Decentralized Coordination of Temperature Control in Multiarea Premises

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Introduction
With local control of a large number of objects that mutually infuence each other, the problem of coordinating local control systems to achieve the best overall result arises. If the structure of the system (the number of control objects and the parameters of interaction) can change frequently, then the process of setting up/training a centralized coordinator will take an unacceptably large part of the action time and require a signifcant amount of resources.
Sets of controlled objects that infuence each other, or multizone distributed objects, are found in many areas.
Examples of such objects are multizone premises. Ensuring a comfortable and safe environment on the premises is an important task, as is energy efciency. Te complexity of the task is due to the ambiguous understanding of the concept of comfort and the presence of conficting requirements for comfort among diferent occupants. In particular, there are many multiarea premises, such as "open space" ofces and studio apartments, with separate areas. Comfortable conditions in such areas demand diferent air temperatures. Although such zones are called thermostatically controlled load (TCL) [1]; however, the static mode in them is observed only for a limited time interval. A characteristic feature of such premises is the rapid and frequent change in requirements: premises are vacated or flled, comfort requirements change following individual needs, etc. In such premises, separate means of maintaining air temperature (air conditioners, convectors, fan heaters, etc.) with local control systems are used. Te mutual infuence between separate areas requires the coordination of local control systems (LCS). Te task of coordinating temperature settings means in connected zones when using movable devices has several general and special properties: a variable number and location of operating heaters or air conditioners; only movable heaters or air conditioners can communicate wirelessly with other devices; and a slight increase in the cost of heaters and air conditioners due to the installation of coordination and data transmission controllers on them. Tese special properties necessitate a new look at the task of coordinating control of TCL movable heaters/air conditioners in the context of the general problem of the decentralized coordinating control of the multizonal distributed objects.

Coordination of Control Systems.
Coordination tasks have a long history. Teoretical research on coordination has been going on for more than a century. H. Fayol is considered the founder of the theory of coordination [2]. However, the problem itself appeared much earlier. Tis is illustrated by the classic formulation of the task of Byzantine Generals [3], which is devoted to the synchronization of the states of two systems over unreliable communication channels.
Te problem of coordination is very popular in scientifc and practical publications. However, the approaches to the coordination of strategic plans and operational decisions of state and corporate management bodies are the main considerations. Unlike the control processes in engineering and technology, these approaches focus on the psychological, legal, and political aspects of a problem. A signifcant contribution to the development of coordination management in man-machine hierarchical systems was made in the series of works by Dmitry Novikov, in particular in [4].
In the control of production and technological systems, the main attention is paid to architectural, informational, and criterion aspects [5]. Te problem of coordination in distributed technological systems focuses on the control of material and energy fows as well as on the characteristics of the dynamics of operations [6].
With the emergence and active development of a multiagent technology for control of distributed systems and interacting objects, the works began to be formulated mainly as the tasks of coordination of agents of a multiagent system [7,8]. In general, coordination tasks are considered while taking into account the balance of local (individual) and global criteria. Various coordination strategies are considered: cooperation, competition, consensus, negotiations, etc. [9][10][11] in this direction. Some authors consider general issues of coordination in systems with a certain structural organization of coordinated processes: parallel [12,13], sequential [14], and serial-parallel [15].
Te architecture of the system and the structure of the coordinators' information connections play an important role in the development of the coordination system. Te most common approaches to the coordination of local technological process control systems (LCS) are based on a centralized or hierarchical system architecture [16,17]. Te centralized architecture is usually used for a small number of LCSs and a small distance between them, and the hierarchical one is for systems with a large number of LCSs and/or a large distance between them. Centralized coordination can be considered the special and simplest one-level case of hierarchical coordination. Te theory of hierarchical coordination is based on the works of M. Mesarovich and coauthors, frst published in 1970 and republished many times [18]. Te advantage of hierarchical systems is the relative simplicity of matching local (individual) and global criteria and the guaranteed stability of control, which is due to the tree-like structure (without cycles) of subsystem connections. However, such systems at each level of coordination presuppose the presence of a top-level coordinator, which must be confgured, administered, and maintained, and its disconnection or failure destroys the entire coordination system. Such systems have a rigid structure of connections, and they are difcult to scale. Tis complicates their application to objects with changing composition and/or connections between coordinated subsystems.

Individual Regulation of the Parameters of the Human
Environment. An important area of application for LCS of distributed objects is to ensure the individual comfort of occupants in the various premises for collective use. Such a task requires coordination of the comfort parameters and LCS settings of the individual systems. In modern scientifc and technical publications, there are many names for systems for individual regulation of the parameters of the human environment on the premises.

Personal environmental control (PEC).
Systems were studied for a long time. Tere are several reviews of PEC systems [19][20][21][22][23]. Most existing research and review articles focus on thermal comfort with PEC systems. For example, PEC's thermal comfort was studied in [20,21], and [22] investigated the efect of PEC on thermal comfort and energy consumption. In [23], research on ways and technologies to infuence thermal comfort in multizonal premises is systematized. It should be noted that the review did not consider algorithms and technologies for coordination (group control) of movable and individual heaters and air conditioners. (TAC). Systems are investigated from the point of view of the tasks to be solved. TAC systems are designed to maintain the thermal regime in a localized area and are controlled individually or by a group of people [24][25][26][27].

Personal Comfort System (PCS)
. PCS is designed both to increase the comfort of individuals and to reduce the energy consumption of heating and cooling systems. Examples of PCS are being handed out, such as spot cooling [28], personal environment module (PEM) [29], individually controlled microenvironment system (ICS) [30], and an ofce partition system with a radiant cooling panel [31]. Many works explore the technological and energy aspects of PCS and their impact on thermal comfort [22].
Te work [32] presents a building control system that takes into account the microzones; a set of algorithms for individual control of devices in microzones was developed, and comfort information is derived from higher-level settings.
In many studies, much attention is paid to the use of communication standards for the organization of multiarea comfort systems [33]. In particular, IPv6, 6LoWPAN, Bluetooth Low Energy (BLE), ZigBee, Wi-Fi, and Z-Wave protocols are considered. Control systems, sensors, and actuators of an existing building, supplemented by the Internet of things (IoT) can be classifed as a general class of "cyber-physical systems" [34,35].
No less attention is paid to research on ways to reduce energy costs for its provision than comfort.
Occupant behaviour and activities have a signifcant impact on the energy efciency of a building, and various researchers have confrmed the human role in building operation [36]. Te results of studies [37,38] show the degree of infuence of user behaviour on energy consumption for heating.
Te methodology for determining the most efective methods of local control focused on the presence of people and was developed in [39]. Te methodology integrates a simulation model with a multicriteria optimization method. In [40], the EnergyPlus simulation platform was used to assess the impact of occupant behaviour on comfort and energy consumption. In [41], the authors also proposed an approach to energy control by monitoring the heating system to ensure comfort.
Some eforts aimed at improving the efciency of PEC are based on simulation. Large and complex work on the development of modelling tools and energy efciency improvements based on this is carried out in the Building Technology and Urban Systems Division of Berkeley Lab (USA). Tey have developed and are developing now such products as EnergyPlus [42], Modelica Buildings Library [43], Building Controls Virtual Test Bed (BCVTB) [44], Generic Optimization Program (GenOpt ® ) [45], Ener-gyPlustoFMU [46], etc. Te EnergyPlus simulation system is the most widely used. Tis modelling system is the basis of most other products by this developer. Te study [47] showed that EnergyPlus makes it possible to evaluate the energy efciency of the behaviours of residents. Te work [48] introduced fuzzy-logic heating, ventilation, and airconditioning (HVAC) controller and used the Building Controls Virtual Test Bed [44] to test the model using EnergyPlus. It is shown that the proposed technique reduces the number of uncomfortable hours by 50%, spending the same amount of energy.
Te work [49] presents a system that includes predictive mechanisms and intelligent heating control algorithms based on an artifcial neural network (ANN) to optimize energy efciency while taking into account the satisfaction of residents. To do this, Berkeley Lab collected data for three years on whole-building and end-use energy consumption, HVAC system operating conditions, indoor and outdoor environmental parameters, and occupant count and created a dataset for analysis and machine learning [50].
Since 2017, the US Environmental Protection Agency has implemented the ENERGY STAR ® [51] performance certifcation program for Internet-connected TCLs. Te authors of the paper [52] analyzed the results of the ENERGY STAR ® data registration and showed that HVAC systems showed stable trends in increasing comfort and energy savings. Moreover, the increase in efciency of the systems of individual suppliers is explained by more successful algorithms and control strategies. In particular, efective ASH-RAE Guideline 36 (G36) control strategies for the multizone operation of a variable air volume (VAV) system are analyzed in [53].
Te abovementioned works are focused on centralized TCL optimal control systems and stationary HVAC devices. For example, the structure and principles of the Alpha-Building ResCommunity system were described in [54], which uses the joint modelling of all community TCLs with the help of EnergyPlus and Modelica for optimal control. Te use of such "heavyweight" systems require a lot of preparatory work for modelling and, accordingly, a developed user interface.
At the same time, it should be noted that in many cases, individual thermal comfort is provided with the help of movable heaters and air conditioners. Issues of installation and use of portable heaters are considered from the points of view of their safety, energy efciency, and ease of use. Te rules of safe use are defned in NFPA 1-Fire Code standards; ANSI/UL Standard 1278 for Movable and Wall-or Ceiling-Hung Electric Room Heaters, etc. At the same time, all studies and standards concern single heaters, and the efciency of groups of mobile heaters is not considered.
However, when using several portable heaters in adjacent areas of the premise, the presence of heat fows between areas with diferent set parameters, complicating their adjustment, and when their location changes, the settings also have to be changed. Te problem of prompt automatic adjustment of the group of movable heaters, which forms a dynamic system with a changing structure, has not yet found an efective solution.

Related Works.
Te review and analysis of distributed systems of coordination control were made in [55].
Te most generalized approaches to the coordination of local control systems are based on a centralized (for a small number of LCS and a small distance between them) or hierarchical (for a large number of LCS or a large distance between them) architecture of the coordination system [56][57][58]. Such systems are widely used, and they provide high efciency when controlling the thermal regime of Complexity 3 individual buildings and groups. In the work [43], the effectiveness of the model predictive control (MPC) HVAC system in a real ofce building using a Modelica-based toolchain was studied. It was shown that MPC saves approximately 40% of HVAC energy over the existing control. However, centralized systems have a rigid structure of connections and are difcult to scale. It makes their application to objects with frequent and rapid changes in requirements more complicated.
A promising way to solve the problem is to use decentralized coordination with Smart coordinators in each local control system. Most often, such a coordination architecture is used with a very large number of relatively autonomous objects, for example, in the energy industry [59,60], in collectives of autonomous robots and engineering objects [61], and in autonomous fying machines (drones) [62,63].
Decentralized systems are the subject of many studies. Many interesting results were obtained in the research Project Control for Coordination of Distributed Systems (CON4COORD and C4C, both acronyms are used) by the Consortium of 12 Research Centers in Europe [5]. A feature of distributed, decentralized systems is signifcant uncertainty in the parameters of subsystems' interaction, the nonfully-connections of the system, and the absence of complete information about the state of other subsystems that are in direct connection with a separate subsystem.
Depending on the type of system, the tasks of decentralized systems control are called synchronization, decentralized stabilization, one-level coordination, peer-to-peer control [64], etc. Linear and nonlinear systems, continuous and discrete, with optimal and adaptive control, robustness, and artifcial intelligence elements are considered [65].
Te adaptive decentralized control with model-based coordination was proposed in 1992 by B M. Mirkin and was being actively further developed by many authors, for example, in [66]. Tis assumes the availability of information among local controllers about the state of the reference models of all local subsystems.
Te concept of distributed optimization of control of a state of the multiarea premise was proposed in the work [67]. Te paper proposes a distributed method for optimizing air conditioning that can be implemented in a parallel way.
Despite a signifcant number of works on the study of decentralized coordination systems, the problem of coordination control of the state of continuous multiarea distributed objects with a dynamic structure and variable requirements has not yet found an efective solution. In addition, the mutual infuence of coordinated objects on each other is rarely taken into account in existing works.

Objectives and Problems.
Let us formulate the main provisions of our study.
Te object of this study is the process of decentralized coordination of local control systems to ensure a comfortable individual thermal environment in multizonal premises under the conditions of dynamic changes in the requirements of zones.
Considering the special features mentioned above of the concept of individual comfort and methods of coordination of local control systems, we formulate the purpose of the study as the improvement of the quality of thermal control in multiarea premises with a dynamic structure of the location and connections of heaters.
In this work, we propose a novel approach to temperature control in multiarea premises through the use of decentralized coordination of movable heaters. Tis provides system fexibility, i.e., the ability to change the number and location of heaters without the need to change the settings of the central control system. At the same time, the characteristics of comfort and energy consumption are not inferior to systems with centralized control of stationary heaters.
Te main contribution of the research is the concept of decentralized coordination of local control systems with a dynamic structure and its implementation in the Movable Smart Heaters (MSH) system. Te criteria for control quality are defned and evaluated. New algorithms for decentralized coordination are proposed. Tese algorithms make it possible to optimize the operating modes of the system automatically when its structure and/or settings are changed. Te implementation of the concept in Movable Smart Heaters allows for an increase in comfort while reducing energy consumption.
Te highlights of the research are as follows: (i) Te statement of the problem of coordination control of the state of continuous multiarea premises with a dynamic structure of the location and connections of control means is made. (ii) Te concept of decentralized coordination of Movable Smart Heaters is proposed. (iii) Criteria for controlling the quality of temperature comfort in multizonal premises are formulated. (iv) Te basic algorithms for coordination and dynamic adjustment of Movable Smart Heaters are developed. (v) A simulation of the MSH system was made, and its results were analyzed.

Statement of Research.
In this section, we will formulate the problem defnition based on an example of the area's locations in the open space ofce plan, as shown in Figure 1. Te plan shows a fragment of the layout of the heaters. Heaters are divided into "Ordinary Heaters" and "Movable Smart Heaters". We propose to equip each Mobile Smart Heater with a coordinator who can communicate with each other by means of a Wi-Fi mesh to optimize the operation of smart heaters in addition to conventional heater controllers. Te comfortable temperature for each area F(k), where k is an area number, is set remotely using a movable device. Te diagram also shows some connections between areas and between coordinators. Te mutual infuences between the areas are "physical," as evidenced by the heat fux between the areas, and following the physical infuences, the information connections between the coordinators should provide optimal control of the thermal condition, taking into account the condition of the surrounding premises. In Figure 1, diferent arrows show that the intensity of mutual infuence can be diferent. In this case, there may be such MSHs that are not connected by mutual infuence with other devices.
It is only necessary to observe the requirement of nonsimultaneity in the coordination procedures of heaters to avoid the possibility of instability in coordination. Terefore, the decentralized coordination process takes more time in total than the centralized one. It is possible to reduce the decentralized coordination time if only those zones that have a signifcant infuence on each other are taken into account at each coordination step. We will call this principle the "principle of short-range action." Tus, the task of coordination is to fnd such a desired state vector (the set of temperatures) of areas T 0 � T 0k , k � 1..M , which provides the minimum deviation of the state of the object from a given function F(k) while taking energy savings into account. Tus, we can formulate the optimization problem that the coordinator solves either as optimization with constraints or as optimization with priority levels: where k ∈ K S is the area number within the controlled area of the premise S; K S is the set of area numbers of the premise; F k (t) is the desired function of change in the time domain of the k-th area state of the multiarea premise; T k (t) is the real state of the k-th area on a time t; R is an average square deviation of the temperature in the controlled area of the premise S from the preset temperature; E k is the energy consumption in k-th MSH; E max is a maximum permissible energy consumption; ΔT max is a maximum permissible temperature deviation; α k is the priority factor of areas; β R , β E are the priorities for comfort and energy consumption; t 0 is the optimization time interval.
In the previous works of the authors [68][69][70][71], research and development of the principles of decentralized coordination in distributed cyber-physical systems for controlling technological processes were carried out. In particular, in [69], a model of the interaction of controlled zones of a distributed technological object was developed. Te structure of the coordinator was proposed in the work [68]. Each coordinator contains the following modules: (i) Object model; (ii) Interface module; (iii) Assessing uncertain parameters module; (iv) Clustering module; (v) Forecasting module; (vi) Criterion optimization module; (vii) Module for controlling the sequence of coordination; (viii) Parameter setting module; (ix) Communicate Wi-Fi mesh module.
Te mentioned modules in the MSH coordinator must satisfy the following several requirements: (1) Ease of software and technical implementation. Te software and additional controller required to implement the mentioned modules should not significantly increase the cost of the heaters. (2) High-speed adjustment and coordination algorithms.
Despite the inertia of thermal processes in living and working premises, the adjustment and coordination processes must occur quickly enough to provide the required comfort during the occupant's stay on the premises. Additional requirements for dynamics are related to the mobility of the heaters; changing their mutual location and turning them on/of leads to a change in the structure of the coordination system and the need for additional adjustment. (3) Full automation. Te user purchasing MSH should not perform any complex procedures for setting up the system. It is enough to start the coordination setting process from the smartphone, set the desired temperature, and select the type of coordination criterion (1). For this, the Connecting module and adjustment module are additionally included in the coordinator.

Basic Algorithms of Coordination and Dynamic
Adjustment. Te frst requirement, "Ease of software and technical implementation," is provided by decentralized coordination. Tis is the main diference and advantage compared to Smart Home systems. Decentralized coordination does not require the existence of a central controller and the implementation of procedures for changing its software and/or hardware confguration when changing the number or location of heaters.
Mobile Smart Heater Ordinary Heater Physical influences (by intensity)

Complexity
With increasing the distance |Z − Z i | of the control point Z i of the Movable Smart Heater from a given area, the infuence of control decreases. Tus, with decentralized coordination, each coordinator should take into account only those controlled elements that are in its immediate surrounding (Figure 2) i.e., the cluster, whose boundaries are determined by the clustering module.
In well-known works, various models are used for modelling thermal processes in premises [72], which are based on the laws of thermodynamics and certain simplifcations due to the peculiarities of premises heating processes. To determine the set of elements of the environment, we used the model of the object [69]. Tis model is based on the equation of thermal energy transfer from an object with temperature T 1 to an object with temperature T 2 : where λ is the heat transfer coefcient; C is the heat capacity, and the known solution of the Burgers transport equation. Te transfer equation is frst-order concerning time and second-order concerning spatial coordinates. In particular, an instantaneous point impact on an element k is propagated to the element j according to the formula v d k , t � P 0k If ε is a signifcant factor in the surrounding of the i-th area K iε , we will consider the set of MSH that satisfy the condition. (4) where 0 ≤ ε ≤ 1; F i is the given state of the i-th area; T 0k is the state of the k-th surrounding area specifed by the coordinator; d k is the distance from the specifed area to the k-th surrounding area; λ k is the heat propagation constant; τ is the heat propagation time constant.
It is difcult to use condition (4) in decentralized coordination, as this requires each coordinator to have information on the state of all MSH. Tis is almost impossible for large distributed multiarea premises. Terefore, it was proposed to introduce an estimation function to determine the set of MSH, which i-th MSH should coordinate.
When choosing an estimation function, we should use the following considerations: (1) If the specifed state function F(Z) � const and controls are located evenly in the distributed object, then the control function satisfes the optimality condition T 0 (Z) � const � F 0 . Tus, from (4), the radius of the ε-area could be estimated in a next way: (2) In the case of asynchronous work of relay heater controllers (3) In the case of synchronous work of relay heater controllers i -th controlled element Area of significant influence of the i -th controlled element Area of significant joint action of two controlled elements ε -area of i -th controlled element Figure 2: Determination of heaters ε-surroundings. 6 Complexity Te analysis of the function T 0k /8(πλt/τ) 3/2 e −d 2 k /4λt/τ to the maximum shows that the maximum is reached at t m � d 2 k τ/6λ. Let substitute t m into condition (5) and obtain Such a case cannot be implemented in practice since it assumes a perfectly uniform, infnitely distributed object with an infnite number of controls and an infnitely small distance between them. However, it can be considered a limiting case for testing coordination algorithms and a certain approximation of real problems.
(1) If F(Z, t) ≠ const, then more distant elements can make a signifcant additional contribution to the state of the considered area. Tis additional increase in the ε-area We get by substitution ΔF � |F k − F i |. In particular, boundary conditions (the state of the environment) are taken into account in this way.
Algorithms for determining the set K kε of MSH with which k-th element must coordinate are worked out. Te algorithm that solves the problem based on the model of heat distribution in a multiarea premise without partitions is shown in Figure 3. Te algorithm involves converting the area number to coordinates in space, calculating the distance between areas, and calculating the coefcients of mutual infuence of areas depending on the distance between them.
Te algorithm for determining the set of MSH in any premise based on the procedure of impact testing is shown in Figure 4. At the frst MSH switch-on, a change in the comfort temperature setting, or a change in MSH position, the testing procedure is started. Te coordinator sends the messages to the rest of the coordinators at MSH, and they register the temperatures in areas. After 5 heating cycles of the relay MSH, the new message is sent. After that, the correlation analysis of temperatures and the calculation of coordination parameters are carried out. Since the heat fows are proportional to the temperature diference, the correlation function was calculated as

Experiments and Results
Te algorithm for determining the coordination parameters was studied on a model in the Scilab system. We have developed a library of modules (Superblocks) in the Scilab system for modelling distributed control systems. Figure 5 shows the part of the simulation module library and the temperature control coordination model in a multiarea premise. Tis model contains 9 MSH areas, each with its own on/of temperature control relay cycle. Te impact of random disturbances (opening/closing doors, windows, etc.) was simulated by generators of random events. Te results of the simulation of the process of parameter testing are shown in Figure 6.
Results (a) and (b) are almost identical, although result (b) is obtained at levels of random efects 10 times higher than in case a). Tis confrms the resistance of the correlation method to interference in determining the coordination parameters.
Results (c) are diferent. Tey were obtained at given random values of temperature in the areas in the range (20°C-24°C), but the results (a) and (b) were at the same set temperature in all areas: 22°C.
However, all the results retain the characteristic features that allow us to determine the parameters of the following coordination: T H is the heater temperature; T a is the air temperature; α is the coefcient that takes into account the design parameters of the heater (area of contact of the heater with the air, speed of airfow near the surface of the heater, etc.); c a is the coefcient that takes into account air parameters (specifc heat capacity and thermal conductivity, which depend on pressure and humidity at a given temperature).
Coordination and adjustment of the parameters of multiple heaters can be performed in diferent sequences. As it was mentioned, it is only necessary to observe the requirement of nonsimultaneity in the coordination procedures of heaters located in the same ε-area to avoid the possibility of instabilities in coordination. To do this, it is necessary to select connected subgraphs in the ε-area graph and solve the problem of traversing all vertices in each subgraph. At the same time, it is possible to use breadth-frst search algorithms, such as Little's algorithm, Prim's algorithm, etc. To control the sequence of local system coordination, we use the wave method. Te wave method can be synchronous or asynchronous. Te synchronous method is implemented by using a wave generator of synchronization pulses. Te asynchronous algorithm is Complexity implemented by passing a token [70]. Te coordinator of each MSH performs the determination of the optimal control taking into account how it will afect the whole ε-environment. To avoid the possibility of instability in coordination due to the confict between local and global optimization, we have modifed the wave algorithm of Lee (1961) and introduced a compromising factor ρ k Tus, the criterion E(T 0i ) of local coordination is Taking into account the model [69], we could write Input number of areas M ; MSH coordinates Z (k) ; comfort temperature F (k) ; system parameters {λ,τ} ; allowable error ε.

Complexity
where w ki is the equivalent transfer coefcient of the transfer function of the system {from k-th MSH to i-th area} [69]. Since the thermal object is linear, then w ki � w ik . Obviously, in the presence of regulators w ii ≈ 1, w kk ≈ 1, and w ki ≪ 1. Te form of integral function under such conditions is shown in Figure 7. Figure 7 shows that the integrand function is bound, positively defnite, tends asymptotically to zero, and therefore is integrated. So, criterion (11) may always be calculated.
Te forms of dependences E(T 0i ) on the parameters d ik (a) and k � λt/τ (b) are shown in Figure 8. Te graphs show that the dependence E(T 0i ) for all parameter values is smooth, at one extreme, and therefore the minimum criterion can be found by the gradient method.
Te Scilab superblock of the local coordinator for the system of three elements in the ε-area is shown in Figure 9. Te coordinator searches for the coordination parameter T 0 for the i-th element by fnding the minimum coordination criterion (11) by the gradient method.
Te correspondence of parameters of the coordinator model to the designation of criterion (11) is given in Table 1.
As the coordination wave passes through the system, the components of the coordination parameter vector T 0 are sequentially changed to improve the coordination criterion. Te convergence and stability of the wave coordination algorithm depend on the ratio of the attenuation index of the exponent d 2 ik /4λt/τ and the coordination coefcient. c: ΔT 0i � −c · (T i − T 0i ).

Te Estimation of Comfort.
Although comfort is a complex concept, we evaluated only one of its components, thermal comfort. It was noted that thermal comfort is defned as "a state of mind that expresses satisfaction with the thermal environment and is evaluated by subjective evaluation" [73]. For a multiarea premise, the desired thermal environment in each area is determined by a given temperature distribution. A method for estimating human thermal comfort based on the predicted mean vote (PMV) 10 Complexity was proposed in 1970 and extended in work [74]. Te level of comfort in a PMV exponentially depends on the energy consumption of the occupant for self-thermoregulation. In our work, we use the PMV model constants and approximate this dependence by an exponential function: where F is the temperature of maximum comfort. General comfort level in multiarea premise: where M is the number of areas of multiarea premises, and i is the area number.

Te Estimation of Energy Consumption.
MSH is a relay system with automatic control. Te method of estimating energy consumption in stationary and transient modes in thermal control systems of multiarea premises was proposed in [71].

Te estimation of coordination.
Many sources consider the concept of the "index of coordination" as the opposite of coordination and chaos. So, there are two understandings of the index of coordination: (1) As a qualitative characteristic of the coordination of the activities of the enterprise, project executors, etc.; (2) As a quantitative characteristic of the location in space (the origin of the concept "coordinates" of molecules in chemical processes or mechanical elements of devices and structures).

Complexity
We can assume that the state of the system changes from completely chaotic to fully coordinated during the process of coordination.
Recently, chaotic processes are increasingly attracting the attention of researchers [75]. In their works, diferent names are given to this phenomenon, such as dynamic chaos, deterministic chaos, and complete chaos. A characteristic feature of chaotic processes is their unpredictability. Te name "deterministic chaos" is used to emphasize that the collective behaviour of many dynamic objects, each of which is deterministic, becomes unpredictable with a certain number of them and a rigid nonlinear model that creates instability in the behaviour prediction algorithm.
Chaos theory identifes the properties of the system, in which it can be chaotic processes [76]: (1) It must be sensitive to the initial conditions. Sensitivity to initial conditions means that small changes in initial conditions can lead to signifcant changes in the state of the system. Tis is typical of a rigid system model. An indoor temperature control system with a sufciently large propagation constant can become nonminimum-phase, so its model is rigid. (2) It must have the property of topological mixing.
Topological mixing means such a scheme of expansion of the system that one of its areas, at some stage of expansion, is superimposed on any other area. While the radius of the ε-area of the element expands with increasing state parameters of the elements, the system corresponds to the condition of topological mixing. (3) It must be nonlinear. According to the Poincare-Bendixson theorem [77], a continuous dynamic system on a plane cannot be chaotic. A discrete dynamical system can exhibit chaotic behaviour even in one-dimensional space.
To assess the potential for coordination of a system, we introduce the concepts of norm and measure of coordination/chaos. On the metric scale shown in Figure 10, states of complete chaos and complete coordination form the boundaries of the range of values of the index of coordination.
However, the use of the index of coordination as an assessment of the consistency of the control subsystems of the distributed system elements is almost nonexistent. An exception is the use of the Lyapunov index to characterize the chaotic motions of a dynamic system [77]. However, for distributed environment control systems, this index is inconvenient because such systems, with the presence of relays and logical conditions, could not be represented by a system of diferential equations, even with the linearization of the characteristics of the elements.
Let us use the inverse normalized RMSE of the vector of the system state as an estimation of the coordination index. If the reduced error is δ T � σ T /max‖T‖, where max‖T‖ is the maximum metric distance between the state vectors of the system (the range of values of the state vector T), then the index of coordination: where τ is a research time interval. Since the deviation of the RMSE cannot take a value beyond the possible range of values of the state vector, then coor ∈ [0; 1]. Tis indicator allows us to determine which of the following three types of systems it belongs to: (2) For a chaotic system without attractors coor � 0; (3) For stochastic system where W T 0i ⟶ T i is the transfer function of the regulator MSH.
Sources of reduced coordination were analyzed in [69]. For the case of periodic wave coordination, expression (9) becomes the following form: Stochastic process Figure 10: Scale of coordination.

Complexity
Te efectiveness of MSH coordination was studied in models of the Scilab system and the EnergyPlus system. Te evolution of the state of the system from switching on to achieving stationarity is shown in Figure 11 ( Figure 11(a) shows the given temperature function F; Figure 11(b) shows the state after the frst cycle of simulation; Figure 11(c) shows the state at the middle of the simulation process; and Figure 11(d) shows the state at the end of the simulation process).
Te results of comparing the efciency of the system with 9 areas with coordination and without one according to the results of modelling in diferent modelling systems are shown in Table 2 and 3. Te plan of the premise with the location of MSH is shown in Figure 12. Te stabilization factor of MSH regulators is 25. Indicators are given in relative units. As a basis, we took the EnergyPlus simulation results without coordination at the same comfort temperature in all areas ∀k: F k � F 0 and external infuence (ambient temperature) u � F 0 , i.e., the ratios of the scatter of external infuence on the area to the power of MSH are σ u /p 0 � 0 and σ F /p 0 � 0. Te simulation data in the Ener-gyPlus system is used as a dataset to adjust the parameters of the Scilab model.
When studying the dependence of characteristics on the spread of comfort temperatures, the distribution of comfort temperatures by areas was generated in the following ways: (1) Random distribution by size and by area number without sorting; (2) Random values are sorted by area number in one direction; (3) Random values are sorted by area number from the centre to the sides. Table 3 shows the average results for all studies. Te results of comparing the efciency of MSH system show an increase in comfort by an average of 3% while reducing energy consumption by 1.5%.
Te results of the simulation of the system of 16 controlled elements with centralized coordination, decentralized coordination, and without it are shown in Figure 13.
Decentralized coordination was carried out cyclically. It can be seen from the diagram that, with each cycle, the value of the criterion gradually approached the global optimum, which is ensured by centralized coordination. Te speed of the approach depends on the size of the ε-area and the parameter c.

. Discussion
Ensuring individual thermal comfort is a difcult task in the conditions of mutual infuence between individual zones of the multizone premises. Te task is further complicated by frequent changes in the set comfort parameters. Usually, this problem is solved with the help of movable heaters and air conditioners. However, you have to reconfgure the heaters in all areas adjacent to the change in comfort settings in one zone or the move of the movable device in one zone.
Te proposed concept of decentralized coordination of local control systems with a dynamic structure and its implementation in the Movable Smart Heaters (MSH) system with algorithms for coordination and automatic detection of the set of adjacent areas makes it possible to simplify the solution of this problem and increase the effciency of the system. Tis approach difers from existing approaches to decentralized control of distributed systems for providing (d) Figure 11: Evolution of the state of an object, t 3 > t 2 > t 1 .

Complexity 13
individual thermal comfort in rooms. Recently, blockchain technology has been used to build decentralized systems [78]; however, this technology involves only decentralization of data collection, storage, and primary processing, while optimal coordination is performed centrally. Te main advantage of MSH and their decentralized coordination is the possibility of their independent production and use at one premise in unlimited quantities without much efort to reconfgure the system. In our work, we have not yet been able to conduct fullscale experiments, but only experiments on a simulation model. Tis is due to the lack of serial MSH, which are currently at the stage of advanced development. Field experiments are planned for the next stage of work. For simulation experiments, the authors have developed a specialized palette (library) of typical blocks on the Scilab platform. Te problem of obtaining a dataset for setting up the model was solved using the EnergyPlus modelling system for building thermal processes, which is recommended by the US Department of Energy and well-tested in practice. Te library of models developed by the authors is a good addition to the EnergyPlus system for the tasks of automatic control of processes to ensure individual thermal comfort.
Te conducted studies of the system model and optimization criteria depicted in Figures 7 and 8 confrmed the possibility of achieving optimal global comfort. However, the process of adjusting system settings when changing its confguration has limitations. Te authors used the correlation method of data processing under the infuence of external random infuences. Tis requires the accumulation of some minimum dataset. Numerical experiments have shown that 6 cycles of operation of the MSH relay control are sufcient. However, for internal zones and small comfort temperature diferences in the zones, these 6 cycles can take a long time. During this time, new changes to the system confguration may occur, and the confguration process cannot complete. At the same time, the fact that the results of comparing MSH system efciency using the simulation model have shown an increase in comfort by an average of 3% while reducing energy consumption by 1.5% indicates that the development is promising.

Conclusions
With local control of a large number of objects that mutually infuence each other, the problem of coordinating local control systems to achieve the best overall result arises. If the structure of the system (the number of control objects and the parameters of interaction) can change frequently, then the process of setting up/training a centralized coordinator will take an unacceptably large part of the action time and require a signifcant amount of resources. In this work, the use of decentralized coordination is proposed to solve the problem. As a basic task for research on decentralized coordination control of objects that mutually infuence each other, stabilizing the comfort temperatures was set in multizone rooms using movable heaters. Providing individual thermal comfort is an important problem. In particular, there are many multiarea premises with conficting requirements for the comfort of habitats. Tis problem can be solved with the help of movable heaters and air conditioners. However, the presence of heat fows between areas with diferent specifed parameters makes it difcult to adjust the movable heaters. To improve the quality of thermal control in multiarea premises with a dynamic structure for the location of movable heaters, we have proposed the concept of Movable Smart Heaters. A group of MSH that could infuence each other and exchange information forms a dynamic system with a changing structure since switching on/ of or moving one MSH to another area changes the mutual infuence and connections in the system. Te criteria for control quality are defned and evaluated. Te proposed decentralized coordination algorithms make it possible to optimize the operating modes of the system automatically when its structure and/or settings are changed. Simulation of the system is performed with the use of a worked-out modelling library in Scilab. Te results of comparing the MSH system's efciency show an increase in comfort while reducing energy consumption. Te study has shown the promise of using decentralized coordination to control a system of interacting objects with a variable structure. Further research is planned to investigate the stability and dynamics of the decentralized coordination in control of a system of interacting objects with a variable structure, in particular the MSH system, taking into account the change in the operating modes of stationary comfort devices and the infuence of heat fows from people and equipment in the premises.

Data Availability
Most data is contained within the article. All the data available from the frst author upon request.

Consent
Informed consent was obtained from all subjects involved in the study.