# A Distributed Optimization Method for Energy Saving of Parallel-Connected Pumps in HVAC Systems

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Rule Based Methods

#### 1.2. Mathematical Programming Methods

#### 1.3. Heuristic Methods

#### 1.4. Distributed Methods

## 2. Problem Formulation

#### 2.1. Pump Models

#### 2.2. Optimization Problem Formulation

**Problem**

**1.**

## 3. Proposed Algorithm

**Assumption**

**1.**

Algorithm 1: Breadth first search random sampling algorithm (BRS) |

## 4. Theoretical Analysis of the Proposed Algorithm

**Assumption**

**2.**

**Problem**

**2.**

**Assumption**

**3.**

**Theorem**

**1.**

**Proof**

**of Theorem 1.**

**Remark**

**1.**

## 5. Results and Discussion

#### 5.1. Set up of Study Cases

#### 5.2. Software Simulation Environment

#### 5.3. Other Algorithms to Be Compared with BRS

#### 5.4. The Violation of Constraint Satisfaction

#### 5.5. Performance Comparison

#### 5.6. Test of Flexibility

#### 5.7. Hardware Test of BRS

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

${a}_{P1,i},{a}_{P2,i},{a}_{P3,i},{b}_{P1,i},{b}_{P2,i},{b}_{P3,i}$ | pump performance parameters |

${H}_{i}$ | head of pump i (m) |

${H}_{0}$ | total demand head (m) |

m | total number of pumps |

${n}_{i}$ | actual speed of pump i |

${n}_{0,i}$ | rated speed of pump i |

${\mathcal{N}}_{i}$ | set of child nodes of node i |

${P}_{k}^{\ast}$ | optimal estimation of total power consumption after the k-th sample |

${P}_{i}$ | power consumption of pump i |

${P}_{T,i,k}$ | partial summation power consumption over subtree of node i |

${Q}_{T,i,k}$ | partial summation flow rate over subtree of node i |

${Q}_{i}$ | flow rate of pump i (L/s) |

${Q}_{0}$ | total demand flow rate (L/s) |

${\mathit{R}}_{T,i,\delta}$ | combination of node’s optimal speed ratio neighborhood over the subtree of node i Greek letters |

${\omega}_{i}$ | speed ratio of node i |

${\omega}^{-}$ | lower limitation of speed ratio |

${\omega}_{i,k}$ | the k-th speed ratio sample of node i |

${\omega}_{T,i,k}$ | the combination of the speed ratio over subtree of node i |

${\omega}_{i}^{\ast}$ | optimal speed ratio |

${\tilde{\omega}}_{i,k}^{\ast}$ | optimal estimation of speed ratio after the k-th sample |

${\omega}^{\ast}$ | optimal combination of speed ratio |

${\tilde{\omega}}_{\mathbf{k}}^{\ast}$ | optimal estimation combination of speed ratio after the k-th sample |

${\xi}_{P}$ | energy saving rate |

${\eta}_{i}$ | mechanical efficiency |

${\eta}_{M,i}$ | motor efficiency |

${\eta}_{VFD,i}$ | variable frequency efficiency |

$\Delta $ | violation of $\sum {Q}_{i}-{Q}_{0}$ |

Subscripts | |

i | pump i |

k | the k-th sample |

T | the subtree |

## Abbreviations

HVAC | Heating, ventilating, and air-conditioning system |

BRS | Breadth first search random sampling algorithm |

BFS | Breadth first search algorithm |

SC | Sequence control strategy |

SLSQP | Sequential least squares programming algorithm |

GA | Genetic algorithm |

TA | Transfer algorithm |

DSP | Distributed simulation platform |

## Appendix A

#### Appendix A.1

Algorithm A1: Breadth First Search algorithm [34] |

#### Appendix A.2

Algorithm A2: Message processing algorithm |

Input: the child nodes set ${\mathcal{N}}_{i}$, the flow rate ${Q}_{i,k}$, the power consumption ${P}_{i,k}$, the speed ratio ${\omega}_{i,k}$, ${Q}_{T,i,k-1},{P}_{T,i,k-1},{\omega}_{T,i,k-1}$; |

Output: every intelligent node’s the partial summation flow rate ${Q}_{T,i,k}$,the partial summation power consumption ${P}_{T,i,k}$ and the combination of the speed ratio ${\omega}_{T,i,k}$ over subtree of node i; |

1 Initialize ${Q}_{T,i,k}={Q}_{i,k}$, ${P}_{T,i,k}={P}_{i,k}$, ${\omega}_{T,i,k}=[{\omega}_{T,i,k},{\omega}_{i,k}]$; |

2 Send ${Q}_{T,i,k-1},{P}_{T,i,k-1},{\omega}_{T,i,k-1}$ message to parent’s input message queue and receive child node $j,j\in {\mathcal{N}}_{i}$ message; |

3 ${Q}_{T,i,k}={Q}_{T,i,k}+{\sum}_{j\in {\mathcal{N}}_{i}}{Q}_{T,j,{k}_{j}}$; |

4 ${P}_{T,i,k}={P}_{T,i,k}+{\sum}_{j\in {\mathcal{N}}_{i}}{P}_{T,j,{k}_{j}}$; |

5 ${\omega}_{T,i,k}=[{\omega}_{T,i,k},\bigcup _{j\in {\mathcal{N}}_{i}}{\omega}_{T,j,{k}_{j}}]$; |

6 return ${Q}_{T,i,k},{P}_{T,i,k},{\omega}_{T,i,k}$; |

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**Figure 2.**The process of constructing a breadth first search algorithm (BFS) tree from the root node marked with the star.

**Figure 6.**The topologies of pump intelligent nodes. (

**a**) Toplogy with all 6 pumps working in normal; (

**b**) Topolgy with Pump 4 out of service; (

**c**) Topolgy with Pump 4 back to normal.

Control Framework | Groups | Methods |
---|---|---|

Rule based | Zhao et al. [8]; Ma et al. [9]; Gao et al. [10] | |

Centralized | Mathematical programming | Bonvin et al. [11]; Horváth et al. [12]; Koor et al. [13] |

Heuristic | Wang et al. [17]; Rong et al. [18]; Olszewski et al. [22] | |

Distributed | Distributed | Dai et al. [24] |

Pump | Performance Parameters | |||||
---|---|---|---|---|---|---|

${\mathit{a}}_{\mathit{P}\mathbf{1}\mathit{,}\mathit{i}}$ | ${\mathit{a}}_{\mathit{P}\mathbf{2}\mathit{,}\mathit{i}}$ | ${\mathit{a}}_{\mathit{P}\mathbf{3}\mathit{,}\mathit{i}}$ | ${\mathit{b}}_{\mathit{P}\mathbf{1}\mathit{,}\mathit{i}}$ | ${\mathit{b}}_{\mathit{P}\mathbf{2}\mathit{,}\mathit{i}}$ | ${\mathit{b}}_{\mathit{P}\mathbf{3}\mathit{,}\mathit{i}}$ | |

PUMP-A | −0.0046 | 0.0696 | 60.271 | −0.0002 | 0.0254 | 0.0616 |

PUMP-B | −0.0112 | 0.1358 | 54.841 | −0.0005 | 0.0316 | 0.2582 |

${\mathit{H}}_{0}$(m) | ${\mathit{Q}}_{0}$(L/s) | |
---|---|---|

Case 1 | 26 | 86 |

Case 2 | 29 | 117 |

Case 3 | 36 | 248 |

Case 4 | 39 | 288 |

Algorithm | Control Framework | Simulation Environment |
---|---|---|

BRS | Distributed | DSP |

SC | Centralized | Python |

SLSQP | Centralized | Python |

GA | Centralized | Python |

TA | Distributed | DSP |

Case | Method | PUMP-A | PUMP-A | PUMP-A | PUMP-A | PUMP-B | PUMP-B | $\mathsf{\Delta}$ (L/s) | ${\mathit{P}}^{\mathbf{*}}$ (KW) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\omega}}_{\mathbf{1}}$ | ${\mathit{Q}}_{\mathbf{1}}$ | ${\mathit{\omega}}_{\mathbf{2}}$ | ${\mathit{Q}}_{\mathbf{2}}$ | ${\mathit{\omega}}_{\mathbf{3}}$ | ${\mathit{Q}}_{\mathbf{3}}$ | ${\mathit{\omega}}_{\mathbf{4}}$ | ${\mathit{Q}}_{\mathbf{4}}$ | ${\mathit{\omega}}_{\mathbf{5}}$ | ${\mathit{Q}}_{\mathbf{5}}$ | ${\mathit{\omega}}_{\mathbf{6}}$ | ${\mathit{Q}}_{\mathbf{6}}$ | ||||

Case 1 | BRS | 0.7340 | 43.476 | 0.7304 | 42.524 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 25.378 |

SC | 0.7199 | 39.623 | 0 | 0 | 0 | 0 | 0 | 0 | 0.9 | 46.377 | 0 | 0 | 0 | 32.970 | |

${Q}_{0}=86$ | SLSQP | 0.7322 | 42.995 | 0.7322 | 42.995 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.01 | 25.377 |

GA | 0.8046 | 59.632 | 0 | 0 | 0 | 0 | 0 | 0 | 0.7528 | 26.342 | 0 | 0 | 0.026 | 26.902 | |

TA | 0.7354 | 43.831 | 0 | 0 | 0 | 0 | 0 | 0 | 0.7269 | 21.296 | 0.7269 | 21.296 | 0.423 | 27.288 | |

Case 2 | BRS | 0.8285 | 58.509 | 0.8284 | 58.491 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 38.757 |

SC | 0.9066 | 74.038 | 0 | 0 | 0 | 0 | 0 | 0 | 0.9 | 42.962 | 0 | 0 | 0 | 45.697 | |

${Q}_{0}=117$ | SLSQP | 0.8285 | 58.509 | 0.8285 | 58.509 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.018 | 38.764 |

GA | 0.7835 | 48.045 | 0.8798 | 68.960 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.005 | 39.539 | |

TA | 0.7790 | 46.901 | 0.7803 | 47.255 | 0 | 0 | 0 | 0 | 0.7710 | 23.201 | 0 | 0 | 0.357 | 39.614 | |

Case 3 | BRS | 0.9051 | 61.208 | 0.9111 | 62.552 | 0.9076 | 61.775 | 0.9107 | 62.465 | 0 | 0 | 0 | 0 | 0 | 101.322 |

SC | 0.9 | 60.036 | 0.9 | 60.036 | 0.9046 | 61.098 | 0 | 0 | 0.9 | 33.415 | 0.9 | 33.415 | 0 | 105.609 | |

${Q}_{0}=248$ | SLSQP | 0.9086 | 61.991 | 0.9086 | 61.991 | 0.9086 | 61.991 | 0.9086 | 61.991 | 0 | 0 | 0 | 0 | −0.036 | 101.317 |

GA | 0.9516 | 71.161 | 0.8425 | 45.289 | 0.9798 | 76.739 | 0.8780 | 54.792 | 0 | 0 | 0 | 0 | −0.019 | 103.741 | |

TA | 0.8833 | 56.087 | 0.8762 | 54.349 | 0.8762 | 54.349 | 0.8832 | 56.076 | 0.8665 | 27.381 | 0 | 0 | 0.242 | 102.540 | |

Case 4 | BRS | 0.9264 | 60.082 | 0.9417 | 63.626 | 0.9597 | 67.610 | 0.9667 | 69.121 | 0.8972 | 27.561 | 0 | 0 | 0 | 129.493 |

SC | 0.9 | 53.510 | 0.9 | 53.510 | 0.9 | 53.510 | 0.9 | 53.510 | 1.0 | 44.156 | 0.9086 | 29.804 | 0 | 134.518 | |

${Q}_{0}=288$ | SLSQP | 0.9742 | 70.722 | 0.9742 | 70.722 | 0.9742 | 70.722 | 0.9742 | 70.722 | 0.8402 | 5.106 | 0 | 0 | −0.006 | 131.494 |

GA | 0.9497 | 65.410 | 0.8780 | 47.457 | 0.9587 | 67.390 | 0.9593 | 67.523 | 0.9717 | 40.189 | 0 | 0 | −0.031 | 131.937 | |

TA | 0.9193 | 58.378 | 0.9193 | 58.378 | 0.9193 | 58.378 | 0.9120 | 56.568 | 0.9018 | 28.494 | 0.8973 | 27.574 | −0.23 | 130.327 |

Case 1 | Case 2 | Case 3 | Case 4 | |
---|---|---|---|---|

${\xi}_{{P}_{SC}}$ | 29.92 | 17.91 | 4.23 | 3.89 |

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## Share and Cite

**MDPI and ACS Style**

Wang, X.; Zhao, Q.; Wang, Y.
A Distributed Optimization Method for Energy Saving of Parallel-Connected Pumps in HVAC Systems. *Energies* **2020**, *13*, 3927.
https://doi.org/10.3390/en13153927

**AMA Style**

Wang X, Zhao Q, Wang Y.
A Distributed Optimization Method for Energy Saving of Parallel-Connected Pumps in HVAC Systems. *Energies*. 2020; 13(15):3927.
https://doi.org/10.3390/en13153927

**Chicago/Turabian Style**

Wang, Xuetao, Qianchuan Zhao, and Yifan Wang.
2020. "A Distributed Optimization Method for Energy Saving of Parallel-Connected Pumps in HVAC Systems" *Energies* 13, no. 15: 3927.
https://doi.org/10.3390/en13153927