Efficient solution strategies for building energy system simulation
Section snippets
Background
Detailed simulation of building energy systems involves the solution of large sets of non-linear algebraic and differential equations. These equations emerge from component-based simulators such as TRNSYS [1] or HVACSIM+ [2], or equation-based tools such as SPARK [3] or IDA [4]. Since each of these tools employs a different solution strategy, the question arises as to which strategy is most appropriate for the kinds of equations encountered in the building simulation domain.
TRNSYS and HVACSIM+
Non-linear equation example
The first benchmark problem derives from a problem in the SPARK User’s Manual consisting of four highly non-linear equations:SPARK finds a solution to these equations by the calculation sequence:Using the default SPARK solution process, Newton–Raphson iteration is performed until the difference between two successive values of x3 is less than a specified tolerance. Thus, it
Laplace’s equation example
The second benchmark problem, purposely chosen to be not well suited to the SPARK methodology, is Laplace’s equation in two dimensions. This equation models many physical phenomena, including heat transfer in a thin, square plate with uniformly distributed heat source and uniform boundary temperature. The problem is discretized by dividing the square into a uniform grid of specified size. Each cell in the grid is represented by a nodal temperature Ti,j and is governed by a heat balance equation:
HVAC benchmarks
Going beyond simple benchmark examples, the numerical methods used in SPARK were also evaluated by modeling an airflow system employing discrete-time controllers. The example used was a typical HVAC airflow network and its associated control loops, a problem involving significant computational burden [16].
A number of steady state component models were implemented as SPARK objects, including variable speed centrifugal fans, flow diverters, flow mixers and control dampers. In modeling air flow, a
Discussion
The above results confirm that the SPARK methodology offers significant reduction in solution times relative to both conventional and sparse matrix methods in the solution of certain kinds of non-linear equation systems. This is borne out most dramatically by the contrived non-linear benchmark problem, but is also quite clear from the HVAC control application. However, in the case of the example involving Laplace’s equation, we observe that without some user intervention, SPARK has difficulty
Conclusions
The principle conclusion that can be drawn from this work is that SPARK outperforms conventional and sparse matrix methods for solution of problems that can be decomposed and/or reduced with graph-theoretical techniques. Roughly speaking, execution time savings will be O(mr3) where r is the ratio of the largest cut set size to the number of equations in the problem, and m is the number of strongly connected components into which the problem partitions. Typical HVAC air flow systems simulation
Acknowledgements
Portions of this work were sponsored by the Japan Ministry of Education, the United Kingdom Royal Academy of Engineering Foresight Award Scheme and the Office of Building Technologies, Building Systems Division, US Department of Energy.
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