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sures [6], [8]–[11]. The advantages of this kind of methods are simple determination of optimal high-side pressure, low computational requirements and usually overall ease of use. The main drawbacks contain the need of measurements under various conditions or the need to select (or assemble) the appropriate equation and the impossibility to response to different conditions changes and disturbances (dirty gas cooler, partially frozen evaporator etc.). Thus the second approach to high-side pressure control was proposed, sometimes referred as an ”online”. These methods are not based on static equation nor on the tabular data, but perform a real-time optimization of high-side pressure to obtain the maximum COP value. According to [12] the online methods can prevent large COP loss in comparison with offline ones. This approach was examined several times and promising results were obtained [13]–[15]. In this paper we propose an online algorithm of optimal high-side pressure optimization based on Fibonacci optimum seeking method. It is said to be the best optimum search method for the first dimension functions [16] and that is the reason for our choice. II. FIBONACCI SEARCH METHOD Fibonacci optimum seeking (FOS) method is a convenient method for finding minimum or maximum of a unimodal function of one variable y(x) in a finite interval x ∈ ⟨a,b⟩. Further we suppose maximum of the function is TABLE I TABLE SHOWING NUMERICAL RESULTS OF MAXIMUM SEEKING. FINAL UI IS h0.4585, 0.5385i WITH xˆ = 0.4985. n 6 5 0 4 0.2308 3 0.3846 2 0.3846 The procedure was discovered and formally described in [17] as the way of determining an interval containing maxi- mum of a unimodal function without any requirements on its continuity, derivatives etc. Thus the method is convenient for seeking maximum outcome of a real experiment that can be considered as a unimodal function of one variable. The method is ”minimax optimal” sequential method among the class of all sequential nonrandomized procedures with fixed number N of the argument values at which the function may be observed. Less formal description of the method can be found in [18]. FOS strategy is based on evaluation of the function (experiments) at N distinct points x1, x2, . . . xN within the interval ⟨a,b⟩. Of course we do not know the value x. For unimodal function we can only claim that x is somewhere in an interval ⟨xK−1,xK+1⟩ and for a given strategy SN we must consider the biggest possible interval. Strategy SN with minimal biggest possible interval is then ”minimax optimal”. Thus FOS method gives better results than other similar methods like dichotomic search or golden section search for the same number N. Next we describe FOS without mathematical details. Mat- lab Central File Exchange function fibonacciSearch can be consulted for implementation details of the minimum seeking method. FOS uses Fibonacci numbers Fn Fn+2 =Fn+1 +Fn, F0 =F1, n=0,1,2,... (2) Then we shall use Fibonacci numbers from sequence {F }N = {F ,F ,...,F ,F } to reduce UI gradually. a p q b f(p) f(q) 0 0.3846 0.2308 0.3846 0.4615 0.4585 0.6154 0.3846 0.4615 0.5385 0.4615 1 0.490902 0.6154 0.437884 0.6154 0.490902 0.6154 0.499659 0.5385 0.499536 0.5385 0.481671 0.490902 0.499659 0.496582 0.499659 0.4585 and Fibonacci sequence y(x) = max y(x). a≤x≤b (1) {Fn}∞n=0 = {1,1,2,3,5,8,13,21,34,55,...}. (3) Contrary to other sequential nonrandomized procedures we must set number of experiments N according to required precision or tolerance tol at the beginning of search. Let us denote interval containing x as uncertainty interval (UI). Thus at the beginning of the search we have the first UI ⟨a, b⟩ with length |b − a|. FOS with N Fibonacci numbers decreases UI to final interval with length Δ Δ= |b−a|. (4) FN From this follows that when we want to find optimum x with given precision or tolerance x ∈ ⟨xˆ − tol; xˆ + tol⟩ using minimum number of experiments, we must find smallest N satisfying |b−a| FN > 2∙tol. (5) nn=1 12 N−1N The algorithm in Fig. 3 describes FOS method for maximum seeking. In the Fig. 2 and Table I we demonstrate results of the algorithm on seeking the maximum of function y = 0.5−(x− 0.48)2 with x = 0.48 in the interval ⟨0, 1⟩ with tol = 0.04 which yields N = 6 and xˆ = 0.4985. III. HEAT PUMP MODEL AND CONTROL LOOP Fibonacci search method was applied on a heat pump model, which was connected to a vehicle cabin. Both the heat pump and the vehicle cabin parameters does not correspond to any real vehicle parameters, but they were chosen to be approximately equivalent to possible real fully electric vehicle. A. Heat pump model The heat pump model was assembled in Dymola tool employing Modelica language and Air Conditioning library by Modelon. All control algorithms were designed in Mat- lab/Simulink as well as simulations were performed using this environment. Matlab/Simulink does not allow utilization of Modelica models and that is why the model was exported into Functional Mock-up Unit (FMU) and then loaded into Matlab using FMUtoolbox [19].PDF Image | high-side pressure of R744 automotive heat pump using Fibonacci search
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