Distributed dimming control for LED lighting.

Distributed dimming control for LED lighting Sang Hyun Lee1 and Jae Kyun Kwon2∗ 1 Department

of Electrical and Computer Engineering, the University of Texas at Austin, TX 78712, USA 2 Department of Electronic Engineering, Yeungnam University, Gyeongsan 712-749, Korea ∗ [email protected]

Abstract: This paper presents a distributed energy-saving lighting strategy for the arrangements of a lighting network consisting of a group of light-emitting diode (LED) lamps and users. LED lamps have a dimming support feature to meet the illuminance requirements imposed by individual users. Both groups interact with each other via visible light communication (VLC) or other wireless communication features. This work aims to identify a configuration of lamps leading to the maximal energy saving in adaptive and distributed ways. To this end, a distributed assignment strategy is developed based on a message-passing framework where only local interactions among lamps and users are allowed for calculations and exchanges of the information on their status. The simulation results show that the proposed algorithm outperforms other distributed algorithms in a range of indoor lighting configurations. © 2013 Optical Society of America OCIS codes: (060.4256) Networks, network optimization; (230.3670) Light-emitting diodes; (220.2945) Illumination design.

References and links 1. Z. Feng, Y. Luo, and Y. Han, “Design of LED freeform optical system for road lighting with high luminance/illuminance ratio,” Opt. Express 18, 22020-22031 (2010). 2. C.-H. Tsuei, W.-S. Sun, and C.-C. Kuo, “Hybrid sunlight/LED illumination and renewable solar energy saving concepts for indoor lighting,” Opt. Express 18, A640-A653 (2010). 3. X.-H. Lee, I. Moreno, and C.-C. Sun, “High-performance LED street lighting using microlens arrays,” Opt. Express 21, 10612-10621 (2013). 4. S. Wang, K. Wang, F. Chen, and S. Liu, “Design of primary optics for LED chip array in road lighting application,” Opt. Express 19, A716-A724 (2011). 5. J. Dong, W. van Driel, and G. Zhang, “Automatic diagnosis and control of distributed solid state lighting systems,” Opt. Express 19, 5772-5784 (2011). 6. T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. Consum. Electron. 50(1), 100-107 (2004). 7. H. Elgala, R. Mesleh, and H. Haas, “Indoor optical wireless communication: potential and state-of-the-art,” IEEE Commun. Mag. 49(9), 56-62 (2011). 8. J. K. Kwon, “Inverse source coding for dimming in visible light communications using NRZ-OOK on reliable links,” IEEE Photon. Technol. Lett. 22(19), 1455-1457 (2010). 9. K.-I. Ahn and J. K. Kwon, “Color intensity modulation for multicolored visible light communications,” IEEE Photon. Technol. Lett. 24(24), 2254-2257 (2012). 10. Y. U. Lee and M. Kavehrad, “Two hybrid positioning system design techniques with lighting LEDs and ad-hoc wireless network,” IEEE Trans. Consum. Electron. 58(4), 1176-1184 (2012). 11. S.-Y. Jung, S. Hann, and C.-S. Park, “TDOA-based optical wireless indoor localization using LED ceiling lamps,” IEEE Trans. Consum. Electron. 57(4), 1592-1597 (2011).

#192524 - $15.00 USD Received 24 Jun 2013; revised 28 Aug 2013; accepted 3 Sep 2013; published 16 Sep 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. S6 | DOI:10.1364/OE.21.00A917 | OPTICS EXPRESS A917

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1.

Introduction

LED lighting has become the prevailing trend as an energy-efficient, durable, and environmentally-friendly way of illumination [1–3]. This is basically digital lighting that has a multitude of major benefits over analog lighting [4,5]. Fast dimming control via digital modulations has brought about a new wireless communication medium of VLC [6–9]. When arranged in a network with an array of lamps, the merits of LED lighting are further elaborated with the improved degree of freedom in immediate controls. This networked lighting has been applied in diverse areas, such as navigation/positioning [10, 11] and intelligent lighting [2, 5, 12, 13]. Self-organization and self-optimization are amongst the key targets of future wireless systems to relax the requirements of human efforts in planning and operation tasks and to reduce the capital and operational expenditure. For such efforts, autonomic management strategies have been considered to capture the personal preferences for the recommendation of certain levels of lighting parameters [14]. For this goal, personal environment modules are introduced for new open space plans to enable users to control their own environment. This can often lead to conflicts from different preferences of users for a variety of environmental parameters. In addition, offering users with the total control results in few options for efficiently managing energy usage. There is thus a trade-off in benefits, choosing between an energy-efficient and a user-centered control strategy. The challenges to develop such a balanced lighting control strategy are three-fold [14]. First, the personal preferences in environments are monitored, as changes in configuration can occur for temporal changes in preference and the migration of


users. Secondly, efficient collection and distribution of the information about immediate environment and control are considered. The last challenge is to quantify and optimize the trade-off between satisfying the users preferences and reducing the energy utilization. Many studies have attempted to address those challenges in dimming control of LED lighting [13–19]: In [13, 15, 16], LED lighting control is formulated in the form of an optimization problem with the objective of minimizing the power and their centralized solutions are found. These centralized strategies require knowledge of global geometry and an additional agent to solve the optimization problems and route the obtained assignments. In [14], the control problem is formulated based on a discrete optimization and its distributed algorithm is determined using distributed decision theory. However, this discrete optimization problem is known to be NP hard and the distributed algorithm provides a suboptimal solution. Continuous-valued optimizations are formulated and the algorithms are developed based on heuristic approaches in [17,18]. In addition, the dimming levels desired by users are formulated in a matrix form and the best values are determined in a distributed manner using a least squares approach in [19]. Many of existing approaches develop distributed algorithms by splitting the centralized optimization formulation into a group of subproblems intentionally such that calculations of the optimal states are carried out in a distributed manner at different sites. Therefore, the solutions of those algorithms sometimes do not capture an explicit physical implication that enables an efficient management of dynamic behaviors of the algorithms and the quality of their solutions. Furthermore, most algorithms require synchronous distributed operations to search the solution correctly. In addition, the global information of the network geometry and knowledge about it by all network entries are necessary before distributed operations can be achieved. To overcome these shortcomings, this paper addresses a low-complexity distributed selforganizing strategy for the most energy-efficient configuration of LED lamps, with a new formulation where the desired illuminance of all users is properly served. Since the lighting network does not require a very sophisticated design, a distributed dimming control strategy enables the quick deployment. Furthermore, each LED lamp and its associated controller do not require (wired/wireless) direct communication with other lamps but obtain knowledge about them indirectly through information exchange with accessible users in terms of the VLC, thereby exempting the careful arrangement and configuration of the illumination. Upon the migration of users into and out of the lighting network, a group of LED lamps adapt their emitting light intensities immediately to the desired illuminance via dimming support cooperation. This triggers other lamps to self-configure for the network-wide distributed adaption of the user association and user-specific dimming control. This strategy yields a practical scheme that determines the optimal energy-saving configuration in a fully distributed manner such that the exact solution is found only by local interactions among lamps and users and global knowledge is unnecessary. 2.

Energy-Saving LED Lighting Control

Consider a two-dimensional space where N LED lamps and M users exist. The N lamps and M users are equipped with VLC or wireless communication features, such as Wi-Fi, Bluetooth, infrared communication, etc., which enables short-range communication with the neighboring counterparts. The lamps and users have their own IDs that distinguish them from all neighboring objects. This allows point-to-point communication between a single lamp and single user. The lamps are assumed to be able to choose their power continuously from the full level to the ‘off’ level but not necessarily indicating physically powered off. The populations of lamps and users depend on the environment. In an indoor environment, such as an office building, lamps normally outnumber users. The opposite situation can occur in an outdoor environment, such as street lighting. However, the subsequent argument is independent of the environment.


OFF

ON

ON

L2

User 1 OFF

ON

User 2 OFF

ON

LED lamps L1

L4

Users User 1

L5

User 2

L3 OFF

L6

User 3 ON

User 3

L7 L8 L9

(a)

(b)

Fig. 1. Example of an LED lighting network and associated bipartite graph.

Although a regular lamp array is considered only, the subsequent argument extends to an arbitrary arrangement. In this setup, which is referred to as a lighting network, it is important to obtain an energy-efficient configuration of lamps such that the minimal desired illuminance of all users is ensured by a subset of lamps and the remaining unnecessary lamps are powered off. All lamps need to adjust their dimming levels adaptively without central controls nor a collection of global information. Figure 1(a) depicts an example of a lighting network. The circles representing the LED lamps are placed regularly in the space. The boxes represent the users distributed randomly over the space. The yellow circles are the LED lamps lighting up the users. The edges represent the lighting contribution of the operating lamps to the connected users. Figure 1(b) presents the physical network configuration into a logical bipartite graph [20] that facilitates the mathematical formulation. To be more specific, in the bipartite graph in Fig. 1(b), there are two groups of nodes where a member of one group is only connected with the members of the other group by an edge. Since an edge represents the interaction (or communication) between its two end nodes, one member does not have any interactions with any other member in the same group, suggesting that the resulting lighting configuration does not invoke the requirement of internal communication among a group of LED lamps or a group of users and enables an asynchronous operation of the lighting control. In addition, since the group of users only is associated with the illuminance requirement, this configuration naturally envisions the constraint of the optimization formulation. In other words, a group of LED lamps defines the set of unknown variables to be determined and the other group of users defines the set of constraint to be satisfied in the optimization problem that follows. Some definitions are made here for the formulation. Let Pi and Ia be the maximum power level of the i-th lamp and the minimal illuminance level ensured for the a-th user, respectively. Let Iai be the maximum illuminance of the i-th LED lamp observed by the a-th user. For a user to measure Iai , each lamp sends the dedicated ID encrypted appropriately and may also transmit either the actual transmit power level encoded or a reference signal consisting of the full-power


levels and turn-off levels in a VLC data frame so that the user’s devices can calculate the current power. This also enables an evaluation of the precise impact of external light sources, such as background lighting which is not under the control and daylight, etc. For concreteness, a continuous variable xi ∈ [0, 1] is introduced to denote the relative power level of the i-th lamp, i.e., xi takes in between zero and one, corresponding to the turn-off and full-power emitting level, respectively. Thus, the actual power of the i-th lamp is given by Pi xi . It is a basic assumption that the intensity is proportional to the power. Correspondingly, the intensity of the i-th lamp observed at the a-th user is given by Iai xi . Therefore, those intensities caused by separate lamps contribute collectively to the overall illuminance that the a-th user undergoes. An adjacent lamp-user pair forms when each other’s IDs are received successfully and distinguished at both sides. Such a relationship depends on the geometry and relative positions to each other, because the light intensity is associated with the physical environment such as the distance and lamp type. Let I denote the matrix with Iai as the (a, i) entry and let Ii and Ia be the i-th column and the a-th row of I, respectively. Furthermore, let I∼i and x∼i be the matrix and the vector constructed by removing only the i-th column and the i-th entry, respectively. Finally, let I be the column vector with Ia as the a-th entry. An optimization is formulated with the objective that the total power consumption is minimized and all users are properly served with their own requirements. To be more precise, the optimal power levels of all LED lamps in the network are determined. The convex optimization formulation is obtained as min

xi ∈[0,1]

subject to

∑ Pi xi i

∑

Iai xi ≥ Ia , ∀a,

(1)

i∈N (a)

where N (a) is the set of adjacent lamps for the a-th user. The constraint dictates that each user be served with the intensity level of at least Ia . It is obvious that Eq. (1) is a linear program (LP), whose solution is obtained using general-purpose LP solvers. However, this approach burdens the network with the need for centralized control: i) It requires an independent device (possibly implemented with microprocessors) equipped with LP solvers for optimization. ii) Local information, such as observed illuminance, minimal requirement, maximum power levels and network geometry, should be collected and forwarded to the control device. iii) An appropriate means of distributing the calculated assignment to the entire network is necessary. In order to mitigate those burdens, this LP is solved in such a way that each lamp decides its power level with only local cooperation among adjacent users. A major advantage of this approach is the robustness against changes in network geometry. Upon a change in the users of the network, the proposed approach handles the local change in the configuration of the lamps and users, and warm start based on the current levels facilitates the rapid determination of new levels, whereas the centralized approach should obtain and solve the new LP from the beginning. In addition, the consideration of an external light source is warranted. Daylight may normally be one of significant external light sources. Thus, the illuminance observed by a user is an additive combination of light from LED lamps and the sun. To distinguish the LED emission, a time interval of turned-off symbols can be used. A series of symbols for which no emission occurs can be placed such that the users’ device can measure the actual observed illuminance and the users do not feel flicker. Since the external light source only emits during the time interval, the amount is evaluated at each user and can be subtracted from the constrained illuminance levels to define the effective contribution by LED lamps. Furthermore, background lighting can be used such that the users do not feel a large contrast between their task area and the background, i.e., all LED lamps may be turned on at some low level and their state variables xi all are nonnegative. In such a case, xi can be reexpressed as xi = xi + x¯i , where x¯i is the power level #192524 - $15.00 USD Received 24 Jun 2013; revised 28 Aug 2013; accepted 3 Sep 2013; published 16 Sep 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. S6 | DOI:10.1364/OE.21.00A917 | OPTICS EXPRESS A921

associated with background lighting. Therefore, optimization problem (1) can be reformulated with respect to new variables xi and remains a formulation that is similar to the original. 3.

Distributed Assignment Strategy

A distributed algorithm that does not require any additional central control is derived here. Although the development of a distributed algorithm for LP formulation (1) is seemingly a simple task, indeed this is not the case. In optimization problems with a linear objective or a monotonically-increasing objective, a class of distributed algorithms that update each unknown variable one by one in a fully asynchronous way are quite likely to fail to converge to a fixed point, even far from being the optimal solution, and only oscillate between two ends of full (xi = 1) and off (xi = 0) levels. The reason is that, compared to synchronous distributed algorithms that can make simultaneous moves for unknown variables to approach the optimal solution, asynchronous distributed algorithms normally update each variable in a best-effort way such that the current objective function is maximized. Therefore, the class of those algorithms is likely to choose the end point associated with the extreme values of the objective function, in particular, at the early stage of the algorithm where few variables converge to correct points. In this lighting control problem, a simpler deployment that does not require a highly synchronized operation is preferred. Hence, an asynchronous distributed algorithm is more suitable for this purpose. Therefore, the development of a distributed assignment strategy is a nontrivial task. To handle this, a regularization term [21] is introduced to the linear optimization formulation for the improvement of the convergence property, and the alternating direction method of multipliers (ADMM) approach [22] is first used to obtain the preliminary operations of individual lamps and users that reside in the network. Then, the dedicated asynchronous operations carried out by individual lamps and users are derived via some algebraic transformation of the preliminary solution such that the resulting algorithm can be deployed physically in a full asynchronous mode. For the convenience of derivation, inequality-constrained LP (1) is recast into the equality-constrained one as

∑ Pi xi

min

xi ∈[0,1] za ≥0

i

subject to

∑

Iai xi − za = Ia , ∀a.

(2)

i∈N (a)

Note that a positive slack variable za is associated with exactly one constraint. This is interpreted as a state variable of the a-th user and acts as a feasibility indicator that becomes nonnegative only when the constraint is met. The derivation begins with augmented Lagrangian formulation [21, 23] max ua ≥0

g(u) ≡ min Lρ (u, x, z) xi ∈[0,1] za ≥0

(3)

where augmented Lagrangian Lρ (u, x, z) is given by

ρ Ix − z − I 2 2 i ρ ≡ ∑ Li (u, xi ) + ∑ La (ua , za ) + Ix − z − I 2 , 2 a i

Lρ (u, x, z) = ∑ Pi xi + ρ uT (Ix − z − I ) +

(4)

where Li (u, xi ) = (Pi + ρ uT Ii )xi and La (ua , za ) = −ρ ua (za + Ia ). Note that the third term is additionally introduced compared to ordinary Lagrangian formulation, in order to enhance the #192524 - $15.00 USD Received 24 Jun 2013; revised 28 Aug 2013; accepted 3 Sep 2013; published 16 Sep 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. S6 | DOI:10.1364/OE.21.00A917 | OPTICS EXPRESS A922

stability in convergence. However, this term prevents Lagrangian Lρ (u, x, z) from being split into the sum of individual functions optimized over the respective variables. To handle this, this term is solved iteratively: At the update step for variable xi , Li (u, xi ) + ρ2 Ix − z − I 2 is minimized for xi with the other variables fixed. Subsequently, at the update step for variable za , La (ua , za ) + ρ2 Ix − z − I 2 is minimized for za with the other variables fixed. Then, a preliminary form of the algorithm is obtained as ρ (t+1) (t) = arg min Pi xi + Ii xi + I∼i x∼i − z(t) − Ia + u(t) 2 x˜i xi 2 ρ (t+1) (t) − za + Ia x(t+1) − Ia + ua 2 zã = arg min za 2 (t+1)

ua

(t) xi (t) za

(t)

(t+1)

=ua + (Ia x(t+1) − za

− Ia )

(t) =T[0,1] (x˜i ) (t) =TR+ (˜za ),

(5)

where TS [v] is a threshold function that limits the value of v within S ⊂ R, i.e., ⎧ ⎪ ⎨max S if v ≥ max S , TS [v] = v if v ∈ S , ⎪ ⎩ min S if v ≤ min S .

(6)

In addition, R+ denotes the set of nonnegative real numbers. By some algebraic operations, the final form of the distributed algorithm is given in a message-passing form [24] as

∑

(t)

ma→i =Ia −

j∈N (a)\i

=T[0,1]

(t+1)

= − Ia +

zâ

(t)

(7)

(t) ∑a∈N (i) Iai ma→i Pi , − ρ ∑a∈N (i) Iai2 ∑a∈N (i) Iai2

(t+1)

xi

(t)

Ia j x j + |˜za |,

∑

i∈N (a)

(t+1)

Iai xi

(t)

+ TR− [ˆza ],

(8) (9)

(t)

where, ma→i is defined to represent the message transferred from the a-th user to the i-th lamp, (t) xi is the message broadcast from the i-th lamp to all neighboring users, N (i) is the set of the i-th lamp’s adjacent users, and R− denotes the set of non-positive real numbers. In addition, (t+1) zâ is a new variable defined by combining two variables za and ua appropriately because the two variables represent the state of the same object (the a-th user) and appear in common in several expressions of Eq. (5). Note that design parameter ρ appears in Eq. (8) only. For a linear objective function, this can be thought of as a constant factor, i.e., Pi /ρ can be considered the individual contribution of xi to the objective, instead of Pi . Therefore, it does not have any impact on the performance of the algorithm presented here and is set to unity for simplicity. (t) For the physical deployment, the i-th LED lamp broadcasts its level xi to all adjacent users (t) and the a-th user sends message ma→i to the i-th lamp at the t-th iteration, respectively. These updates continue until all messages converge to fixed values. Here are the physical interpre(t) tations of the variables: Message ma→i is an estimate of the i-th lamp’s intensity observed at the a-th user. The a-th user requests the i-th lamp to emit so that it observes this level of the (t+1) is an estimate of the i-th lamp’s relative power level. illuminance at its position. Message xi The first term of Eq. (8) amounts to the weighted average of the i-th lamp’s relative power level. #192524 - $15.00 USD Received 24 Jun 2013; revised 28 Aug 2013; accepted 3 Sep 2013; published 16 Sep 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. S6 | DOI:10.1364/OE.21.00A917 | OPTICS EXPRESS A923

The second term of Eq. (8) adjusts the level based on its contribution to the objective so that (t+1) the objective is minimized. Variable zâ is an internal state rather than a message because it is not sent to any lamp. The negative value indicates that the desired illuminance has not been met. The outline of the algorithm is summarized in Algorithm 1. Algorithm 1 Distributed assignment (0)

Initialize: Set ma→i = Ia for all (a, i), t ← 0. repeat repeat (t+1) using Eq. (8) and broadcast to all adjacent Lamp: At the i-th lamp, update message xi users. (t+1) User: On receipt of messages from the i-th lamp at the a-th user, update message ma→i using Eq. (7) and send it back to the i-th lamp. until All user-lamp pairs exchange messages at least once. (t+1) using Eq. (9). Adjust: At all users, update state zã Increment: t ← t + 1. until All messages converge or maximum number of iterations is reached. (t) Assign: The i-th lamp adjusts its power level according to xi . The convergence property of the proposed algorithm is warranted here. The algorithm based on the ADMM framework converges to the optimum if all associated functions are closed and convex [23]. Since all functions in Eq. (1) are linear, thereby satisfying all those conditions, the algorithm is guaranteed to converge. Note that the power level may have a more complicated γ relationship with the light intensity such as Pi xi with γ > 1 for relative illuminance level xi , because additional input power is needed to drive the output intensity in the higher intensity regime compared to the lower regime for certain types of the LED lamp. Therefore, the objecγ tive of Eq. (1) can be given by ∑i Pi xi . When γ > 1, the convexity of the objective still holds, and the algorithm and its convergence remain intact. 4.

Simulation Results

The performance of the proposed algorithm is tested and compared with other algorithms by simulation. For the simulation setup, an indoor environment with dimensions of 10m×10m×2.5m is considered. The LED lamps are placed regularly every 2m atop, i.e., 25 LED lamps in total (N = 25). Each lamp is made up of an arrangement of four off-the-shelf white dimmable LED bulbs, each with the maximum power of 17W and maximum luminous flux of 1100lm, considered as a single point light source. That is, the i-th light source has max= 4400lm. The simulation imum power of Pi = 68W and maximum luminous flux of Φmax i considers M users ranging from 1 to 30 distributed uniformly indoors. The user devices are located at the height of desks, 0.7m high. Let Lai and θai be the distance and angle between user-lamp pair (a, i), respectively. The possible range of illuminance for the pair is calculated 2 2 based on Φmax i /(π Lai ) cos θai [6]. More precisely, Lai is the length of the line connecting the i-th lamp and the a-th user and θai is defined as the angle between the vertical line from the i-th lamp and the line connecting the i-th lamp and the a-th user. The maximum luminous intensity m+1 (in cd) in the vertical direction is given by Φmax i 2π , where m is the order of Lambertian emiscos θai )/π . The sion [6]. When m = 1 is set, the intensity in the direction of θai becomes (Φmax i −2 and the received direction cos θai yield the overall expression. consideration of the path loss Lai Then, the M × N matrix I is obtained for the corresponding network dimensions. First, the adaptation of the algorithm to the temporal change in topology is examined. Figures #192524 - $15.00 USD Received 24 Jun 2013; revised 28 Aug 2013; accepted 3 Sep 2013; published 16 Sep 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. S6 | DOI:10.1364/OE.21.00A917 | OPTICS EXPRESS A924

Lamps’ output 1 L3 L4 L8 L9 L13 L14 L19 L20 L24 L25

0.9

Relative lamp power

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

1000

2000 3000 Time unit

4000

5000

Fig. 2. LED lamps’ relative intensity levels for 15 users and 25 lamps.

Users’ illuminance 700 A B C D E F G H I J K L M N O

600

Illuminance (lx)

500 400 300 200 100 0 0

1000

2000 3000 Time unit

4000

5000

Fig. 3. Users’ observed illuminance levels for 15 users and 25 lamps.


Optimal configuration (950 time units) 1

C 6

1

11

Vertical dimension (m)

2 3 4 5

2

7

12

10 0

500

17

22

I

400

B 3

8

F 13

18

23

D9

14

19

24

L

M

15

20

300

K 4

O

8 G 9

21

H

6 7

600

N

16

200

E J5

10

Illuminance (lx)

0

100

25 A

2

4 6 Horizontal dimension (m)

8

10

0

Fig. 4. Optimal configuration at the 950th time unit.

2 and 3 illustrate the temporal trajectories of the output power of a subset of lamps (the value of (t) (t) xi ) and the input illuminance level of users (the value of ∑i Iai xi ), obtained by the proposed algorithm for an instance with 15 users and the minimum required illuminance of 400lx during the period of 5000 time instants. Furthermore, Figs. 4-8 show the corresponding optimal configurations at five different time instants (950, 1950, 2950, 3950, and 4950th instants). The lamps are represented in ten colors varying smoothly from black through shades of red, orange, and yellow, to white, corresponding to the relative lamp power from 0 to 1. A user and a lamps is connected with an edge if the lamp contributes to more than 10% of illuminance of the user. Initially, 25 LED lamps (labeled numerically) and 15 users (labeled alphabetically from ‘A’ to ‘O’) are located at the positions shown in Fig. 4. At every one-thousand time instant, the users move slightly as shown in Figs. 5-8 and, at the 4000 time instant, all lamps are requested to turn off. Thus, the corresponding illuminance for all users varies at every change. The time instant indicates the time duration that corresponds to the set of message update iterations where at least one message exchange occurs between each neighboring pair of lamps and users. From Eqs. (7) and (8), the lamps broadcast their new updates and the users can respond immediately. Since a VLC data frame is communicated at an operating frequency of at least a few megaHertz according to indoor PHY II mode of IEEE 802.15.7 VLC standard [25], this unit of time lasts for 8.33μ s - 267μ s with the assumption that the exchange of all messages takes roughly 1000 optical pulses at most. In Fig. 2, the trajectories of only LED lamps that turn on in the course of the simulation are presented and the optimal solutions of all lamps’ states obtained by the centralized LP are represented in dotted lines. Therefore, all lamps can adapt to the optimal states within at most 500 time instants which correspond to the time duration within 4.17ms - 133ms in indoor PHY II mode of the IEEE 802.15.7 VLC system. The above figures show that the proposed algorithm works correctly and has performance as good as the optimal solution. In addition, the algorithm successfully adapts to the temporal changes in the network configuration. The performance of the proposed algorithm (denoted by ‘MP’) is presented by simula#192524 - $15.00 USD Received 24 Jun 2013; revised 28 Aug 2013; accepted 3 Sep 2013; published 16 Sep 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. S6 | DOI:10.1364/OE.21.00A917 | OPTICS EXPRESS A926


C 6

1

11


2 3 4 5

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7

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tion over a large number of different random configurations. However, the comparison with works [13–19] is not straightforward because all existing approaches are based on the respective optimization formulations and the corresponding algorithms are developed specific to those distinct formulations. Instead, distributed algorithms based on the coordinate descent approach [26], which can operate fully asynchronously for lighting control, are presented. Such a class of algorithms pick variable xi either by turns or at random for the update at each iteration. The resulting algorithm (denoted by ‘CD’) is given by (t)

λ =T[0,1] max ( ai − Pi ) , a∈N (i) Iai

(t) (t) λai =TR+ Ia − ∑ Ia j x j .

(t+1)

xi

(10)

j∈N (a)\i

As in the previous algorithm, the i-th LED lamp broadcasts its update xi and the a-th user sends back corresponding request λai . This set of operations continues until everything converges or the maximally allowed iteration is reached. However, the algorithm can be further enhanced by applying a log-barrier function based approach [27] and the algorithm (denoted by ‘CDLB’) with ε > 0 is given by Pi Iai (t+1) , xi =T[0,1] max χ : ∑ = (t) χ ε a∈N (i) Iai χ − λai

(t) (t) (11) λai =TR+ Ia − ∑ Ia j x j . j∈N (a)\i

In this algorithm, each LED lamp is needed to solve its associated fractional equation to obtain a new update for xi . Note that another well-known algorithm based on dual decomposition [22] never works because of severe oscillations between both bounds. All algorithms are set to stop if there is no significant change in the update or the number of iterations exceeds the limit, and it is set that ρ = 1 and ε = 0.03. The objectives of all algorithms are evaluated with respect to more than 100 random instances for each dimension. Figures 9-12 show the resulting objective values and the normalized errors between the optimal objective value and algorithms’ output (defined by 10 log10 ((∑i Pi xi − ∑i Pi xiopt )/ ∑i Pi xiopt )) at the 100th and 500th iteration for the desired minimum illuminance of 200lx and 400lx, respectively. The optimal solution (denoted by xiopt ) is obtained using convex optimization solver CVX [28]. The proposed algorithm has the minimal objective values among asynchronous distributed algorithms and improves its performance for increasing number of the iteration. On the other hand, two other algorithms do not improve much as the number of iteration increases, thus the performance curves for two different iterations almost overlap. The insets in Figs. 9 and 11 show the performance of the algorithm in Eq. (10), as it has much poorer performance than other algorithms. The reason is that it is highly likely to be stuck at an intermediate suboptimal solution where all LED lamps’ desires conflict and no lamp wants to decrease its utility i.e., power saving, before converging to the correct solution. This is indeed a common phenomenon that occurs much more frequently in asynchronous distributed algorithms than in synchronous counterparts. The normalized error indicates the relative difference of the objective value obtained by a distributed algorithm with respect to the optimal solution xiopt . Note that the objective values are compared for the normalized error instead of the solutions themselves, in that several instances may have multiple optimal solutions with identical objective values. The simulation results show that the proposed algorithm outperform the other algorithms consistently for all dimensions at given iterations of 100 and 500. The other algorithms do not


Avg. Idesired=200 500 450

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Optimal MP (t=100) CDLB (t=100) CD (t=100) MP (t=500) CDLB (t=500) CD (t=500) 25

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Fig. 9. Objective values for the minimum required illuminance of 200lx.

Avg. I

=200

desired

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0 −10 −20 MP (t=100) CDLB (t=100) CD (t=100) MP (t=500) CDLB (t=500) CD (t=500)

−30 −40 −50

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25

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Fig. 10. Normalized errors from the optimal solution for the minimum required illuminance of 200lx.


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Optimal MP (t=100) CDLB (t=100) CD (t=100) MP (t=500) CDLB (t=500) CD (t=500) 25

30

Fig. 11. Objective values for the minimum required illuminance of 400lx.

Avg. Idesired=400 10

Normalized error (dB)

0 −10 −20 MP (t=100) CDLB (t=100) CD (t=100) MP (t=500) CDLB (t=500) CD (t=500)

−30 −40 −50

5

10


25

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Fig. 12. Normalized errors from the optimal solution for the minimum required illuminance of 400lx.


improve the optimal cost as quickly as the proposed algorithm and often do not approach the optimum closely enough, even after 500 iterations. Although the error of the proposed algorithm becomes progressively larger as the size increases, it decreases as the number of iterations increases. In addition, it is observed that, if the iteration is not limited, the error for the proposed algorithm eventually decays to zero, while other algorithms fail to do so. This suggests that the quadratic augmentation in Eq. (4) implicitly contributes to better convergence properties than the other approaches. The convergence can be accelerated if small-valued variables are fixed to zero during the algorithm, since a large portion of iterations are devoted to decide the powered-on lamps. The complexity of the algorithms is considered here. A brief analysis reveals that, at a single iteration, the overall complexity of the algorithm is given by O(NE[|N (i)|]), i.e., linear in the number of LED lamps over the configurations of sparse bipartite graph where each user can interact with only a subset of LED lamps in the lighting network. It is also known that the other algorithms have the same order of computational complexity. However, since the log-barrierbased algorithm (11) calculates solutions of the fractional equation and their maximum at every update, it is more demanding in computation. Additional issues warranted for the application to lighting design practice are finally addressed. Although realistic environments include indirect illumination along with direct illumination, the simulation presented in this work considers only direct illumination because of simple construction of input parameters and fast measurement of output parameters in the simulation configuration that reflects temporal changes of the environment. However, the algorithm is expected to work still correctly in the environments that include indirect illumination. Since indirect illumination only results in the increase of the value of Iai , the algorithm remain intact and still valid for the new setup. In addition, the convergence property is not impaired by indirect illumination. To see this, we consider the values of the observed illuminance levels. Since this additional component of the illumination only causes the increase in illuminance level at each user, it contributes favorably to the satisfaction of the minimum illuminance requirement for the user. Subsequently, the space of feasible configurations of lamps is expanded and the existence of the solution becomes more apparent. Therefore, this leads to the improvement of the convergence of the proposed algorithm. However, to evidence the usefulness of the proposed algorithm in more physical environments, additional consideration of indirect illumination is required for future research. 5.

Conclusion

A distributed LED lighting strategy is developed to determine the energy-efficient configuration. An optimization formulation is presented with the objective of minimizing the energy and the requirement of minimally allowed light intensity. With this formulation, a distributed algorithm is derived based in a message-passing form such that all LED lamps and users calculate the optimal assignment in a fully collective and asynchronous way. The simulation results show the advantages of the proposed algorithm in performance and convergence for various network topologies. In addition, the proposed algorithm successfully adapts to temporal changes of the lighting configuration. Acknowledgments This research was conducted under the industrial infrastructure program for fundamental technologies (10033630) which is funded by the Ministry of Trade, Industry & Energy (MOTIE, Korea).


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