We propose a linear electrolyzer model for steady-state load flow analysis of multi-carrier energy networks, where the electrolyzer is capable of producing hydrogen gas and heat. For our electrolyzer model, we show that there are boundary conditions that lead to a well-posed problem. We derive these conditions for two cases, namely with a known and unknown heat efficiency parameter. Furthermore, the derived conditions are validated numerically. Moreover, we investigate the extensibility of our model by including nonlinear models from electricity, gas, and heat. In this setting, we derived boundary conditions based on our previous findings. Due to the involvement of nonlinearity, it is a challenge to prove that the boundary conditions lead to a well-posed problem. Therefore, we simulated the electrolyzer connected with an electricity, gas, and heat system. Additionally, we considered a known and unknown heat efficiency parameter. The numerical results support that the linear electrolyzer model is solvable in a multi-carrier energy network.

To integrate renewable energies with our current energy systems, we require interaction between gas and electrical networks. The coupling of networks results in a larger system of equations to be solved. Henceforth, scalable solvers are more suitable for large coupled networks. In this paper, a preliminary research is done, by investigating Krylov solvers on gas networks from the GasLib library. The networks are simulated with steady-state models. The models yield a nonlinear system, which is solved with the Newton-Raphson method. The corresponding Jacobian is non-symmetric, indefinite and sparse. We have considered the following Krylov solvers: GMRES, Bi-CGSTAB and IDR(s). We compare the performance with a direct solver, which is the LU factorisation. Our results show that basic Krylov solvers are ineffective in solving the networks, because most networks have a large condition number and an unfavourable distribution of the eigenvalues. Hence, we have explored several preconditioners, such as Jacobi, Gauss-Seidel and ILU methods. Only the ILU preconditioner with the use of the COLAMD reordering scheme leads to convergence of all networks. For this preconditioner, the fill ratio has to be taken large enough, otherwise the ILU factorisation breaks down due to a zero pivot. The minimum required fill ratio leads to a similar amount of work as the direct solver. Thus the combination of ILU and Krylov solver does not perform better than direct solvers for these medium sized problems.

In this paper, we consider a block Jacobi preconditioner and various deflation techniques applied in the Deflated Preconditioned Conjugate Gradient (DPCG) method for solving a sparse system of linear equations derived from a statistical linear mixed model that analyses simultaneously phenotypic and pedigree information of genotyped and ungenotyped animals with Single Polymorphism Nucleotide genotypes of genotyped animals. In livestock production systems, evaluating the genetic merit of the animals through such a model is a key process to ensure an improvement of animals for some characteristics of interest at each generation. First, we propose to define the deflation vectors using a subdomain deflation approach that considers some biological properties of the genotypes. Using simulated data, this approach reduces the number of iterations by up to 87% in comparison to a Preconditioned Conjugate Gradient method with a Jacobi preconditioner. Furthermore, compared to a DPCG method with same number of subdomains but defined randomly, this approach reduces the number of iterations by up to 20% for the same computational costs of one DPCG iteration. The properties of the resulting systems show that this approach annihilates the largest eigenvalues of the preconditioned coefficient matrix. Second, we propose the use of solution vectors of 12 systems of equations that include between 0.25% and 3% less data, as deflation vectors. For reducing the computational costs, we also consider a Proper Orthogonal Decomposition-reduced set of these 12 vectors. The properties of the resulting systems show that this recycling information approach annihilates the smallest eigenvalues of the preconditioned coefficient matrix, and results in a reduction of up to 39% in comparison to the PCG method. Finally, based on our experiment, the combination of the subdomain deflation approach relying on biological properties and of the POD-based approach to recycle previous solution vectors, for defining the deflation vectors, results in annihilating both the smallest and largest eigenvalues, and in a reduction of up to 88 % of the number of iterations in comparison to the PCG method.