## 1 Introduction  
In the recent decades, great progress has been made in light-emitting diode (LED) manufacturing where LED lighting is replacing the traditional incandescent and fluorescent lighting due to its low energy consumption, long lifetime, and high efficiency [1; 2]. Meanwhile, taking advantage of the superior modulation capability of LEDs, visible light communication (VLC) can provide data transmission and illumination simultaneously, which has become a promising complementary wireless communication technique for the traditional radio frequency (RF) due to the notable advantages of huge bandwidth, high security, and no electromagnetic interference [3; 4; 5]. White light illumination from LEDs can be realized by combinations of phosphors excited by blue LED emission or by the mixture of multi-color LEDs, and the latter one is regarded as a promising approach for VLC systems to provide potential high-speed data transmission and high-quality illumination compared with the phosphor-converted LEDs [6; 7; 8]. Essentially, VLC can provide wireless data transmission, as well as the inherent illumination function, so it is essential to meet the actual illumination requirements while achieving the communication function where the brightness and chromaticity are two basic characteristics measuring the LED lighting quality. Accordingly, many research works have been done in this field for the purpose of boosting the data rate and improving illumination quality. For the multi-color LED-based VLC system, the wavelength-division multiplexing (WDM) technique can be utilized for multiple parallel data transmission, while the receivers recover the transmitted signal of different sub-channels equippedwith optical filters. In [9], the RGB LED-based WDM-VLC system has been experimentally realized with an aggregate data rate of 4.5 Gbps. Moreover, an off-line 8-Gbps data rate VLC system can be achieved based on RGBY LEDs [10]. Unfortunately, these works merely pursue a higher transmission rate, ignoring the practical illumination requirements. Recently, some works analyzed the illumination quality with brightness and chromaticity constraints while guaranteeing the communication functions. The work in [11] proposed the power and transmission rate optimization design for the LED-based color-division VLC systems under the chromaticity constraints. Utilizing RGB LEDs, the optimal constellation design was formulated subject to some practical lighting constraints in [12]. An optical power allocation scheme was proposed in [13] to maximize the multi-user sum-rate for the multi-chip LED-based VLC systems in consideration of the luminance, chromaticity, amplitude, and bit error rate constraints. However, the practical multi-color LED-based VLC system have the channel crosstalk problem due to the overlapping spectra of the LEDs and the imperfection of optical filters [14], which were ignored in the previous works. Besides, as a major source of energy consumption, the lighting energy saving problem has attracted considerable attention. Therefore, an energy-efficient VLC system should be established and deserves further research. In this paper, the model of the multi-color LED-based VLC system is established, where the crosstalk problem among parallel channels is considered. More specifically, an optimal power allocation scheme is proposed to maximize energy efficiency via collaboratively meeting the luminance, chromaticity constraints and signal to interference plus noise ratio (SINR) requirements. It is worth noting that the chromaticity specification for LED products has been established in [15], where the chromaticity tolerance is modified to the quadrangle range to replace a fixed point in the Commission Internationale de I'Eclairage (CIE) 1931 chromaticity diagram, which provides more degrees of freedom to improve the performance of the system. Correspondingly, the optimization problem is convex, which can be solved by the efficient optimization toolbox CVX in MATLAB, which is utilized to solve convex programs.