Method for controlling the speed of cryogenic compressors arranged in series for cooling cryogenic helium

Abstract

A method for controlling speeds of compressors arranged in series for compressing a fluid. The desired inlet pressure is predefined and the actual inlet pressure is detected. The actual discharge pressure of the fluid is recorded and the actual total pressure ratio is recorded. A proportional integral value is determined from the deviation of the actual inlet pressure from the desired inlet pressure and a capacity factor is determined from the proportional integral value and the actual total pressure ratio. A model total pressure ratio is determined from the actual total pressure ratio and the capacity factor. A reduced desired speed for each compressor is determined as a function value of the control function associated with the respective compressor. The control function assigns a reduced desired speed to each value pair of capacity factor and model total pressure ratio and is used to adjust the speed of each compressor.

Claims

1. A method for controlling the speeds of compressors arranged in series that are formed to compress a fluid, comprising: Specifying a desired inlet pressure which the fluid should have at an entry of the compressor arranged the furthest upstream, Recording an actual inlet pressure of the fluid at said entry, Recording an actual discharge pressure of the fluid at an output of the compressor arranged the furthest downstream, Establishing an actual total pressure ratio, wherein the actual total pressure ratio corresponds to the quotient of the actual discharge pressure and the actual inlet pressure, Determining a proportional integral value based on the deviation of the actual inlet pressure from the desired inlet pressure, Determining the capacity factor of the proportional integral value and the actual total pressure ratio, Establishing a model total pressure ratio based on the actual total pressure ratio and the capacity factor, Determining a reduced desired speed for each compressor, wherein the respective reduced desired speed is determined as a function value of a control function associated with the respective compressor, which control function assigns a reduced desired speed to each value pair consisting of capacity factor and model total pressure ratio, and Converting the reduced desired speeds into target speeds and adjusting the speed of each compressor to the respectively assigned target speed.

2. The method according to claim 1, wherein the proportional integral value is less than or equal to the sum of the natural logarithm of a design total pressure ratio and of a choke capacity factor, wherein the choke capacity factor is 1 and wherein the design total pressure ratio is the total pressure ratio which results when all compressors of the series are operated at their design points, wherein the design point of a compressor defines the operating state in which the respective compressor has its highest efficiency.

3. The method according to claim 1, wherein the capacity factor corresponds to the difference between the proportional integral value and the natural logarithm of the actual total pressure ratio.

4. The method according to claim 1, wherein a maximum value and minimum value for the capacity factor is defined, wherein the maximum value is between 0.8 and 1 and/or the minimum value is between 0 and 0.1.

5. The method according to claim 4, wherein the model total pressure ratio corresponds to the actual total pressure ratio multiplied by a saturation function that is dependent on the capacity factor, wherein the saturation function is 1 when the capacity factor is between the minimum value and the maximum value, and wherein the saturation function corresponds to an exponential function of the difference of the capacity factor and the minimum value, when the capacity factor is less than the minimum value, and wherein the saturation function corresponds to an exponential function of the difference of the capacity factor and the maximum value when the capacity factor is greater than the maximum value.

6. The method according to claim 5, wherein, when the capacity factor is greater than the maximum value, the capacity factor is equated to the maximum value, after the model total pressure ratio is determined, and that, when the capacity factor is less than the minimum value, the capacity factor is equated to the minimum value, after the model total pressure ratio is determined.

7. The method according to claim 1, wherein the discharge temperature of the fluid at the output of the respective compressor is equal to the inlet temperature of the fluid at the entry of the compressor of the series arranged respectively downstream of the respective compressor, and that the discharge pressure of the fluid at the output of the respective compressor is equal to the inlet pressure of the fluid at the entry of the compressor of the series arranged respectively downstream of the respective compressor.

8. The method according to claim 7, wherein the discharge temperature and the discharge pressure for each compressor are established based on the inlet pressure and the inlet temperature of the compressor of the series arranged the furthest upstream, in particular using an Euler equation of a turbo machine equation, wherein the reduced speed for each compressor and the reduced mass flow rate is established by means of the respective compressor as a function of the total pressure ratio and the capacity factor of the series.

9. The method according to claim 8, wherein a plurality of capacity lines are set for each compressor, wherein each capacity line is a function of the total pressure ratio for each compressor, and a function of the reduced mass flow rate and of the reduced speed of the respective compressor, and wherein the capacity factor along the respective capacity line is constant for each compressor.

10. The method according to claim 9, wherein the control function establishes the reduced desired speed based on a pre-calculated table, wherein the table for each capacity factor, which is located on a capacity line and for each total pressure ratio, exhibits the respective reduced speed and wherein for capacity factors and for total pressure ratios that are not listed in the table, the corresponding values for the reduced speeds of the respective compressor are established by means of an interpolation method.

11. The method according to claim 9, wherein the capacity lines exhibit those pairs of values of reduced mass flow and reduced speed that effect for the actual inlet pressure to adapt to the desired inlet pressure, when the control function from the model total pressure ratio and the capacity factor establish a reduced desired speed for each compressor, from the pre-calculated table, and the control is carried out with the established reduced speeds.

12. The method according to claim 9, wherein the capacity lines are located between a surge and a choke characteristic, wherein the surge characteristic comprises operating states of the respective compressor in which, in case of a given reduced speed and a given reduced mass flow, a single pressure ratio to be reached cannot be maintained, and wherein the choke characteristic comprises operating states of the compressor, in which, in case of a defined reduced desired speed of the respective compressor, a decrease of the respective single pressure ratio does not result in a significantly increased reduced mass flow through the respective compressor.

Description

(1) Further details and advantages of the invention shall be illustrated by the following Figure descriptions of exemplary embodiments by means of Figures. Shown are

(2) FIG. 1: Performance map with capacity lines of a first compressor, which is arranged the furthest upstream of four compressors connected in series.

(3) FIG. 2: Performance map with capacity lines of a second compressor, which is arranged downstream of the first compressor;

(4) FIG. 3: Performance map with capacity lines of a third compressor, which is arranged downstream of the second compressor;

(5) FIG. 4: Performance map with capacity lines of a fourth compressor, which is arranged downstream of the third compressor;

(6) FIG. 5: Control field with the capacity lines of FIG. 1 for the first compressor;

(7) FIG. 6: Control field with the capacity lines of FIG. 2 for the second compressor;

(8) FIG. 7: Control field with the capacity lines of FIG. 3 for the third compressor;

(9) FIG. 8: Control field with the capacity lines of FIG. 4 for the fourth compressor;

(10) FIG. 9: Performance map with evenly distributed capacity lines of the first compressor of the compressors connected in series;

(11) FIG. 10: Performance map with evenly distributed capacity lines of the second compressor;

(12) FIG. 11: Performance map with evenly distributed capacity lines of the third compressor;

(13) FIG. 12: Performance map with evenly distributed capacity lines of the fourth compressor;

(14) FIG. 13: Control field with the capacity lines of FIG. 9 for the first compressor;

(15) FIG. 14: Control field with the capacity lines of FIG. 10 for the second compressor;

(16) FIG. 15: Control field with the capacity lines of FIG. 11 for the third compressor;

(17) FIG. 16: Control field with the capacity lines of FIG. 12 for the fourth compressor;

(18) FIG. 17: Plugging chart for carrying out the method according to the invention;

(19) FIG. 18: Flow chart for the determination of the capacity factor and of the model total pressure ratio.

(20) FIG. 19: Establishing a capacity line in the performance map of compressor V.sub.1

(21) FIGS. 1 to 4 show performance maps of four compressors V.sub.1, V.sub.2, V.sub.3, V.sub.4 connected in series for the compression of cryogenic helium in the area around 4K. The method according to the invention can also be used for the control of more or less than four compressors, depending on the total pressure ratio that is to be generated. Below a series arrangement of four compressors V.sub.1, V.sub.2, V.sub.3, V.sub.4 is being discussed as an example.

(22) The performance map of a compressor represents operating states of the compressor, which operating states can be described by a reduced mass flow and an individual pressure ratio assigned to said reduced mass flow, wherein each operating state is assigned a desired speed in the performance map, which desired speed is required to achieve the operating state.

(23) Such a performance map can be created for each compressor or is available for each compressor. A performance map can be created by measuring a plurality of different operating states and thereby characterizing the performance map, or also by a suitable software, which can display the compressor virtually.

(24) In the so-called equilibrium operation, i.e. when the system is running in the planned state, the helium, e.g., is compressed from approximately 15 mbar to 600 mbar. That is, around the design point (operating state for which the compressor series or the compressor system is designed) the compressor system has a total pressure ratio of approximately 40 (600 mbar/15 mbar).

(25) In each performance map of the four FIGS. 1-4, five capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 are drawn in that comprise the capacity factor X=0 (X.sub.00), 0.25 (X.sub.02), 0.5 (X.sub.05), 0.75 (X.sub.07), and 1 (X.sub.10). The distribution of these lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 is different for each compressor V.sub.1, V.sub.2, V.sub.3, V.sub.4 of the series. The specific distribution of the capacity lines X.sub.00, X.sub.02, X.sub.059 X.sub.07, X.sub.10 in the respective performance map of the compressor ensures that an increase of the capacity factor X generally leads to an increase in the total surge performance, such that a stable system operation is guaranteed. The transverse dotted lines represent states that show the same reduced speeds n.sub.1, n.sub.2, n.sub.3, n.sub.4 of the respective compressor V.sub.1, V.sub.2, V.sub.3, V.sub.4.

(26) During the operation of a system according to the method according to the invention, the compressors V.sub.1, V.sub.2, V.sub.3, V.sub.4 are run on the same capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07 X.sub.10 in each of the performance maps or control fields assigned to the respective compressor. I.e. that all compressors are run with the same capacity factor X.

(27) It further applies that the discharge state, in particular the discharge pressure p.sub.1 and the discharge temperature T.sub.1 of the first compressor V.sub.1 represent the inlet state of the second compressor V.sub.2. The discharge state of the second compressor V.sub.2 represents the inlet state of the third compressor V.sub.3 and the discharge state of the third compressor V.sub.3 in turn represents the inlet state of the fourth compressor V.sub.4.

(28) The product of the individual pressure ratios q.sub.1, q.sub.2, q.sub.3, q.sub.4 forms the actual total pressure ratio .sub.actual. By varying the capacity factor X, the distribution of the actual total pressure ratio .sub.actual across the compressor series changes. In other words, the different capacity factors X influence the distribution of the individual pressure ratios q.sub.1, q.sub.2, q.sub.3, q.sub.4 via the respective compressors V.sub.1, V.sub.2, V.sub.3, V.sub.4, which are composed differently, according to each capacity factor X, whereby the common mass flow changes throughout all compressors V.sub.1, V.sub.2, V.sub.3, V.sub.4 of the series.

(29) FIG. 1 shows the performance map of the first compressor V.sub.1 of the series arrangement of the four compressors V.sub.1, V.sub.2, V.sub.3, V.sub.4 with the five capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10. The capacity line X.sub.00 is located on the surge characteristic S.

(30) In order to guarantee stable operation of the compressor V.sub.1 and in order for the compressor V.sub.1 not to be run into surge state S, the capacity factor X, which is passed to the control function F, is limited to a minimum value X.sub.min of 0.05, if necessary. However, the capacity line X.sub.10 is not located on the choke characteristic C of compressor V.sub.1.

(31) FIG. 2 shows the performance map of the second compressor V.sub.2 of compressors V.sub.1, V.sub.2, V.sub.3, V.sub.4 arranged in series. Again, the capacity line X.sub.00 is located on the surge characteristic S and the capacity lines X.sub.05, X.sub.07 and X.sub.10 run similarly as in the performance map of compressor V.sub.1. Only the capacity line X.sub.02 runs further to the left here, close to capacity line X.sub.00.

(32) FIG. 3 shows the performance map of the third compressor V.sub.3 of compressors V.sub.1, V.sub.2, V.sub.3, V.sub.4 arranged in series. The capacity line X.sub.05 runs in the left area of the performance map here.

(33) FIG. 4 shows the performance map of the fourth compressor V.sub.4 of compressors V.sub.1, V.sub.2, V.sub.3, V.sub.4 arranged in series of the compressor system. Here, only the capacity line X.sub.10 is still located in the right area of the performance map. The capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07 are concentrated around the surge characteristic S. The reduced desired speeds n.sub.1, n.sub.2, n.sub.3, n.sub.4 for these capacity factors are calculated comparatively close to the so-called surge speed, that is, the reduced speed n.sub.4, at which compressor V.sub.4 moves into the surge state. To ensure stable operation of compressor V.sub.4, said reduced desired speeds n.sub.4 are limited to a range between 90-95% of the surge speed.

(34) The described distribution of capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 in the performance map or control field for each compressor V.sub.1, V.sub.2, V.sub.3, V.sub.4 effects that in case of an actual total pressure ratio .sub.actual that is too low, only the last compressor V.sub.4 downstream contributes to the generation of the actual total pressure ratio .sub.actual. All of the foregoing compressors V.sub.1, V.sub.2, V.sub.3 rotate just fast enough as not to create a flow resistance.

(35) FIGS. 5 to 8 show control fields of compressors V.sub.1, V.sub.2, V.sub.3, V.sub.4. In the control field, the reduced speed n.sub.1, n.sub.2, n.sub.3, n.sub.4 is applied as a function of the natural logarithm of the total pressure ratio . Furthermore, the capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 are recorded in the control field, the run of which lines and their distribution are in particular predefined by the run and the distribution of the capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 in the performance maps of compressors V.sub.1, V.sub.2, V.sub.3, V.sub.4.

(36) For the transformation of the capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 from the performance map into the control field, a plurality of working points on the capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 of the compressor V.sub.1 are arithmetically followed via the capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 of all subsequent compressors (V.sub.2, V.sub.3 and V.sub.4) to the system output. This calculation is always based on the assumption that the state of the fluid at the entry of each additional compressor corresponds to the discharge state of the preceding compressor. For each working point the total pressure ratio 7 and the associated reduced speeds n.sub.1, n.sub.2, n.sub.3, n.sub.4 are determined.

(37) Based on these control fields the reduced (desired) speeds n.sub.2, n.sub.3, n.sub.4 for the respective compressor can be determined for each given value pair consisting of the capacity factor X and the total pressure ratio . These reduced (desired) speeds n.sub.1, n.sub.2, n.sub.3, n.sub.4 are converted by means of measured temperatures into absolute desired speeds. Along the capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 the control function F can be explicitly read from the control field.

(38) Controlling the Series in Different States:

(39) The method according to the invention is suitable for controlling compressors V.sub.1, V.sub.2, V.sub.3, V.sub.4, in particular during the equilibrium operation, during which only low or slow inlet pressure fluctuations and discharge pressure fluctuations are to be expected. However, this method is also suitable for the so-called pump-up (the desired inlet pressure p.sub.desired is higher than the actual inlet pressure p.sub.actual), or pump-down (the desired inlet pressure p.sub.desired is lower than the actual inlet pressure p.sub.actual) from states that are comparatively far from equilibrium, which is an indicator of the stability of the method.

(40) Example Equilibrium Operation:

(41) In the equilibrium operation, only small fluctuations of the actual inlet pressure p.sub.actual occur, which is at about 20 mbar. At the output of the series, the output pressure p.sub.4 fluctuates between, for example, 450 mbar and 500 mbar. The fluctuations are caused, for example, in the variable mass flow and the subsequent reaction of downstream volumetric machines after the compressor series. The actual total pressure ratio p.sub.actual lies consequently between 450 mbar/20 mbar=22.5 or 500 mbar/20 mbar=25. Thus, the natural logarithm of the actual total pressure ratio .sub.actual is in the value range of 3.11-3.22. In this total operating state, the capacity factor X is about 0.5. For the fluctuating actual total pressure ratio .sub.actual of 3.11-3.22, the respectively reduced desired speeds n.sub.1, n.sub.2, n.sub.3, n.sub.4 four the four compressors V.sub.1, V.sub.2, V.sub.3, V.sub.4 can be found in FIGS. 5 to 8.

(42) Example Pump Down:

(43) The actual inlet pressure p.sub.actual is bigger than the desired inlet pressure p.sub.desired (e.g, actual inlet pressure p.sub.actual=100 mbar, desired inlet pressure p.sub.desired=20 mbar). The actual discharge pressure p.sub.4 fluctuates between 450 mbar and 500 mbar, i.e. the logarithm of the actual total pressure ratio .sub.actual fluctuates between 1.5 and 1.6. Due to the high deviation of the actual inlet pressure p.sub.actual from the desired inlet pressure p.sub.desired the capacity factor X is continuously increased (e.g., from 0.5 to 1).

(44) In a logarithmic actual total pressure ratio .sub.actual of 1.5, an increase of the capacity factor from 0.5 to 1 effects the following: The reduced desired speed n.sub.1 for the first compressor V.sub.1 is slightly increased (FIG. 5). The reduced desired speed n.sub.2 for the second compressor V.sub.2 is increased by more than the reduced desired speed of the first compressor V.sub.1 (FIG. 6). The reduced desired speed n.sub.3 for the third compressor V.sub.3 is reduced for the capacity factors X between 0.5 and 0.75, and is increased again between 0.75 and 1 (FIG. 7). The reduced desired speed n.sub.4 is increased for capacity factors X between 0.5 and 0.75 and again reduced between 0.75 and 1 (FIG. 8).

(45) Through this control, the actual inlet pressure p.sub.actual adjusts to the desired inlet pressure p.sub.desired, wherein the capacity factor X, depending on the actual total pressure ratio .sub.actual is adapted, and eventually, when reaching the desired inlet pressure p.sub.desired, drops to approximately 0.5 again.

(46) In FIGS. 5 to 8 this control can be seen as follows: By increasing the capacity factor X, one moves at first generally vertically in the control field, i.e. the reduced desired speeds n.sub.1, n.sub.2, n.sub.3, n.sub.4 are (generally) increased for each compressor V.sub.1, V?, V.sub.3, V.sub.4. This causes a change of the total pressure ratio .sub.actual because an increased reduced mass flow {dot over (m)}.sub.1, {dot over (m)}.sub.2, {dot over (m)}.sub.3, {dot over (m)}.sub.4 results as a consequence of the increased speeds n.sub.1, n.sub.2, n.sub.3, n.sub.4. Thereby, the actual total pressure ratio .sub.actual increases, since the actual inlet pressure p.sub.actual decreases. An increasing actual total pressure ratio .sub.actual is now causing a substantially horizontal movement in the control field, such that the capacity factor X is generally down-regulated again. If the actual inlet pressure p.sub.actual now corresponds to the desired inlet pressure p.sub.desired, the capacity factor X is approximately 0.5. At this value of the capacity factor X and the logarithm of the design total pressure ratio .sub.design (of, for example, 3.5), the series of compressors is running with the highest efficiency.

(47) Example Pump Up:

(48) The actual inlet pressure p.sub.actual is smaller than the desired inlet pressure p.sub.desired (e.g. actual inlet pressure p.sub.actual=20 mbar, desired inlet pressure p.sub.desired=100 mbar). The actual discharge pressure p.sub.4 fluctuates between 450 mbar and 500 mbar, i.e. the logarithm of the actual total pressure ratio .sub.actual fluctuates between 3.11 and 3.22. Due to the deviation of the actual inlet pressure to the desired inlet pressure, the capacity factor X is reduced (e.g. from 0.5 to 0).

(49) If the capacity factor X at a logarithmic actual total pressure ratio .sub.actual of 3.11 decreases from 0.5 to 0, the following occurs: The reduced desired speed n.sub.1 of the first compressor V.sub.1 is increased (FIG. 5). The reduced desired speed n.sub.2 for the second compressor V.sub.2 is initially increased for the capacity factors X between 0.5 and 0.25 and then again reduced between 0.25 to 0 (FIG. 6). The reduced desired speed n.sub.3 of the third compressor V.sub.3 is reduced (FIG. 7). The reduced desired speed n.sub.4 of the fourth compressor V.sub.4 is reduced (FIG. 8).

(50) In this way, the actual total pressure ratio .sub.actual is reduced and therefore actual inlet pressure p.sub.actual strives towards the desired inlet pressure p.sub.desired.

(51) This type of regulation is especially advantageous for operating states or actual total pressure states .sub.actual close to the design total pressure ratio .sub.design. During the control of states that deviate significantly from the design total pressure ratio .sub.design, the capacity factor X is run in saturation (i.e. 0 or 1, or 0.05 or 0.9), yet the actual total pressure ratio .sub.actual does not necessarily change, since, for example, two capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 overlap at these states. An increase or reduction of the capacity factor X does not result in a change of the reduced desired speed n.sub.1, n.sub.2, n.sub.3, n.sub.4 there. In this case, the method according to the invention controls as follows:

(52) Example Pump Down:

(53) The actual discharge pressure p.sub.4 is, for example, at 450 mbar, and the actual inlet pressure p.sub.actual is at 350 mbar. The logarithm of the actual total pressure ratio .sub.actual is therefore approximately at 0.25. The desired inlet pressure p.sub.desired is 20 mbar. The capacity factor X is thus increased due to the difference between the actual and the desired inlet pressure.

(54) In FIG. 8, which shows the control field of the fourth compressor V.sub.4, it can be seen that an increase of the capacity factor X from 0.75 to 1 (or to 0.9, due to the limitation to the maximum value X.sub.max) does not necessarily result in an increase of the reduced desired speed n.sub.4. The actual total pressure ratio .sub.actual would not change any further then. In this case, the actual total pressure ratio .sub.actual is replaced or adapted by a model total pressure ratio .sub.model (cf. above).

(55) However, it could also occur that the increase leads to higher desired speeds in two compressors of the series, and to a reduction of the desired speed in the other two compressors. If the total reaction of the chain can reduce the actual inlet pressure p.sub.actual, the system can still continue to work with the actual total pressure ratio .sub.actual, otherwise the actual total pressure ratio .sub.actual is replaced by the model total pressure ratio .sub.model, as described.

(56) The model total pressure ratio .sub.model is slightly larger than the actual total pressure ratio .sub.actual. Thus, in the control field of the fourth compressor V.sub.4, movement occurs horizontally along the capacity line X.sub.10 of 1 (or 0.9). Consequently, one moves out of the overlapping region of the capacity lines X.sub.07 and X.sub.10, such that the control based on the model total pressure ratio .sub.model and the capacity factor X continues to work effectively. Once the capacity factor X is no longer in saturation, i.e. when the proportional integral value PI is no longer above the maximum value X.sub.max of the capacity factor X, the model total pressure ratio .sub.model equals the actual total pressure ratio .sub.actual.

(57) Rational Arrangement of the Capacity Lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 in the Performance Map of each Compressor V.sub.1, V.sub.2, V.sub.3, V.sub.4:

(58) FIGS. 9 to 12 show an even distribution of the capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 in the performance map for each compressor V.sub.1, V.sub.2, V.sub.3, V.sub.4. This type of distribution involves several disadvantages, the elimination of which results, for example, in distributions such as shown in FIGS. 1 to 4.

(59) Along the capacity line X.sub.10, the second and the third compressor V.sub.2, V.sub.3 (FIGS. 10 and 11) display very high reduced mass flows {dot over (m)}.sub.2, {dot over (m)}.sub.3 on top of very high reduced speeds n.sub.2, n.sub.3 of the respective compressor. In this area, the efficiency of the two compressors V.sub.2, V.sub.3 drops significantly and the discharge temperature increases, thus increasing the risk, in particular at the third compressor V.sub.3, of a too high speed (overspeed).

(60) Furthermore, by increasing the capacity factor X (i.e., in particular, if the actual inlet pressure p.sub.actual significantly deviates from the desired inlet pressure p.sub.desired), according to expectations, a higher reduced desired speed n.sub.2, n.sub.3 should be achieved. FIG. 13 shows, however, that during an even distribution of the capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 in the performance map, the reduced speeds n.sub.1 in the performance map, for example, for the first compressor V.sub.1 (FIG. 9), are reduced to a pressure ratio of approximately 3 for increasing capacity factors X, which would be an undesired control. The goal is, after all, an increase of the reduced speeds n.sub.1, in order to reduce the actual inlet pressure p.sub.actual.

(61) Furthermore, the fourth compressor V.sub.4 has temporarily very high reduced speeds n.sub.4 at low actual total pressure ratios .sub.actual (FIG. 16). This applies, in particular, to capacity line X.sub.10. Very high reduced speeds n.sub.1 point to very high speeds and high temperatures, which characterizes an inefficient operational state.

(62) An uneven distribution of the capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 in the performance map, on the other side, effects the following favorable characteristics: FIG. 5 shows that the reduced speeds m from the first compressor V.sub.1 increase along the capacity lines X.sub.02, X.sub.05, X.sub.07 and X.sub.10, in each case in the actual total pressure ratio .sub.actual. Thereby, the undisturbed operation of the most important compressor with the highest single pressure ratio is ensured during a pump down.

(63) FIGS. 1 to 4 show that no compressor is driven into the choke state (i.e. to the choke characteristic C), thus guaranteeing a high efficiency.

(64) FIGS. 5 to 8 show further, that each compressor V.sub.1, V.sub.2, V.sub.3, V.sub.4, at a certain actual total pressure ratio .sub.actual (or the logarithm thereof) reaches a reduced desired speed n.sub.1, n.sub.2, n.sub.3, n.sub.4 of 1 and remains in this area. (At low pressure conditions .sub.actual the fourth compressor V.sub.4, at medium pressure conditions, the second and third compressors V.sub.2, V.sub.3, and at high pressure conditions .sub.actual, as can be found at the design point, the first compressor V.sub.1). This behavior ensures an undisturbed pump down and reduces the risk of overspeeding.

(65) High capacity factors X do not always lead to higher reduced speeds n.sub.1, n.sub.2, n.sub.3, n.sub.4, neither at evenly distributed capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10, nor at unevenly distributed capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10. However, in the case of unevenly distributed capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10, there is the possibility to achieve a consistent increase of the reduced desired speeds n.sub.1, n.sub.2, n.sub.3, n.sub.4 by increasing the total pressure ratio n to a model total pressure ratio .sub.model.

(66) The criteria for the distribution of the capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 in the performance map of each compressor V.sub.1, V.sub.2, V.sub.3, V.sub.4 may be derived from the following principles. By defining/distributing the capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10, the regulation function F for the reduced speeds n.sub.1, n.sub.2, n.sub.3, n.sub.4 is also determined.

(67) The most critical compressor, usually the first compressor V.sub.1 in the series, must display reduced speeds n.sub.1 for increasing capacity factors X from X=0 to X=1, which reduced speeds should be as steady and continuously increasing as possible, as well as increasing reduced speeds n.sub.1 for increasing total pressure ratios . No compressor is to be operated on the choke or surge characteristic C, S. No compressor must be controlled to an overspeed, since otherwise machine safety is not guaranteed.

(68) The compressors V.sub.1, V.sub.2, V.sub.3, V.sub.4 must reach their design points (economic operating states) successively and, upon reaching the design point, the reduced desired speed value n.sub.1, n.sub.2, n.sub.3, should stay around 1 (at a tolerance of approximately 5%). I.e., during low total pressure ratios, the total pressure ratio should be generated by the fourth (last) compressor V.sub.4 of the series, wherein, during an increasing total pressure ratio, when the fourth compressor is already running at the design point, the third compressor V.sub.3, and in case of a further increase of the total pressure ratio, the second compressor V.sub.2 is connected, and finally the first compressor V.sub.1, such that in the end all compressors are operated on their respective design points.

(69) At the design point, at which all the compressors V.sub.1, V.sub.2, V.sub.3, V.sub.4 run at approximately the reduced speed n.sub.1, n.sub.2, n.sub.3, n.sub.4 equal to 1, the compressors with the highest single pressure ratio q.sub.1, q.sub.2, q.sub.3, q.sub.4 must (if possible) display increasing reduced desired speeds n.sub.1, n.sub.2, n.sub.3, n.sub.4 for increasing capacity factors X, such that the control around the design point can be carried out quickly. At the design point, the first compressor V.sub.1 usually shows the highest single pressure ratio q.sub.1.

(70) Furthermore, the capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 must in particular meet the following conditions in the performance map: They must be located between a surge characteristic S and a choke characteristic C. Each capacity line X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 assigns each reduced mass flow {dot over (m)}.sub.1, {dot over (m)}.sub.2, {dot over (m)}.sub.3, {dot over (m)}.sub.4 exactly one single pressure ratio q.sub.1, q.sub.2, q.sub.3, q.sub.4. Each capacity line X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 extends along the entire value range of the reduced mass flow {dot over (m)}.sub.1, {dot over (m)}.sub.2, {dot over (m)}.sub.3, {dot over (m)}.sub.4.

(71) FIG. 19 shows, using a performance map for the first compressor V.sub.1 as an example, that for a given reduced mass flow {dot over (m)}.sub.1 a plurality of individual pressure ratios q.sub.1 is possible. A capacity line, for example, capacity line X.sub.05, then determines how the compressor for a certain capacity factor within the series is to be regulated for the series to work as efficiently as possible. It can be seen that the capacity line X.sub.05 does not run exactly in the center of the maximum and minimum single pressure ratios (marked by circles in FIG. 19) that are possible for a given reduced mass flow {dot over (m)}.sub.1.

(72) Example for Calculating the Input and Discharge States Above a Compressor and for Determining the Control Behavior and the arrangement of the Capacity Lines of the Compressor Series:

(73) For a given inlet temperature (4.05K) and a given inlet pressure (24 mbar) at the first compressor V.sub.1 and a given capacity factor X=0, which lies on the capacity line X.sub.00, the single pressure ratio q.sub.1 for all reduced mass flows and all (reduced) speeds is calculated: Based on FIG. 19, a reduced mass flow {dot over (m)}.sub.1 of 0.3 results in a reduced speed n.sub.1 of 0.53.

(74) The (absolute) speed is calculated according to the formula for the conversion from reduced to absolute speeds to 600 Hz:

(75) $n_{\mathrm{abs}}=n_{\mathrm{red}}.Math.n_{\mathrm{Design}}.Math.\sqrt{\frac{T_{\mathrm{actual}}}{T_{\mathrm{Design}}}}$
Wherein n.sub.abs is the absolute speed, n.sub.red is the reduced speed (in this case n.sub.1) and n.sub.design, is the speed, for which the compressor was designed. TT.sub.actual is the actual temperature of the fluid, and T.sub.design is the delivery temperature or design temperature of the compressor. Based on the reduced mass flow {dot over (m)}.sub.1 the (absolute) mass flow is calculated from the following formula to 16 g/s:

(76) ${\stackrel{.}{m}}_{\mathrm{red}}=\frac{m_{\mathrm{actual}}}{m_{\mathrm{Design}}}.Math.\frac{p_{\mathrm{Design}}}{p_{\mathrm{actual}}}.Math.\sqrt{\frac{T_{\mathrm{actual}}}{T_{\mathrm{Design}}}}$
wherein {dot over (m)}.sub.red is the reduced mass flow through the compressor, {dot over (m)}.sub.ist is the current mass flow, {dot over (m)}.sub.Design refers to the mass flow, for which the respective compressor is designed, p.sub.design constitutes the design pressure at the respective compressor, T.sub.design is the design temperature, and p.sub.actual is the actual inlet pressure on the respective compressor.

(77) The assumption is that the diameter of the compressor wheel of the compressor V.sub.1 is, for example, 100 mm. Now, based on the diameter and the absolute speed, a peripheral speed at *100 mm*60 Hz=188.49 m/s is calculated.

(78) Hereinafter, a flow rate is calculated, in particular the tangential flow rate in compressor V.sub.1. Since the exit face of the compressor wheel is known, the flow rate can be calculated by means of the fluid density at the output of compressor V.sub.1. However, the density is a function of the discharge conditions (in particular of the pressure and the temperature). Therefore, this step is calculated iteratively, as will be explained in the following. The density is assumed to be 0.27 kg/m.sup.3 for example. I.e. based on 16 g/s, the density of 0.27 kg/m.sup.3 and the exit face of the compressor, a flow rate of the fluid can be calculated. By adopting a flow angle (for example, based on the geometry of the compressor wheel), the tangential flow rate is based on the flow rate of the fluid.

(79) By means of the turbo machine equation (Euler equation) the enthalpy increase is calculated based on the product of the tangential flow rate and the peripheral speed of the compressor wheel.

(80) The enthalpy increase at compressor V.sub.1 is converted into a temperature increase by means of the known heat capacity of the fluid. Furthermore, the efficiency of compressor V.sub.1 at the respective operating state (reduced speed n.sub.1, reduced mass flow {dot over (m)}.sub.1) is established in the performance map. The pressure increase results from the temperature increase and the efficiency of the compressor at the respective operating state.

(81) Thus, the discharge temperature T.sub.1 and the discharge pressure P.sub.1 of the first compressor V.sub.1 of the series are established. Next, the density of the fluid is calculated based on these two variables, and then compared with the originally assumed density value. If the density values deviate from one another, the previous steps for calculating the density (in particular by variation of the assumed density) are repeated until the calculated density corresponds to the assumed density. As already mentioned above, discharge pressure P.sub.1 and discharge temperature T.sub.1 form the inlet state of the subsequent compressor V.sub.2.

(82) It is assumed that T.sub.1=9K and p.sub.1=100 mbar. The (absolute) mass flow is the same for all compressors, i.e. equals 16 g/s. Based on these variables and capacity factor X), discharge temperature T.sub.2, and discharge pressure p.sub.2 of the second compressor V.sub.2 of the series are calculated analogously to the procedure above. Using this model, the behavior of the compressor series V.sub.1, V.sub.2, V.sub.3, V.sub.4 can be pre-calculated for all capacity factors X and distributions of capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 in the performance maps. The run and the arrangement of capacity lines X.sub.00, X.sub.02, X.sub.05, X.sub.07, X.sub.10 in the respective performance map of the respective compressor V.sub.1, V.sub.2, V.sub.3, V.sub.4 can thus be optimized using this calculation model of the compressor series.

(83) Alternatively, in order to use the Euler equation, tables can be generated by generating a table from each performance map and reading the pressure ratios out of this table as a function of the reduced mass flow and the reduced speed.

(84) Examples for the Calculation of the Proportional Integral Value PI:

(85) In the method according to the invention, a PI controller (proportional integral controller) establishes in particular a proportional value prop from the difference between the desired inlet pressure p.sub.desired and the actual inlet pressure p.sub.actual at the first compressor V.sub.1 of the series. In transient operation, i.e., for example, when starting up the system (pump down), the desired inlet pressure p.sub.desired is smaller than the actual inlet pressure p.sub.actual. Now, the proportional value prop is the difference between the desired and the actual inlet pressure, multiplied by an amplification factor k:
prop=k(p.sub.desiredp.sub.actual).
In addition, the PI controller calculates an integral value int.sub.t=n+1 based on this proportional value. Hereby, the proportional value prop is multiplied by a cycle time t, divided by an integral time T, and added to the integral value of the preceding cycle int.sub.t=n:

(86) ${\mathrm{int}}_{t=n+1}={\mathrm{int}}_{t=n}+\mathrm{prop}.Math.\frac{t}{T}$

(87) Theoretically, the capacity factor X can adopt values between 0 (X.sub.surge=0, surge regime) and 1 (X.sub.choke=1, choke regime). In order for the compressor to not be driven into these regimes, the capacity factor X is limited to values between the minimum value X.sub.min=X.sub.surge+0.05 and the maximum value X.sub.max=X.sub.choke01.

(88) In the same way, an upper and a lower limit value int.sub.max and int.sub.min of the integral value int are derived from X.sub.max or X.sub.min and from the natural logarithm of the total pressure ratio ln(.sub.actual):
int.sub.max=X.sub.max+ln(.sub.actual),
int.sub.min=X.sub.min+ln(.sub.actual).

(89) Since the measured actual total pressure ratio .sub.actual continuously increases in transient operation (pump down) (the actual inlet pressure p.sub.actual continuously decreases), the limit values of the integral value consequently continuously increases as well. In the reverse case (pump up), i.e. when the desired inlet pressure p.sub.desired is smaller than the actual inlet pressure p.sub.actual, these limit values continuously decrease.

(90) When the integral value int.sub.t=n+1 becomes bigger or smaller than the upper or lower limit value int.sub.max, int.sub.min, it is limited to the respective limit value.

(91) Proportional value prop and integral value int.sub.t=n+1 are added to generate the proportional integral value PI.
PI=prop+int.sub.t=n+1

(92) When all compressors V.sub.1, V.sub.2, V.sub.3, V.sub.4 run in series at their design points, the compressor system reaches its design or working point at a design total pressure ratio .sub.design.

(93) When the proportional integral value PI is smaller than the sum of the maximum value of the capacity factor X.sub.max and of the natural logarithm of the design total pressure ratio .sub.design, the capacity factor X is calculated based on the difference of the proportional integral value PI and the natural logarithm of the actual total pressure ratio .sub.actual. Otherwise, the proportional integral value PI is limited to the sum of the natural logarithm of the design total pressure ratio .sub.design and the maximum value of the capacity factor X.sub.max, in particular for the calculation of the capacity factor X, i.e.:

(94) $\begin{array}{cc}X=\mathrm{PI}-\ln \left({}_{\mathrm{actual}}\right),& \mathrm{if}\mathrm{PI}<\ln \left({}_{\mathrm{design}}\right)+X_{\mathrm{choke}}\\ X=\ln \left({}_{\mathrm{design}}\right)+X_{\mathrm{choke}}-\ln \left({}_{\mathrm{actual}}\right)& \mathrm{otherwise}\end{array}$

(95) Based on the thus calculated capacity factor X, a decision is made, based on the method according to the invention, as to how the model total pressure ratio .sub.model is established. As described above, the model total pressure ratio .sub.model equals the actual total pressure ratio .sub.actual, when the thus determined capacity factor X lies between the minimum value and the maximum value X.sub.min, X.sub.max. If the capacity factor X is outside this value range, the model total pressure ratio .sub.model changes as described above by means of a saturation function. Next, the capacity factor X is limited to its minimum value or maximum value X.sub.min, X.sub.max, and then, in particular together with the model total pressure ratio .sub.model passed on to the control function F, which determines the reduced desired speed n.sub.1, n.sub.2, n.sub.3, n.sub.4 for the respective compressor V.sub.1, V.sub.2, V.sub.3, V.sub.4 based on these arguments.

(96) The reduced desired speeds n.sub.1, n.sub.2, n.sub.3, n.sub.4 for each compressor V.sub.1, V.sub.2, V.sub.3, V.sub.4 can, in particular, be recorded in a table (look-up table). This table can, in particular, be created by means of model calculations. According to the capacity factor X and the model total pressure ratio .sub.model, in particular a software for reading out the reduced desired speeds n.sub.1, n.sub.2, n.sub.3, n.sub.4 from the table can be used.

(97) Instead of a PI controller, a PID (proportional integral derivative) controller can be used. This is in particular advantageous when the mass flow volumes to be controlled are smaller than the volumes in cooling systems of the type described above, as rapid fluctuations in these relatively large volumes are rather rare. When small volumes are to be controlled, it is advantageous to also have a fast-reacting controlling component, such as a PID controller, which reacts faster than a PI controller due to its differentiating component.

LIST OF REFERENCE SYMBOLS

(98) TABLE-US-00001 V.sub.1, V.sub.2, V.sub.3, V.sub.4 first, second, third and fourth compressor n.sub.1, n.sub.2, n.sub.3, n.sub.4 first, second, third and fourth reduced (desired) speed n.sub.abs absolute speed n.sub.design design speed p.sub.actual actual inlet pressure at the first compressor p.sub.desired desired inlet pressure at the first compressor p.sub.1, p.sub.2, p.sub.3, p.sub.4 discharge pressure downstream of the first, second, third, and fourth compressor T.sub.actual actual temperature at the inlet of the first compressor T.sub.design design temperature T.sub.1, T.sub.2, T.sub.3, T.sub.4 temperature at the output of the first, second, third, and fourth compressor {dot over (m)}.sub.1, {dot over (m)}.sub.2, {dot over (m)}.sub.3, {dot over (m)}.sub.4 reduced mass flow through the first, second, third, and fourth compressor X capacity factor X.sub.max maximum value of the capacity factor X.sub.min minimum value of the capacity factor X.sub.choke choke capacity factor X.sub.surge surge capacity factor X.sub.00, X.sub.02, X.sub.05, X.sub.10 capacity lines prop proportional value int integral value PI proportional integral value total pressure ratio .sub.model model overall pressure ratio .sub.actual actual total pressure ratio .sub.design design total pressure ratio F regulation function S surge characteristic C choke characteristic