Power flow computations are important for operation and planning of the electricity grid, but are computationally expensive because of nonlinearities and the size of the system of equations. Linearized methods reduce computational time but often have the disadvantage that they are not applicable to general grids. In this paper we propose a novel linearized power flow (LPF) technique that is able to handle PQ- and PV-buses, and works on both transmission and distribution networks. This technique is based on previous work on handling PQ-buses by connecting them to artificial-additional ground buses. We extend this idea to PV-buses. Test-cases show that the novel LPF method leads to similar accuracy as nonlinear power flow (NPF) methods while significantly reducing computation time. Therefore, the general LPF methods is a good alternative to NPF methods.@en

Coupling single-carrier networks into multi-carrier energy systems (MESs) has recently become more important. Various formulations of the single-carrier load flow problem (LFP) are used. Moreover, different coupling models lead to different integrated systems of equations for the LFP of MESs. Both could affect the convergence of the Newton-Raphson method (NR) used to solve the nonlinear system of equations. This paper considers the steady-state LFP for example MESs of varying size, with various coupling models and topologies, and various formulations in the single-carrier parts. Based on numerical experiments, this paper compares the convergence behavior of NR for the various single- and multi-carrier systems. For these examples, NR of the steady-state LFP of the MESs is independent of the size of the network and of the coupling, and NR requires at most as many iterations as the slowest single-carrier network.@en

Energy systems are vital in modern society, and reliable operation is crucial. Multi-carrier energy systems (MESs), which couple two or more single-carrier systems, have recently become more important, as the need for sustainable energy systems increases. Important tools for the design and operation of energy systems are steady-state simulation and optimization. Steady-state simulation, which involves solving the load flow (LF) problem, is concerned with determining the flow of energy through the system and the values of other quantities throughout the system, such as voltages and pressures, for given demands. In operational optimization, which involves solving the optimal flow (OF) problem, the distribution of generation over the various sources and the set points of controllable elements are determined such that some objective is optimized and such that the system is operated within physical limits. LF problems and OF problems have been widely studied for single-carrier (SC) systems. However, conventional LF models for the separate single-carrier networks (SCNs) are not able to capture the full extent of the coupling. Recently, different LF models for MESs have been proposed, either using the energy hub (EH) concept, or using a case specific approach. Yet, they do not state how the graphs of the SCNs can be combined into one multi-carrier network (MCN). A good description of integrated networks of multiple energy carriers is very important. Some couplings between energy systems, while possible in practice, can lead to model problems. Although the EH concept can be applied to a general MES, it is unclear in the existing models how the EH should be represented in the graph of the MES. On the other hand, the case specific approaches are not easily applicable to general MESs. Moreover, the effect of the coupling on solvability and well-posedness of the system of nonlinear LF equations for a MES has had little attention in these models. Operational optimization requires the detailed LF equations to be incorporated into the optimization problem. Nonlinearities of these equations cause issues with convexity and solvability of the OF problem. Hence, the formulation of the LF equations, and the way they are incorporated in the OF problem, greatly influence the solvability of the OF problem and the convergence of the optimization algorithms. In this thesis, we address some of the existing issues and possibilities to improve on the available models. We present a graph-based framework for steady-state load flow analysis of general MESs that consist of gas, electricity, and heat. The framework is based on connecting the SCNs to heterogeneous coupling nodes, using homogeneous dummy links, to form one connected MCN. Load flow equations are associated with each network element, including the coupling nodes, which are combined with boundary conditions to form one integrated system of nonlinear equations, that needs to be solved to find the solution to the LF problem. This is the integrated approach to formulate the LF problem of a MES. Alternatively, the model of the connected MCN can be reformulated, such that a MES is represented by a disconnected MCN that consists of the SC networks and a coupling network. This allows for a more decoupled approach to the LF problem, in which the system of nonlinear equations, now consisting of interface conditions connecting the coupling network with the SC networks and the LF equations per SC network, can be solved making use of individual solves for each SC network. The model framework is validated using a small example MES. Using the integrated approach, we formulate the LF problem of various example MESs, of varying size, with various coupling models and topologies, and various formulations in the single-carrier parts, and solve their LF problems using the Newton-Raphson method (NR). Using these examples, we investigate the effect of coupling on the system of LF equations and discuss the problems arising due to the coupling of SC networks on the solvability of the LF problem. Based on numerical experiments, we compare the convergence behavior of NR for the various single- and multi-carrier systems. Finally, we formulate and solve the LF problem of MESs using the integrated approach and using the decoupled approach. We compare the systems of equations, and we compare the convergence of the solution methods for the two approaches. Furthermore, in this thesis, we consider two ways to include the LF equations in the OF problem for general MESs, called formulation I and formulation II. In formulation I, optimization is over the combined control and state variables, with the LF equations included explicitly as equality constraints. In formulation II, optimization is over the control variables only, and the LF equations are included as a subsystem, which is solved to obtain the state variables for given control variables. We compare the two formulations theoretically, and we illustrate the effect of the two formulations on the solvability of the OF problem by optimizing two MESs. This study shows that the graph-based framework can be used to formulate and solve the steady-state LF problem for general MESs that consist of gas, electricity, and heat, both with the integrated approach and with the decoupled approach. Moreover, the framework can be used with different components and models, both in the SCNs and for the coupling units. Therefore, our framework includes and extends the currently available LF models for MESs. Furthermore, the model framework provides guidelines to obtain a solvable steady-state LF problem for MESs. We find that using the decoupled approach to perform LF analysis is slower than using the integrated approach. For the LF problem of an example MES with a tree-like structure, NR is independent of the size of the network and of the coupling, and NR requires at most as many iterations as the slowest single-carrier network. Both formulation I and formulation II result in a solvable OF problem. For the two example MESs, the optimization algorithms require significantly fewer iterations with formulation II than with formulation I.@en

Optimization is an important tool for the operation of an energy system. Multi-carrier energy systems (MESs) have recently become more important. Load flow (LF) equations are used within optimization to determine if physical network limits are violated. Due to nonlinearities, the solvability of the OF problem and the convergence of the optimization algorithms are influenced by how the LF equations are included in the optimal flow (OF) problem. In addition, scaling greatly influences the practical solvability of OF problems. This paper considers two ways to include the LF equations within the OF problem for general MESs. In formulation I, optimization is over the combined control and state variables, with the LF equations included explicitly as equality constraints. In formulation II, optimization is over the control variables only. The state variables are solved from the LF equations in a separate subsystem, for given control variables. Hence, the LF equations are included only implicitly in formulation II. The two formulations are compared qualitatively, from a theoretical perspective and based on numerical experiments. Both formulation I and formulation II result in a solvable OF problem. Formulation I is easier to implement and more efficient in terms of CPU time. However, formulation II ensures feasibility and can be used for optimization in combination with dedicated load flow solvers. Both matrix scaling and per unit scaling can be used to solve the OF problem, but they are not equivalent.@en

The coupling of single-carrier network into multicarrier energy systems (MES) has recently become more important. Conventional single-carrier steady-state load flow models are not able to capture the full extent of the coupling. Different models for multi-carrier networks have been proposed, either based on the energy hub concept or using a case specific approach. However, the effect of the coupling on solvability and well-posedness of the integrated system of non-linear equations has not been discussed. Using a general load flow model on a small example MES, this paper discusses the problems arising due the coupling of single-carrier networks, and provides guidelines to obtain a solvable steady-state load flow model for MES.@en

Coupling single-carrier networks into multi-carrier energy systems (MESs) has recently become more important. Conventional load flow models for the separate single-carrier networks are not able to capture the full extend of the coupling. Recently, different models for multi-carrier energy networks have been proposed, either using the energy hub (EH) concept, or using a case specific approach. Although the EH concept can be applied to a general MES, it is unclear how the EH should be represented in the graph of the MES. On the other hand, the case specific approaches are not easily applicable to general MESs. This paper presents a graph-based framework for steady-state load flow analysis of general MESs. Furthermore, the effect of coupling on the resulting integrated system of equations is investigated. The proposed framework is validated using a small MES. This example shows that our framework is applicable to a general MES, and that it generalizes both the EH concept and the case specific approach.@en

Optimization is an important tool for the operation of an energy system. Multi-carrier energy systems (MESs) have recently become more important. Load ow (LF) equations are used within optimization to determine if physical network limits are violated. The way these LF equations are included in the optimal ow (OF) problem, influences the solvability of the OF problem and the convergence of the optimization algorithms. This paper considers two ways to include the LF equations within the OF problem for general MESs. In the first formulation, optimization is over the combined control and system-state variables, with the LF equations included explicitly as equality constraints. In the second formulation, optimization is over the control variables only. The system-state variables are solved from the LF equations in a separate subsystem, given the control variables. Hence, the LF equations are included only implicitly in the second formulation. The two formulations are compared theoretically. The effect of the two formulations on the solvability of the OF problem is illustrated by optimizing two MESs. Both formulation I and formulation II result in a solvable OF problem. For the two example MESs, the optimization algorithms require significantly fewer iterations with formulation II than with formulation I. For formulation II, the direct and the adjoint approach can be used to determine the required derivatives within the optimization algorithms. Scaling is needed to solve the OF problem for MESs. Both matrix scaling and per unit scaling can be used, but they are not equivalent. @en

Coupling single-carrier networks (SCNs) into multi-carrier energy systems (MESs) has recently become more important. Steady-state load flow analysis of energy systems leads to a system of nonlinear equations, which is usually solved using the Newton-Raphson method (NR). Due to various physical scales within a SCN, and between different SCNs in a MES, scaling might be needed to solve the nonlinear system. In single-carrier electrical networks, per unit scaling is commonly used. However, in the gas and heat networks, various ways of scaling or no scaling are used. This paper presents a per unit system and matrix scaling for load flow models for a MES consisting of gas, electricity, and heat. The effect of scaling on NR is analyzed. A small example MES is used to demonstrate the two scaling methods. This paper shows that the per unit system and matrix scaling are equivalent, assuming infinite precision. In finite precision, the example shows that the NR iterations are slightly different for the two scaling methods. For this example, both scaling methods show the same convergence behavior of NR in finite precision.@en

Energy systems are becoming more complex due to increased coupling between different networks, resulting in multi-carrier energy networks. Conventional models for the separate networks are not able to capture the full extent of the coupling. Recently, different models for multi-carrier networks have been proposed, either using the energy hub concept or using a case specific approach. Although the energy hub concept can be applied to a general integrated net- work, it is unclear how the energy hub should be represented in the graph of the multi-carrier network. This paper presents a graph-based framework for steady-state load ow models of multi-carrier energy systems. Furthermore, the effect of coupling on the integrated system of equations is investigated. The proposed framework is tested on two small multi-carrier net- works, for comparison with models in literature. Results show that our framework is applicable to a general system, and that it generalizes both the energy hub concept and the case specific approaches.    @en