2] C. Huang et al., “Modeling and optimal operation of reversible solid oxide cells considering heat recovery and mode switching dynamics in microgrids,” Appl. Energy, vol. 357, p. 122477, Mar. 2024.