Convergence of explicitly coupled simulation tools (co-simulations)

Abstract

In engineering, it is a common desire to couple existing simulation tools together into one big system by passing information from subsystems as parameters into the subsystems under influence. As executed at fixed time points, this data exchange gives the global method a strong explicit component. Globally, such an explicit cosimulation schemes exchange time step can be seen as a step of an one-step method which is explicit in some solution components. Exploiting this structure, we give a convergence proof for such schemes. As flows of conserved quantities are passed across subsystem boundaries, it is not ensured that systemwide balances are fulfilled: the system is not solved as one single equation system. These balance errors can accumulate and make simulation results inaccurate. Use of higher-order extrapolation in exchanged data can reduce this problem but cannot solve it. The remaining balance error has been handled in past work by recontributing it to the input signal in next coupling time step, a technique labeled balance correction methods. Convergence for that method is proven. Further, a proof for the lack of stability of such methods is given for cosimulation schemes with and without balance correction.

Spectral Complexity of Directed Graphs and Application to Structural Decomposition

“…The energy for this graph by using equation (14) is equal to 28.3401 (sum of singular values is equal to 7.9352). If the matrix is symmetrized, then the energy for this graph by using equation (14) is equal to 33.9041 (sum of singular values is equal to 9.4931).…”

“…For example, these cycles can give rise to positive feedback loops [13] which lead to system instabilities. Cycles in engineering systems also make design and analysis challenging from a simulation convergence perspective [14,15].…”

On Meeting Energy Balance Errors in Co-Simulations

“…The coupling equations (9) -(10) together with the algebraic equations (6) and (8) of the subsystems form a global system of algebraic equations. If the graph of the information flow through the subsystems has no loops and one applies an according Gauss-Seidel scheme, then the exchanged data is consistent in the sense that the equations (9), (10), (6) and (8) are fulfilled -in the general case they are not if one just solves (9), (10) with respect to the input.…”

“…S 1 :ẋ 1 = f 1 (x 1 , z 1 , u 12 ) (5) 0 = g 1 (x 1 , z 1 , u 12 ) (6) S 2 :ẋ 2 = f 2 (x 2 , z 2 , u 21 ) (7) 0 = g 2 (x 2 , z 2 , u 21 ) (8) where u i are given by coupling conditions that have to be fulfilled at exchange times T k 0 = h 21 (x 1 , z 1 , u 21 ) (9) 0 = h 12 (x 2 , z 2 , u 12 )…”

“…From seeing instability as a rise in systems energy as mentioned in [8,Sec.4.2.2] and [7,Sec.3.2] and considering the power that is given by variables at the subsystems interfaces, in this contribution a stable explicit coupling scheme is derived.…”