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.

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Power ow simulations form an essential tool for electricity network analysis but conventional models are designed to work on a separated transmission or distribution network only. The continuing growth of electricity consumption, demand side participation, and renewable resources makes the electricity net- works co-dependent. Integrated models incorporate the coupling of the net- works and interaction that they have on each other, representing the power ow within this changing environment accurately. Several numerical methods are available to solve the power ow problem on integrated networks. They can be categorized as a untied or as a splitting method and networks can be modelled as a homogeneous or hybrid network. In this paper, we review and assess these methods on the network models by running simulations on small test networks and comparing the outcome on their numerical performance, ie on convergence rate and CPU-time. The re- view shows that the convergence rate is comparable for most of the methods, but that hybrid networks have a slight advantage in computational time. Realistic network models, running on millions of buses and with large distribution networks, should give a better insight into the speed of the computations. @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

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.

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Electrical power systems are complex systems and traditionally modeled in two separate systems. Power is generated at the transmission system and at several substations converted to the distribution systems. The increasing amount of generation produced at distribution level can eventually effect the transmission network. An integrated model of both systems can help studying these effects and prevent harmful events on the power system. Transmission and Distribution systems differ significantly from each other. Where transmission systems are assumed to be balanced and therefore modeled as a single-phase system, the distribution systems are in general unbalanced and should be modeled in three-phase. Furthermore, high R/X ratios of distribution lines, the lower voltage level, the radial structure and the presence of unbalanced loading lead to different solution techniques. Connecting these two systems pose complications for both the solution method and the connection method. In this report, we present several methods to solve the integrated Transmission-Distribution system. One of them is to omit the simplifications we can make in a transmission system and solve both systems as a three-phase system. Another method is to use a master-slave splitting approach and solve both systems iteratively, using a boundary state. A last method is building an interconnected network which solves the system at once, respecting both the transmission and distribution conditions. Some artificial currents and voltages have to be injected on the boundary. We compare the different methods on CPU-time, convergence, accuracy and complexity and present the preferable method for the specific network criteria. @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

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