A simple but energy-efficient HVAC control synthesis for data centers

TL;DR

This paper presents a simple, energy-efficient HVAC control method for data centers using model-free control synthesis, which simplifies tuning and achieves excellent temperature tracking under various conditions.

Contribution

It introduces a novel model-free control approach for HVAC systems in data centers, eliminating the need for complex mathematical modeling.

Findings

Excellent temperature tracking in simulations

Robust performance under CPU load variations

Simplified controller tuning

Abstract

The air conditioning management of data centers, a key question with respect to energy saving, is here tackled via the recent model-free control synthesis. Mathematical modeling becomes useless in this approach. The tuning of the corresponding intelligent proportional controller is straightforward. Computer simulations show excellent tracking performances in various realistic situations, like CPU load or temperature changes.

Equations

$$\dot{y}=F+\alpha u$$

$$u=-\frac{\hat{F}-{\dot{y}}^{\ast }+K_{P}e}{\alpha }$$

$$\dot{e}+K_{P}e=0$$

$$sY=\frac{\Phi }{s}+\alpha U+y(0)$$

$$Y+s\frac{dY}{ds}=-\frac{\Phi }{s^2}+\alpha \frac{dU}{ds}$$

$$\hat{F}(t)=-\frac{6}{{\tau }^3}{\int }_{t-\tau }^t\left[(\tau -2\sigma )y(\sigma )+\alpha \sigma (\tau -\sigma )u(\sigma )\right]d\sigma$$

$$\{\begin{array}{l}{\dot{T}}_{\text{IT}}={\alpha }_{11}{P}_{\text{IT}}-{\alpha }_{12}(T_{\mathrm{IT}}-T_{\text{Rack}})\\ {\dot{T}}_{\mathrm{Rack}}={\alpha }_{21}(T_{\mathrm{cAisle}}-T_{\mathrm{Rack}})+{\alpha }_{22}(T_{IT}-T_{\mathrm{Rack}})\\ {\dot{T}}_{\mathrm{cAisle}}={\alpha }_{31}(T_{\mathrm{Air},\mathrm{in}}-T_{\mathrm{cAisle}})+{\alpha }_{32}(T_{\mathrm{cAisle}}-T_{\mathrm{cWall}})\\ {\dot{T}}_{\mathrm{cWall}}={\alpha }_{41}(T_{\mathrm{out}}-T_{\mathrm{cWall}})+{\alpha }_{42}(T_{\mathrm{cWall}}-T_{\mathrm{cAisle}})\\ {\dot{T}}_{\mathrm{hAisle}}={\alpha }_{51}(T_{\mathrm{Rack}}-T_{\mathrm{hAisle}})+{\alpha }_{52}(T_{\mathrm{hAisle}}-T_{\mathrm{hWall}})\\ {\dot{T}}_{\mathrm{hWall}}={\alpha }_{61}(T_{\mathrm{out}}-T_{\mathrm{hWall}})+{\alpha }_{62}(T_{\mathrm{hWall}}-T_{\mathrm{hAisle}})\end{array}$$

Full text

Key words— Data centers, cloud computing, HVAC, PID, model-free control, intelligent proportional controller, tracking.

I Introduction

Two exciting advances in cloud computing [2], a fast growing industry in information technology, have been recently derived by the authors:

• •

improving resource elasticity [7] thanks to model-free control in the sense of [17],

• •

workload forecasting [18] via time series analysis as in [20].

Data centers, which are fundamental in this context, consume a huge amount of electrical energy [8, 27, 44]. Almost half of it is devoted to their cooling. The aim of this communication is to show that model-free control might provide also a most efficient control tool with respect to air conditioning.

Remark 1

HVAC*, i.e., heating, ventilation, and air conditioning, which is defined by Wikipedia as “the technology of indoor and vehicular environmental comfort” (see, e.g., [29]), plays therefore a key rôle (see, e.g., [11, 28]). The corresponding numbers of publications and patents are increasing rapidly.* 2. 2.

From an applied control engineering perspective, on/off and PID controllers seem to be widely used (see, e.g., **[9, 14, 15, 32, 37, 40]**, and the references therein). To a large extent this situation is explained by their conceptual simplicity. Nevertheless their tuning, which is too often a quagmire, might lead to poor performances. 3. 3.

Most of the model-based approaches rest on various optimization techniques (see, e.g., **[6, 12, 26, 34, 38, 43]**). Let us add that predictive control (see, e.g., **[13, 16, 31, 35, 36, 41, 45]**) is essential in that respect. Deriving sound mathematical modeling necessitates complex parameter identification and/or machine learning procedures in order to get convincing results (see, e.g., **[21]**).

Remark 2

Besides excellent existing results on the HVAC of greenhouses [30] and buildings [1, 5, 33, 39], model-free control has already given birth to many successful concrete applications (see the references in [17] and [4, 22]) including some patents.

Our paper is organized as follows. Model-free control is summarized in Section II. A simplified mathematical modeling via ordinary differential equations is sketched in Section III for the purpose of computer simulations. The performances of our control synthesis are presented and discussed in Section IV. Section V is devoted to some concluding remarks.

II What is model-free control?111See [17] for more details.

II-A Generalities

Replace the unknown or poorly known SISO system by ultra-local model

[TABLE]

where

• •

$u$ and $y$ are the input (control) and output variables,

• •

the derivation order of $y$ is $1$, like in most concrete situations,

• •

$\alpha\in\mathbb{R}$ is chosen by the practitioner such that $\alpha u$ and $\dot{y}$ are of the same magnitude.

The following explanations on $F$ might be useful:

• •

$F$ subsumes the knowledge of any model uncertainties and disturbances,

• •

$F$ is estimated via the measures of $u$ and $y$.

II-B Intelligent controllers

The loop is closed by an intelligent proportional controller, or iP,

[TABLE]

where

• •

$y^{\star}$ is the reference trajectory,

• •

$e=y-y^{\star}$ is the tracking error,

• •

$K_{P}$ is the usual tuning gain.

Combining equations (1) and (2) yields:

[TABLE]

where $F$ does not appear anymore. Local exponential stability is ensured if $Kp>0$:

• •

The gain $K_{P}$ is thus easily tuned.

• •

Robustness with respect to different types of disturbances and model uncertainties is achieved.

Remark 3

See [17] for a discussion about the equivalence between the iP (2) and proportional-integral controllers (PIs).

II-C Real-time estimation of $F$

The term $F$ in Equation (1) is estimated in real-time according to recent algebraic identification techniques [19]. It may be assumed to be “well” approximated by a piecewise constant function $\hat{F}$ (see, e.g., [10]). Rewrite then Equation (1) in the operational domain (see, e.g., [42]):

[TABLE]

where $\Phi$ is a constant. We get rid of the initial condition $y(0)$ by multiplying both sides on the left by $\frac{d}{ds}$:

[TABLE]

Noise attenuation is achieved by multiplying both sides on the left by $s^{-2}$. It yields in the time domain the real-time estimate, thanks to the equivalence between $\frac{d}{ds}$ and the multiplication by $-t$,

[TABLE]

where $\tau>0$ might be quite “small.”

III A simple mathematical model for

computer simulations

Our model, which is essential for computer simulations, is to a great extent borrowed from [13]. Figures 1-(a) and 1-(b) represent respectively the server air flow circulation and the simplified data center. Figure 2 is sketching the controller and permits to define various important variables. Basic thermodynamic laws lead to the differential equations

[TABLE]

where

• •

$P_{\rm IT}$ is the input power which corresponds to the CPU load,

• •

$T_{\rm Air,in}$ (resp. $T_{\rm IT}$) is the control (resp. output) variable,

• •

$\alpha_{\rm ij}$ are suitable parameters.

From a classic control-theoretic viewpoint,

• •

Equations (6) yields a system ($\Sigma$) with a single input $u=T_{\rm Air,in}$ and a single output $y=T_{\rm IT}$ (see also Figure 2),

• •

$P_{\rm IT}$ may be viewed as an external disturbance.

IV Some computer simulations

IV-A Basic facts

The following values of the parameters are inspired by [13]: $\alpha_{\rm 11}=2.7248$, $\alpha_{\rm 12}=-32.6975$, $\alpha_{\rm 21}=4.2997.10^{3}$, $\alpha_{\rm 22}=2.9632.10^{4}$, $\alpha_{\rm 31}=537.4670$, $\alpha_{\rm 32}=131.6406$, $\alpha_{\rm 41}=514.2857$, $\alpha_{\rm 42}=153.5354$, $\alpha_{\rm 51}=335.9169$, $\alpha_{\rm 52}=7.7166$, $\alpha_{\rm 61}=12$, $\alpha_{\rm 62}=9.6000$. Following again [13], the output $y$ of System ($\Sigma$) is assumed to track the setpoint $20.9\textdegree$ (degree Celsius). In Formulae (1)-(2), set $\alpha=10$, $K_{p}=1$. The sampling period is $1$ min.

Remark 4

Note that forecasting results via time series were used in [13]. They become pointless here.

IV-B Four preliminary scenarios

IV-B1 Sudden CPU load change

Figure 3-(a) exhibits a sudden change of the CPU load $P_{\rm IT}$. Figure 3-(d) shows that the setpoint is well tracked.

IV-B2 A more realistic CPU load change

It is given by lnagral, i.e., the Company to which two authors, M. Bekcheva and A. Moradi, belong, and is depicted in Figure 4-(a). Figures 4-(d) confirms a great tracking.

IV-B3 Sudden temperature change

Figure 5-(b) exhibits a sudden temperature of the data center temperature $T_{\rm out}$. Here again Figure 5-(d) depicts an excellent tracking.

IV-B4 Another reference trajectory

Some situations may necessitate, contrarily to Section IV-A, to replace the setpoint, i.e., a constant reference trajectory, by a more general one. As demonstrated by Figure 6, the tracking remains exceptional.

IV-C Sudden model change

Represent a sudden model change at time $t=2.7$ h by multiplying $\alpha_{21}$, $\alpha_{31}$, $\alpha_{51}$ in Equations (6) by $0.5$ and $1.5$.333In accordance with [13], those variations may be justified by the change of a coefficient called $\kappa$. If those changes would occur at time $t=0$, they should be interpreted as a model mismatch. The variables $P_{\rm IT}$ and $T_{out}$ remain unaltered and constant. Although the model-free control synthesis of Section IV-A remains unchanged, Figures 7 et 8 display excellent performances.

V Conclusion

The power usage effectiveness, or PUE, of data centers, although heavily criticized [24], seems to be the only measure for checking the energy saving quality today. It would however be meaningless to try applying this indicator here, in the context of such a paper. Section IV, which demonstrates that the tracking works well in rather stringent conditions, may convince the reader that our approach should be nevertheless quite efficient with respect to energy saving. The most important for future developments is of course the application of our method to real data. A positive outcome would lead to a critical simplification of the HVAC control management of data centers:

• •

irrelevance of complex and time-consuming mathematical modeling, which is inherently uncertain,

• •

forthright tuning.

Promising experiments with a greenhouse [30] and a building [33] comfort this hope. It would confirm thanks also to [7] that model-free control should become important in computer science (compare with [3, 23, 25]).