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A fully distributed optimal control strategy for the parallel variable speed pumps in heating, ventilation, and air-conditioning (HVAC) systems is proposed. Compared with the traditional centralized method, the efficient control and coordination are scattered to every updated smart pump without the need for a monitoring host. Similar to the structure, mechanism, and characteristics of biological communities, a smart pump can communicate with adjacent nodes and operate collaboratively to complete pumps group operation with the least total power consumption under load demand and system constraints. And a decentralized optimization method that is decentralized estimation of distribution algorithm (DEDA) under local interaction games framework has been transplanted to the proposed structure to optimize the pumps working in parallel mode. Besides, convergence property of the two novel mechanisms is analyzed theoretically. Finally, simulation studies have been conducted based on the models of a physical pumps system, and the performance of the proposed algorithm is compared with centralized algorithms in terms of both effectiveness and stability. The results provide support for the validity of the proposed algorithms and control structure.

A pumping system, which generally consists of several circulation pumps, is one of the main energy consuming parts in a chilled water based HVAC system. According to the International Energy Agency, electrical motors have consumed 46% of global electricity demand by end-use [

Currently, frequency converters are widely used with induction motor-driven pumps, enabling the pumps speed variation to control the head and flow rate of pumps under different working conditions in an energy-efficient way, that is, variable speed pumps (VSPs). However, it makes the moderating process more complicated in that not only the combination of pumps but also the frequency of each pump needs to be determined compared to the nonvariable-speed pumps.

On the one hand, the physical models of pumps operation are difficult to be designed accurately and the pumps unit commitment is always a mixed integer programming problem, which leads to conventional optimization methods difficult to be used. In this case, the majority of existing researches focus on adopting different kinds of heuristic algorithms to solve such problems. Kecebas et al. utilized the artificial neural network to optimize the geothermal district heating systems [

On the other hand, in most recent studies, the keynote has been put on updating the existing optimization methods or developing more effective ones[

Structure of the centralized control system of a typical HVAC system.

Structure of the physical system

Structure of the centralized control system

From the engineering perspective, the centralized control structure is not consistent with the physical system, so a mapping mechanism between the two systems is needed, such as configuring and commissioning. Therefore, the development incurs high maintenance and labor costs and somehow diminishes the flexibility and universality of a construction.

After that the key procedure is to write the control model and algorithm into the monitoring center. The mathematical model for the total system is always complex and difficult to be obtained. In the algorithm writing process, a lot of work is needed such as system recognizing, software developing, and operating debugging. Moreover, traditional centralized methods require information transmission to the supervising computer during operations, which can cause severe link congestion and operational lag. Besides, since all the devices are managed by the master controller, malfunctions in one device will also bring about influence on other devices which will cause the entire control system running abnormally or even paralysis.

Recently, distributed multiagent control systems which are being focused on by many learners have provided a possible solution to this problem [

In this paper, a novel distributed control structure where each pump and chiller are updated to a smart unit by an embedded controller is proposed. These independent units can work independently and cooperate together to fulfill the global optimizing task.

To achieve the global optimization in a fully distributed way, a local interaction game is introduced into cognitive ratio networks in [

The rest of this paper is outlined as follows: In Section

The pump head and flow rate of a single VSP_{0} is the rated pump speed and_{i} is the flow rate,_{i} is the pump head of each branch, and_{i} is the efficiency.

The speed ratio

And, the fitting formulas of pumps running at nonrated speed can be expressed as_{set} is the demanded pressure difference on the main pipe._{i}=_{i}_{i}/_{i}._{i},_{i},_{i} and_{i},_{i},_{i} are static physical characteristics of different branches for pump

If the detected pressure difference_{o} is not equal to_{set}, the control system will calculate the demanded total flow rate_{p} as shown in (_{i} and make the calibration of each_{i} satisfy_{p}._{o} and_{o} are the current pressure difference and flow rate detected by the sensors.

In the largest pump systems, the pump groups are composed by several identical big pumps and one or two small pumps for energy saving. As shown in Figure

Decentralized control structure of a parallel pump system.

Each smart pump is connected hand to hand with its neighbor building up a chain-like form which corresponds to its actual engineering installation layout as shown in Figure

The operation object can be described as finding a combination of the running pumps and different pump can adopt a number of speed ratio strategies that can satisfy_{p} with the least energy consumption. Thus, it can be built up as

Implementation of the centralized model (

The decentralized optimization problem for each node

Indeed, (_{i} of the pump_{p} with the least energy consumption.

Thus, the key problem lies in designing the mechanism to achieve global optimization via local interaction among nodes (the terms of agent, node, and smart pump refer to the same concept in this paper) only with direct communication links. As we all know, it is local interaction game. In this manner, optimal load distribution can be completed on the field level. In this paper, any method that solves (

Coordination and combinatorial optimization, when viewed from a game theory perspective, can form a subset of the class of potential games.

Player action sets _{i}_{-i} is the complimentary set of_{i}.

And a number of learning algorithms can be found for the potential game. Most of the learning algorithms for potential games guarantee convergence to a (pure) Nash equilibrium.

A joint strategy profile,

It is easy to see that, in potential games, any action profile maximizing the potential function is a pure Nash equilibrium. Hence, every potential game possesses at least one such equilibrium.

Consequently, each smart pump can be treated as a player and the energy consumption (

However, there may also exist suboptimal pure Nash equilibrium that do not maximize the potential function[

In fact, there exist many evolutionary and swarm intelligent algorithms for coordination and combinatorial optimization in engineering[

Estimation of distribution algorithm (EDA) is an evolutionary optimization method that guides the search for the optimum by building and sampling explicit probabilistic models of promising candidate solutions[_{i},_{i} is the vector of parameters for the probability density of factor_{i}.

The EDA is adapted into decentralized fashion in this study. DEDA is different from traditional evolutionary or swarm intelligent algorithms, where the populations of each smart pump only cover its own variable as shown in Figure

Comparison between EDA and decentralized optimization algorithm.

And the detailed description of DEDA is illustrated in Figure

Flow chart of the DEDA in each smart pump and executed in serial mode.

The convergence proof of the DEDA is presented in Proposition

The solution x^{de}= [x_{1}^{de}, …, x_{i}^{de}, …,x_{N}^{de}] for the decentralized optimization (

Assume the global solution space is defined as_{1} _{2} _{N}, where_{i},_{i}. An elite-preserving operator is used in the presented algorithm. This operator favors the

Firstly, the optimality convergence of each agent_{i} of problem (_{i}(_{i}), and the optimal solution set is expressed as_{i}(_{i}): _{i}

For agent

(I) At least one chromosome belongs to_{0}, which is written as_{0}.

(II) All_{1}, which is written as_{1}, where

And if_{ij} (_{i}(_{j} when the_{i}(_{i}, then the following conditions hold:

(I)_{i} (_{0} and_{00} =1.

Condition (I) is obviously valid because the elite-preserving operator is used in this paper.

(II)_{i} (_{1} and_{11} ≤

Conditions (II) is analyzed as follows: For any_{i0}

Define

The probability that _{0} can be defined by_{i}), with

For

For simplicity,

Define

As

Further, for any

Then the following theorem is introduced to complete the proof process.

_{1}_{2}

_{ k }

According to Theorem 5,_{i}^{t}(_{i}(_{i}) is continuous on the bounded region_{i}, the optimal solution

According to the above analysis, the solution_{i}^{de} can converge to the optimum _{i}, that is,

According to (

For the coupled variables, a similar result is also obtained:

Then the present study directly obtains^{de} converges to an optimal solution of the centralized optimization problem with a probability of 1 as the iteration step

This case study is carried out to validate the performance of DOABP by applying it to the model of a physical pump system under different load demands. The physical parameters (PAR) corresponding to performance curves of the pumps are shown in Table ^{3}/

Pumps parameters.

PAR | PUMP | |||
---|---|---|---|---|

1.KP1020-3 406 | 2.KP1020-3 406 | 3.KP1020-3 457 | 4.KP1020-3 457 | |

| -6.07e-6 | -5.36e-6 | -8.54e-6 | -8.41e-6 |

| -0.00822 | -0.00949 | -0.00048 | -0.00046 |

| 60.41767 | 60.61789 | 74.54932 | 73.55836 |

| -4.08e-7 | -3.96e-7 | -3.87e-7 | -3.82e-7 |

| 0.000940 | 0.000911 | 0.000999 | 0.000985 |

| 0.316357 | 0.328377 | 0.249898 | 0.259137 |

The strategy space is discretized, and it becomes a limited strategy space to ensure that the potential game is a finite game. And, the speed ratio (SR) of each pump, as selected action set, is shown in Table

Pumps discreted strategy space.

SR | PUMP | |||
---|---|---|---|---|

1.KP1020-3 406 | 2.KP1020-3 406 | 3.KP1020-3 457 | 4.KP1020-3 457 | |

1 | 0 | 0 | 0 | 0 |

2 | 0.93 | 0.94 | 0.825 | 0.825 |

3 | 0.94 | 0.945 | 0.85 | 0.85 |

4 | 0.945 | 0.95 | 0.875 | 0.875 |

5 | 0.95 | 0.955 | 0.9 | 0.9 |

6 | 0.96 | 0.96 | 0.915 | 0.915 |

7 | 0.97 | 0.97 | 0.925 | 0.925 |

8 | 0.98 | 0.98 | 0.95 | 0.95 |

9 | 0.99 | 0.99 | 0.975 | 0.975 |

10 | 1 | 1 | 1 | 1 |

Note. The speed ratio of 0 means that the pump is off.

In the case study, the proposed algorithms will be triggered as follows: when the pipe net has been changed, the smart unit will detect that the pressure difference_{o} is not equal to_{set} and_{o} which can be got through the flow meter on the trunk; thus_{i} can be solved in (

DEDA can be downloaded to microprocessor chip in each smart pump

Iterative process in decentralized optimization.

As shown in Figure

The iterative process of each pump

Iterative process of cost on each pump in decentralized optimization.

Iterative process of total cost in decentralized optimization.

Comparing Figure

Iterative process in centralized optimization.

Further, a more detailed calculation result is shown in Table

Comparison between centralized and decentralized results.

#Working condition | DEDA | EDA | ||||
---|---|---|---|---|---|---|

Demand: 1496.9 m^{3}/h | | ^{ 3 } | | ^{ 3 } | ||

| ||||||

Pump 1 | 47.00 | 700.17 | 0.79 | 47.00 | 700.17 | 0.79 |

| ||||||

Power/kW | 227.4804 | 227.4804 | ||||

| ||||||

Difference | 0.0169 | 0.0169 | ||||

| ||||||

Demand: 2583.4m^{3}/h | | ^{ 3 } | | ^{ 3 } | ||

| ||||||

Pump 1 | 48.50 | 886.98 | 0.83 | 48.50 | 886.98 | 0.83 |

| ||||||

Power/kW | 392.2045 | 392.2045 | ||||

| ||||||

Difference | 0.9772 | 0.9772 | ||||

| ||||||

Demand: 2952.9m^{3}/h | | ^{ 3 } | | ^{ 3 } | ||

| ||||||

Pump 1 | 47.25 | 733.01 | 0.80 | 48.00 | 827.24 | 0.82 |

| ||||||

Power/kW | 435.5667 | 439.1761 | ||||

| ||||||

Difference | 0.8373 | 0.0181 | ||||

| ||||||

Demand: 3234.7m^{3}/h | | ^{ 3 } | | ^{ 3 } | ||

| ||||||

Pump 1 | 50.00 | 1054.53 | 0.85 | 48.50 | 886.98 | 0.83 |

| ||||||

Power/kW | 475.8191 | 485.6048 | ||||

| ||||||

Difference | 0.7344 | 1.1869 | ||||

| ||||||

Demand: 3858.3m^{3}/h | | ^{ 3 } | | ^{ 3 } | ||

| ||||||

Pump 1 | 47.00 | 700.17 | 0.79 | 48.50 | 886.98 | 0.83 |

| ||||||

Power/kW | 572.8274 | 561.5738 | ||||

| ||||||

Difference | 0.6517 | 0.4537 | ||||

| ||||||

Demand: 4901.2 m^{3}/h | | ^{ 3 } | | ^{ 3 } | ||

| ||||||

Pump 1 | 49.00 | 944.62 | 0.84 | 48.00 | 827.24 | 0.82 |

| ||||||

Power/kW | 778.3465 | 777.4647 | ||||

| ||||||

Difference | 0.4889 | 1.2384 |

Most of the optimizing methods for parallel pump systems in HVAC system are based on a centralized structure under which all the management tasks have to be settled with one master controller. In this structure, the computation efficiency and stability are relatively low in that all the data and inspection have to be processed by one controller and that one malfunctioning device will influence the whole system. Besides, the control strategy is a case-by-case one which cannot be transplanted easily to other systems.

In this study, a decentralized control strategy has been applied to the parallel pump systems in which the traditional master controllers have been substituted by a list of independent smart units embedded in each pump.

(i) A new fully distributed mathematical model for pumps optimal distribution and decentralized flat architecture are presented. With the rapid development of the electronic industry, smart hardware has been widely utilized in different fields. According to the vision of the decentralized method, traditional units can be upgraded to smart agents by incorporating a microcontroller chip inside. The decentralized algorithm can be written into the chip. The smart unit communicates with neighboring nodes such that they can work collaboratively to achieve the optimal load distribution task. In this case, the complicated onsite modeling, configuration, commissioning, and other secondary developing work can be simplified to the wiring of communication connection among different nodes because the smart unit can self-organize and plug-and-play.

(ii) Decentralized optimization is used to solve pumps optimal load distribution under potential game framework. And, to achieve effectiveness of global goals, DEDA using local interaction is introduced. The convergence property of the novel method is also analyzed theoretically. Simulation results presented in the preceding paragraph indicate that the proposed method can achieve optimal load distribution.

This scenario is extremely motivating for further investigation in the field of distributed networked control in other large-scale interconnected nonlinear systems. And, there are still several potential issues of the presented method that needed to be emphasized here. Firstly, the proposed mechanism is executed in sequence. Therefore, the proposed distributed algorithms should in principle be applicable to chain-like networks. Each smart pump is connected hand to hand with its neighbor, but the meshed network should also be considered. Secondly, future work will consider the hardware test and implementation on practical engineering.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is supported by National Key Research and Development Program of China No. 2017YFC0704100 (entitled New Generation Intelligent Building Platform Techniques). And the authors are grateful for the support provided by the Tsinghua University-UTC Research Center for Integrated Building Energy, Safety, and Control Systems.