Problem variables and parameters.
\r\n\tCyanobacteria are an interesting group of bacteria for their unique characteristics and potential biotechnological applications. They are photosynthetic autotrophs and are among the earliest forms of life to colonize the earth. Cyanobacteria are very important for life on earth because they are oxygenic organisms that also play crucial roles in the cycles of carbon, nitrogen, and oxygen. Interestingly, they have unique specialized cells called heterocysts for nitrogen fixation, have circadian rhythms and can survive in different hostile habitats. Therefore, they are ideal model organisms for studying photosynthesis, nitrogen fixation and other biological processes. In addition, cyanobacteria are well recognized for their potential for a variety of biotechnological applications. They were reported to produce a wide array of novel antimicrobial and anticancer compounds that are attracting interest from the pharmaceutical industry for drug development. Also, they have various applications in agriculture as potential biofertilizers or in the industry for biofuel production in addition to many other biotechnological applications. Though cyanobacteria are very important microorganisms, their classification and phylogenetic relationships among other bacterial lineages were puzzled. Given the recent advances in molecular data, bioinformatics and available completed genomes sequences, the evolution and taxonomy of these bacteria can be revealed. Another aspect of interest is the role cyanobacteria play in the ecosystem functionality and interactions with other organisms. In this regard, it is interesting to understand how the cyanobacteria adapt to such a wide range of environments and environmental stresses as well as the resistance mechanisms.
\r\n\r\n\tThis book will be essential for anyone with an interest in cyanobacteria and its topics will cover different aspects including their taxonomy, ecology and applications. It is meant as a useful resource for students, researchers and professionals studying or working with cyanobacteria.
",isbn:"978-1-83962-490-2",printIsbn:"978-1-83962-489-6",pdfIsbn:"978-1-83962-491-9",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"2fec78743d3f973c80881957ce3e6d79",bookSignature:"Prof. Wael N. Nabil Hozzein",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10442.jpg",keywords:"Structure, Morphology, Microstructure, Forms, Molecular Biology, Sequence Analysis, Genome Sequences, Comparative Genomics, Applications, Habitats, Resistance Mechanisms, Extreme Environments",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"October 1st 2020",dateEndSecondStepPublish:"November 16th 2020",dateEndThirdStepPublish:"January 15th 2021",dateEndFourthStepPublish:"April 5th 2021",dateEndFifthStepPublish:"June 4th 2021",remainingDaysToSecondStep:"2 months",secondStepPassed:!0,currentStepOfPublishingProcess:4,editedByType:null,kuFlag:!1,biosketch:"Prof. Hozzein is the author of more than 100 publications, an editorial board member for Microbiology Insights (SAGE journals), and a reviewer for Antonie van Leeuwenhoek Journal, International Journal of Systematic and Evolutionary Microbiology, World Journal of Microbiology and Biotechnology, etc. In the past, he also worked as a visiting scientist at Newcastle University, UK, and Michigan State University, USA.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"189233",title:"Prof.",name:"Wael N.",middleName:"Nabil",surname:"Hozzein",slug:"wael-n.-hozzein",fullName:"Wael N. Hozzein",profilePictureURL:"https://mts.intechopen.com/storage/users/189233/images/system/189233.jpeg",biography:"Prof. Wael N. Hozzein is a professor of microbiology at the\nFaculty of Science, Beni-Suef University, Egypt. He received his\nPhD from Cairo University, Egypt, in 2003, and then worked as\na visiting scientist at Newcastle University, UK, and Michigan\nState University, USA. Recently, he worked as the chair professor\nof the Bioproducts Research Chair at King Saud University, Saudi\nArabia. He has vast experience in the field of bacterial taxonomy\nwith research interests in microbial biodiversity and applications. Prof. Hozzein is\nthe author of more than 140 publications and a guest editor and reviewer for several\ninternational journals. Additionally, he has been involved in many academic activities\nand educational reform projects and initiatives. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"878",title:"Phytochemicals",subtitle:"A Global Perspective of Their Role in Nutrition and Health",isOpenForSubmission:!1,hash:"ec77671f63975ef2d16192897deb6835",slug:"phytochemicals-a-global-perspective-of-their-role-in-nutrition-and-health",bookSignature:"Venketeshwer Rao",coverURL:"https://cdn.intechopen.com/books/images_new/878.jpg",editedByType:"Edited by",editors:[{id:"82663",title:"Dr.",name:"Venketeshwer",surname:"Rao",slug:"venketeshwer-rao",fullName:"Venketeshwer Rao"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"52340",title:"A Continuous-Time Recurrent Neural Network for Joint Equalization and Decoding – Analog Hardware Implementation Aspects",doi:"10.5772/63387",slug:"a-continuous-time-recurrent-neural-network-for-joint-equalization-and-decoding-analog-hardware-imple",body:'\nEnergy efficiency has been increasingly attracting more interest due to economical and environmental reasons. Mobile communications sector has currently a share of 0.2% in global carbon emissions. This share is expected to double between 2007 and 2020 due to the ever-increasing demand for wireless devices [1, 2]. The sustained interest in higher data rate transmission is strengthening this impact. While major resources are being invested in increasing the energy efficiency of digital circuits, there is, on the other hand, a growing interest pointing at alternatives to the digital realization [3], including a mixed (analog/digital) approach. In such an approach, specific energy consuming (sub)tasks are implemented in analog instead of a “conventional” digital realization. The analog implementation possesses a high potential to significantly improve the energy efficiency [4] because of the inherent parallel processing of signals that are continuous in both time and amplitude. This has been shown in the field of error correction coding with a focus on decoding of low-density parity-check (LDPC) codes. Our ongoing research on equalization reveals similar results. We do not intend “analog” for linear signal processing with all its disadvantages like component inaccuracies and susceptibility to noise and temperature dependency [5] but for nonlinear processing instead. The work of Mead [6] and others on Neuromorphic analog very-large-scale integration (VLSI) has shown that “analog signal processing systems can be built that share the robustness of digital systems but outperform digital systems by several orders of magnitude in terms of speed and/or power consumption” [5].
\nThe nonlinearity makes the analog implementation of an algorithm as robust as its digital counterpart [3, 5]. This profits from the match between the needed nonlinear operations for the algorithm and the physical properties of analog devices [7].
\nThe capability of artificial neural networks (in the following neural networks) to successfully solve many scientific and engineering tasks has been shown oftentimes. Moreover, mapping algorithms to neural network structures can simplify the circuit design because of the regular (and repetitive) structure of neural networks and their limited number of well-defined arithmetic operations. Digital implementations can be considered precise (reproducibility of results under similar circumstances) but accurate (closeness of a result to the “true” value) only to the extent to which they have enough digits to represent [8]. This means, accuracy in digital implementations is achieved at the cost of efficiency (e.g., relatively larger chip area and more power consumption) [9]. An analog implementation is usually efficient in terms of chip area and processing speed [9], however, at the price of an inherent lack of the reproducibility of results [8] (because of a limited accuracy of the network components as an example [9]). However, by exploiting the distributed nature of neural structures the precision of the analog implementation can be improved despite inaccurate components and subsystems [8][1] -. In other words, it is the distributed massively parallel nonlinear collective behavior of an analog implementation (of neural networks) which offers the possibility to make it as robust as its digital counterpart but more energy efficient[1] - (additionally to smaller chip area). Particularly for recurrent neural networks (the class we focus on when considered as nonlinear dynamical systems), the robustness can be additionally achieved by exploiting “attracting” equilibrium points. In the light of this discussion, we map in this chapter a joint equalization and decoding algorithm into a novel continuous-time recurrent neural network structure. This class of neural networks has been attracting a lot of interest because of their widespread applications. They can be either trained for system identification [10], or they can be considered as dynamical systems (dynamical solver). In the latter case, there is no need for a computationally complex and time-consuming training phase. This relies on the ability of these networks (under specific conditions) to be Lyapunov stable.
\nEqualization and channel decoding (together, in the following detection) are processes at the receiver side of a digital transmission. They aim to provide a reliable and efficient transmission. Equalization is needed to cope with the interference caused by multipath propagation, multiusers, multisubchannels, multiantennas and combinations thereof [11]. Channel (de)coding is applied for further improving the power efficiency. Equalization and decoding are nonlinear discrete optimization problems. The optimum solutions, in general, are computationally very demanding. Therefore, suboptimum solutions are applied, often soft-valued iterative schemes because of their good complexity-performance trade-off.
\nFor high data rates, the energy consumption of equalization and decoding algorithms is expected to become a limiting factor. The need for floating-point computation and the nonlinear and iterative nature of (some of) these algorithms revive the option of an analog electronic implementation [12, 13], embedded in an essentially digital receiver. This option has been strengthened since the emergence of the “soft-valued” computation in this context [4] since soft-values are a natural property of analog signals. In contrast to analog decoding, analog equalization did not attract that amount of attention.
\nFurthermore, joint equalization and decoding (a technique where equalizer and decoder exchange their local available knowledge) further improves the efficiency of the transmission as an example in terms of lower bit error rates, however, at the cost of more computational complexity [14]. Most of the work related to joint equalization and decoding is limited to the discrete-time realization. One of the very few contributions focusing on continuous-time joint equalization and decoding is given in [13]. The consideration in [13] is not “neural networks-based”. Stability and convergence are observed but not “deeply” considered.
\nWe introduce in this chapter a novel continuous-time joint equalization and decoding structure. For this purpose, continuous-time single-layer recurrent neural networks play an essential role because of their nonlinear and recursive characteristic, special suitability for analog VLSI and since they serve as promising computational models for analog hardware implementation [15]. Both, equalizer and decoder are modeled as continuous-time recurrent neural networks. An additional proper feedback between equalizer and decoder is established for joint equalization and decoding. We also review individually, both continuous-time equalization and continuous-time decoding based on recurrent neural network structures. No training is needed since the recurrent neural network is serving as a dynamical solver or a computational model [15, 16]. This means, transmission properties are used to define the recurrent neural network (number of neurons, weight coefficients, activation functions, etc.) such that no training is needed. In addition, we highlight challenges emerging from the analog hardware implementation such as adaptivity, connectivity and accuracy. We also introduce our developed circuit for analog equalization based on continuous-time recurrent neural networks [3]. Characteristic properties of recurrent neural networks such as stability and convergence are addressed too. Based on the introduced model, we show by simulations that the superiority of joint equalization and decoding can be preserved in the analog “domain”.
\nThe main motivation for performing joint equalization and decoding in analog instead of using conventional digital circuits is to improve the energy efficiency and to minimize the area consumption in the VLSI chips [17]. The proposed continuous-time recurrent neural network serves as a promising computational model for analog hardware implementation.
\nThe remainder of this chapter is organized as follows: In Section 2, we describe the block transmission model. Sections 3 and 4 are dedicated to the equalization process, the application of continuous-time recurrent neural networks and the analog circuit design and its corresponding performance and energy efficiency. Sections 5 and 6 are devoted to the channel decoding and the application of continuous-time recurrent neural networks for belief propagation (a decoding algorithm for LDPC codes). For both equalization and decoding cases, analog hardware design aspects and challenges and the behavior of the continuous-time recurrent neural network as a dynamical system are discussed. The continuous-time joint equalization and decoding based on recurrent neural networks is presented in Sections 7 and 8. Simulation results are shown in Section 9. We finish this chapter with a conclusion in Section 10.
\nThroughout this chapter, bold small and bold capital letters designate vectors (or finite discrete sets) and matrices, respectively.[1] - All nonbold letters are scalars. diagm{B} returns the matrix B where the nondiagonal elements are set to zeros.
The block transmission model for linear modulation schemes is shown in Figure 1. For details, see [18]:
\nBlock transmission model for linear modulation schemes. SRC (SNK) represents the digital source (sink). DET is the detector. COD performs the encoding process (adding redundancy). M\n\n maps encoded bits to complex-valued symbols. R\n is the block transmit matrix.
SRC (SNK) represents the digital source (sink). SRC repeatedly generates successive streams of k bits, i.e., q1, q2, ⋯ ,qM.
\n
\n
COD performs a bijective map from q to qc where
x ∈ ψN is the transmit vector of length N.
N is the block size. Successive transmit vectors are separated by a guard time to avoid interference between different blocks. Thus, Figure 1 describes the transmission for a single block and stays valid for the next block (possibly with a different R).
ψ
\n
We distinguish:\n
– For an uncoded transmission
– For a coded transmission and
– For a coded transmission and
– For a coded transmission and
R
\n
DET is the detector including equalization and decoding.
The model in Figure 1 is a general model and fits to different transmission schemes like orthogonal frequency division multiplexing (OFDM), code division multiple access (CDMA), multicarrier CDMA (MC-CDMA) and multiple-input multiple-output (MIMO). The relation with the original continuous-time (physical) model can be found in [11, 18]. The model in Figure 1 can be described mathematically as follows [11]:
By decomposing R into a diagonal part Rd = diagm{R} and a nondiagonal part R\\d = R−Rd, Eq. (1) can be rewritten as:
For the j-th element of the receive vector
We notice from Eqs. (2), (3) that the nondiagonal elements of R describe the interference between the elements of the transmit vector at the receiver side. For interference-free transmission R\\d = 0. For an interference-free transmission over an additive white Gaussian noise (AWGN) channel R = I.
\n\nFigure 2 shows the channel matrix for a MIMO transmission scheme for different number of transmit/receive antennas. Figure 3 shows the channel matrix for OFDM with/without spreading. Figure 4 shows the channel matrix for MIMO-OFDM. In Figures 2–4, the darker the elements, the larger the absolute values of the entries of the corresponding matrix R, and hence larger the interference [21].
\nVisualization of the channel matrix for a MIMO transmission scheme with eight transmit antennas and different receive antennas.
Visualization of the channel matrix for OFDM with 16 subcarriers and spreading over four subcarriers with/without interleaving.
Visualization of the channel matrix for a MIMO-OFDM transmission scheme with eight subcarriers and three transmit antennas.
Remark 1. For a clear distinction between channel matrix and block transmit matrix, we refer to [11, 18]. Generally speaking, the block transmit matrix R is a block diagonal matrix of “many” channel matrices.
\nThe detector DET in Figure 1 has to deliver a vector
Detection: EQ is the equalizer, DEC is the decoder, DECI is a hard decision function. Notice the feedback from the decoder to the equalizer in (b), i.e., the turbo principle.
DECI in Figure 5 is a hard decision function. For a coded transmission, DECI is a unit step function. For an uncoded transmission, COD and DEC are removed from Figure 1 and Figure 5, respectively. DECI in this case is a stepwise function depending on the symbol alphabet ψ which maps the (in general complex-valued) elements of the equalized vector
For an uncoded transmission DECI: \n
For a coded transmission DECI:
For an uncoded transmission, the detection DET reduces to a vector equalization EQ as shown in Figure 6.
\nUncoded block transmission model. Neither encoding at the transmitter nor decoding at the receiver. The detection reduces to a vector equalization EQ.
The optimum vector equalization rule (the maximum likelihood one) is based on the minimum Mahalanobis distance and is given as [21]
For each receive vector
The dynamical behavior of continuous-time single-layer recurrent neural networks of dimension N′, abbreviated in the following by RNN[1] -, is given by the state-space equations [22]:
In Eq. (5), ϒe is a diagonal and positive definite matrix of size N′ × N′. v(t) is the output, u(t) is the inner state, e is the external input.
Continuous-time single-layer real-valued recurrent neural network. v(t) is the output, u(t) is the inner state, e is the external input and φ(⋅) is the activation function. This model is known as “additive model” or “resistance-capacitance model” [23].
As a nonlinear dynamical system, the stability of the RNN is of primary interest [16]. This has been proven under specific conditions by Lyapunov’s stability theory in [24] for real-valued RNN and in [22, 25] for complex-valued ones, among others. The RNN in Eq. (5) represents a general purpose structure. Based on N′, φ, W, W0 a wide range of optimization problems can be solved. First and most well-investigated applications of the RNN include the content addressable memory [24, 26], analog-to-digital converter (ADC) [27] and the traveling salesman problem [28]. In all these cases, no training is needed since the RNN is acting as a dynamical solver. This feature is desirable in many engineering fields like signal processing, communications, automatic control, etc., and has first been exploited by Hopfield in his pioneering work [24, 29], where information has been stored in a dynamically stable RNN. We focus in the following on the vector equalization.
\nRemark 2. The dimension of a real-valued RNN is the same as the number of neurons.
\nRemark 3. Two real-valued RNNs each of N′ neurons are required to represent one complex-valued RNN (with dimension N′). This is possible by separating Eq. (5) into real and imaginary parts. However, this doubles in general the number of connections per neuron (and hence the number of multiplications) because of the required connections (represented by Wi) between the two real-valued RNNs as it can be seen from the following equation:
ϒe in this case is a diagonal positive definite matrix of size 2 · N′ × 2 · N′ and
The usage of the RNN for vector equalization became known for multiuser interference cancellation in CDMA environments [30, 31]. However, this was limited to the binary phase-shift keying (BPSK) symbol alphabet ψ = {−1, +1}. This has been generalized to complex-valued symbol alphabets in [21] by combining the results of references [20], [22], [32][1] -. Based thereon, it has been proven that the RNN ends in a local minimum of Eq. (4) if the following relations are fulfilled [21], cf. Eqs. (1), (2), (5) and Figures 6 and 7.
shows an example of an eight quadrature amplitude modulation (8 QAM) symbol alphabet and its corresponding DECI function. The relations in Eq. (7) are obtained by the comparison between the maximum likelihood function of the vector equalization and the Lyapunov function of the RNN.
\nAn example of an 8 QAM symbol alphabet and its corresponding DECI function. Each element of the symbol alphabet (marked with ×) has its own “decisions region” visualized by different colors. The function DECI delivers that element of the symbol alphabet, where the input argument lies in its corresponding decision region.
The dynamical behavior of the vector equalization based on RNN can be given as, cf. Eqs. (1), (5), (7)\n
The locally asymptotical stability of Eq. (8) based on Lyapunov functions has been proved in [21] (based on [22]) for separable symbol alphabets ψ(s). When Eq. (8) reaches an equilibrium point uep, i.e.,
If additionally, a correct equalization is achieved, i.e.,
Thus, the RNN as vector equalizer, Eq. (8) acts as “analog dynamical solver” and there is no need for a training. The covariance matrix of ne is
In Eq. (7), θ(opt) (·) is the optimum activation function and depends on the symbol alphabet ψ. For BPSK (a real-valued case)
Remark 4. For separable symbol alphabets,
The analog signal processing as a matter of topical importance for modern receiver architectures was recognized in [34], where an analog vector equalizer—designed in BiCMOS technology—was considered as a promising application for the analog processing of baseband signals. The equalizer accepts sampled vector symbols in analog form with an advantage that the equalizer does not require an ADC at the input interface. At very high data rates, the exclusion of an ADC softens the trade-off between chip area requirement and overall power consumption. We discuss in the following section the main features/challenges of the analog implementation of the vector equalizer based on RNN.
\nStructure: An RNN of dimension N′ (in general 2 ·N′ neurons) is capable to act as a vector equalizer as long as the block size at the transmitter side N (over all possible symbol alphabets, coding schemes and block sizes) is as maximum as N′, i.e., N ≤ N′.
\nActivation function: The definition of the optimum activation function θ(opt)(·) is not general, but depends on the symbol alphabet under consideration. Different symbol alphabets need different activation functions. However, we have proven in [20] that for square QAM symbol alphabets—the most relevant ones in practice—θ(opt) (·) can be approximated as a sum of a limited number of shifted and weighted hyperbolic tangent functions. Square QAM symbol alphabets are separable ones, cf. Remark 4. The analog implementation of the hyperbolic tangent well befits the large-signal transfer function of transconductance stages based on bipolar differential amplifiers [3, 34].
\nAdaptivity: A vector equalizer must be capable to adapt to different and time-variant interference levels. The adaptivity is regulated by the measurement of the block transmit matrix R, a task performed by a “channel estimation unit” (CEU). The weight matrices W and W0 are then computed as in Eq. (7) and forwarded to the RNN (Figure 9). Thus, the weight matrices W and W0 are not the outcome of any training algorithm but related directly to R, cf. Eq. (7). This represents a typical example for the mixed-signal integrated circuit, where the weight coefficients are (obtained and) stored digitally, converted into analog values, later used as weight coefficients for the analog RNN [8].
\nUncoded block transmission model. The detection reduces to a vector equalization EQ. The channel estimation unit (CEU) estimates the block transmit matrix R.
For the j-th neuron in the additive model Figure 7, the ratio between two resistors Rj and Rjj′ (Rj and Rj0) is used to configure each weight coefficient wjj′ (wj0). According to the additive model, Rjj′ and Rj0 can assume both positive and negative values, and the absolute value theoretically extends from Rj to infinite (for wjj′ ∈ [−1, +1]). This puts serious limitations to the direct implementation of the model. In [3], we showed how this difficulty can be overcome by using a Gilbert cell as a four-quadrant analog multiplier. A Gilbert cell [35] is composed of two pairs of differential amplifiers with cross-coupled collectors, and is controlled by a differential voltage input Gji applied at the base gate of the transistors. When biased with a differential tail current
Accuracy: Locally asymptotical Lyapunov stability can be guaranteed for the RNN in Eqs. (5), (8) if, among others, the hermitian property is verified for the weight matrix W (the symmetric property in the real-valued case). Inaccuracies in the weights’ representation may jeopardize the Lyapunov stability and impact the performance of the vector equalizer. The first cause of weights’ inaccuracy may arise from the limited accuracy of the analog design in terms of components’ parasitics, devices’ mismatch, process variation, just to name a few. Those inaccuracies (if modest) are expected to slightly degrade the performance without causing a catastrophic failure, thanks to the high nonlinearity of the equalization algorithm. Moreover, it has been shown in [8], [36] that in some cases, they produce beneficial effects: These imperfections incorporate some kind of simulated annealing which enables escaping local minima by allowing occasionally “uphill steps” since the Lyapunov stable RNN is a gradient-like system. This feature is emulated in discrete-time by stochastic Hopfield networks [23]. Non-precision of the weights may also arise from an insufficient resolution of the digital-to-analog converter (DAC) (Figure 9). On the other hand, an overzealous DAC design increases the chip area, the power consumption and adds complexity to the interface between the analog vector equalizer and the digital CEU. In this case, a conservative approach suggests to use a DAC with enough resolution to match the precision used by the CEU.
\nInterneuron connectivity and reconfigurability: Scaling the architecture of an analog VLSI design is not straightforward. A vector equalizer based on recurrent neural networks is composed by the repetition of equal sub-systems, i.e., the neurons. Using a bottom-up approach, the first step to scale the system involves the redesign of the single neuron in order to handle more feedback inputs. In a successive step, the neurons are connected together and a system-level simulation is performed to check the functionality of the system. However, several design choices must be made during the process and it is not guaranteed that the optimum architecture for a certain number of neurons is still the best choice when the number of neurons changes. For large N, the block transmit matrix R, defining the weight matrix W, is usually sparse. If a maximum number of nonzero elements over the rows of R is assumed, the requirement for a full connectivity between the neurons in Figure 7 can be relaxed, and only a maximum number of connections per neuron will be necessary. In this case, however, in addition to the “adaptivity”, the RNN must be reconfigured according to the position of the nonzero elements in R. The hardware simplification given by the partial connectivity may be counterbalanced by the necessity of a further routing (e.g., multiplexing/demultiplexing) of the feedback. For special cases, where the block transmit matrix can be reordered around the diagonal, more independent RNNs can be simply used in parallel. In Figures 3(b) and 3(c), four independent RNNs, each of dimension four, can be used in parallel. Additionally, for specific transmission schemes such as MIMO-OFDM in Figure 4, the connectivity can be assumed limited (number of transmit antennas minus one) and fixed (crosstalk only between same subcarriers, when used simultaneously on different transmit antennas).
\nExample 1. In Figure 4, eight RNNs (number of subcarriers) each of dimension of three (number of transmit antennas) can be used in parallel. Each neuron has two feedback inputs.
\nWe review here the main features of the analog circuit design of an RNN as vector equalizer working with the BPSK symbol alphabet and composed of four neurons. Detailed explanation can be found in reference [3]. The RNN is realized in IHP 0.25 μm SiGe BiCMOS technology (SG25H3). A simplified schematic of a neuron is shown in Figure 10. Schematics of gray boxes are presented in Figure 11.
\nA simplified schematic of a single neuron as a part of a (four neurons) RNN analog vector equalizer. u′j\n\n is the inner state, e′j is the external input and Gji\n is used for adapting the weight coefficient wji\n from the output of the i-th neuron to the input of the j-th neuron. The circuit is fully differential [3].
Details of the circuit building blocks. Gilbert cell used as a four-quadrant analog multiplier, buffer stages, BJT differential pairs for the generation of the hyperbolic tangent function and a metal-oxide-semiconductor field-effect transistor (MOSFET) switch used as a sequencer [3].
The dynamical behavior of the circuit in Figures 10 and 11 is described as [3]
which is equivalent to Eq. (5). τ = R · C is the time constant of the circuit. R is shown in Figure 10 and C is a fictitious capacitance between the nodes and
(1) Performance: Simulation results based on the above described analog RNN are shown in Figure 12. The interference is described by the channel matrix Rtest.
\nBER vs. Eb/N0\n for the analog RNN vector equalizer. Evolution time equals 10⋅τ\n. BPSK symbol alphabet and channel matrix Rtest\n.
\n
The black dashed line shows the bit error rate (BER) for a BPSK symbol alphabet in an AWGN channel (an interference-free channel). Performance achieved by the maximum likelihood algorithm in Eq. (4) is included as a solid black line. The performance of the analog RNN vector equalizer[1] - is presented in a solid red line with square markers. Compared to the optimum algorithm, the signal-to-noise ratio (SNR) loss for the analog RNN vector equalizer can be quantified in approximately 1.7 dB at a BER of 10−4. This loss in SNR emphasizes the suboptimality of the RNN as vector equalizer and depends on the channel matrix. Figure 13 shows an example of a transient simulation for the analog RNN vector equalizer. The time constant is approximately τ = 40 ps. The SNR ratio is set to 2 dB and a series of three receive vectors are equalized in sequence. Because of the channel matrix and noise, the sampled vectors at the input of the equalizer
An example of a transient simulation for the analog RNN vector equalizer.
Remark 5. The evolution and reset times are the two limiting factors for the maximum throughput of the analog RNN vector equalizer. However, they cannot be unlimitedly minimized since the RNN needs a minimum evolution time to reach an equilibrium point representing a local minimum of the Lyapunov function, i.e., a local minimum of Eq. (4).
\n(2) Energy efficiency: The energy efficiency of a hardware “architecture” is the ratio between the power requirement (Watt) of the architecture and its achievement in a given time period. In our case, the throughput of the equalizer represents the achievement. Combining the value of τ and the power consumption, the abovementioned analog vector equalizer is expected to win the competition versus common digital signal processing, thanks to three to four orders of magnitude better energy efficiency [3].
\nChannel coding (including encoding at the transmitter side COD and decoding at the receiver side DEC) aims to enable an error-free transmission over noisy channels with maximum possible transmit rate. This is done by adding redundancy (extra bits) at the transmitter side, i.e., the bijective map from q to qc (Figure 14,) such that the codewords qc are sufficiently distinguishable at the receiver side even if the noisy channel corrupts some bits during the transmission. Figure 14 shows a coded transmission over an AWGN channel.
\nFor every received codeword, the optimum decoding (the maximum likelihood one) needs to calculate the distance between the received codeword and all possible codewords
Coded transmission over an BER channel.
One of the largest drawbacks of RNNs is their quadratic Lyapunov function [39]. Optimization problems associated with cost functions of higher degree cannot be solved “satisfactorily” by RNNs. Increasing the order of the Lyapunov function leads to a nonlinear feedback in the network. In doing so, we obtain the single-layer high-order recurrent neural network, named differently in literature, depending on the nonlinear feedback [39–42].
\nRemark 6. High-order recurrent neural networks are in the literature exclusively real-valued.
\n\nFigure 15 shows the continuous-time single-layer high-order recurrent neural network, abbreviated in the following by HORNN[1] -.
\nThe dynamical behavior is given by
\n\n
Continuous-time single-layer real-valued high-order recurrent neural network. vˇ(t) is the output, uˇ(t) is the inner state, eˇ is the external input and φˇ(⋅)\n is the activation function. fˇ(vˇ)\n is a real-valued continuously differentiable vector function with fˇ(0nˇ)=0nˇ [21].
The parameters in Eq. (16) can be linked to Figure 15 in the same way as Eq. (5) linked to Figure 7.
Remark 7. In the special case
In order to apply HORNNs to solve optimization tasks, their stability has to be investigated. A property without which the behavior of dynamical systems is often suspected [39]. This was the topic of many publications [39–42]. A common denominator of the locally asymptotical stability proof of the HORNN based on Lyapunov functions is
\n
The right side of the first line of Eq. (16) can be rewritten as a gradient of a scalar function.
Tanner graph of the systematic Hamming code n = 7 and k = 4.
Originally proposed by Gallager [37], belief propagation is a suboptimum graph-based decoding algorithm for LDPC codes. The corresponding graph is bipartite (n parity nodes and n – k check nodes) and known as Tanner graph [43]. This is shown in Figure 16 for the Hamming code with the parity check matrix
For every binary linear block code characterized by the binary parity check matrix H of size (n − k) × n for n > k, three binary matrices
\n
\n
\n
In the last relation,
\n
\n
\n
\n
The dynamical behavior of belief propagation can be described based on Eqs. (16), (18) and Figures 14, 15, and 16 [45]
\nIn a series of papers, Hemati et. al. [12, 17, 46–49] also modeled the dynamics of analog belief propagation as a set of first-order nonlinear differential equations Eq. (20). This was motivated from a circuit design aspect, where
The absolute stability of belief propagation Eqs. (20), (21) was proven for repetition codes (one of the simplest binary linear block codes) in [44, 45]. In this case
A simple model for analog decoding as presented in [46].
Far away from repetition codes, it has been noticed that iterative decoding algorithms (belief propagation is one of them) exhibit depending on the SNR a wide range of phenomena associated with nonlinear dynamical systems such as existence of multiple fixed points, oscillatory behavior, bifurcation, chaos and transit chaos [50]. Equilibrium points are reached at “relatively” high SNR. The analysis in reference [50] is limited to the discrete-time case.
\nRemark 8. The HORNN in Figure 15, Eqs. (18), (20) for belief propagation acts as a computational model.
\nMany analog hardware implementation aspects have been already mentioned in Section 4-B. We mention here only additional aspects exclusively related to the analog belief propagation based on HORNN.
\nStructure: In practice, different coding schemes (different parity check matrices H) with various (k, n) constellations are applied to modify the code rate
Adaptivity: No training is needed.
Interneuron connectivity: No full connection is needed since the matrix P for LDPC codes is sparse. The number of connections per neuron must equal the maximum number of nonzero elements in P row-wise over all considered coding schemes and equals
Vector function connectivity: For different coding schemes, the number of the arguments
Remark 9. For a specific coding scheme, the interneuron connectivity can be made fixed. The resulted HORNN structure in this case is valid also for all codeword lengths resulted after performing a puncturing of the original code.
\nRemark 10. Both, the interneuron connectivity and the weight adaptation play a significant role, in the equalization as well as in the decoding. It can safely be said that they represent the major challenge of the circuit, since the analog circuit must be capable to perform equalization and decoding for a given number of possible combinations of block size, symbol alphabet, coding scheme, etc. Particularly for the decoding, the advantage of having a non-full connectivity is counterbalanced by a double (and very complex) (de)multiplexing of the signals (once for the vector function
Two examples for joint equalization and decoding. Notice the feedback from the decoder to the equalizer, i.e., turbo principle.
Joint equalization and decoding as described in [52]. Lext\n represents the “knowledge” obtained by exploiting the redundancy of the code.
Turbo equalization is a joint iterative equalization and decoding scheme. In this case, a symbol-by-symbol maximum aposteriori probability (s/s MAP) equalizer exchanges in an iterative way reliability values L with a (s/s MAP) decoder [51, 52]. This concept is inspired from the decoding concept of turbo codes, where two (s/s MAP) decoders exchange iteratively reliability values [53]. Despite its good performance, the main drawback of the turbo equalizer is the very high complexity of the s/s MAP-equalizer for multipath channels with long impulse response (compared with symbol duration) and/or symbol alphabets with large cardinality. Therefore, a suboptimum equalization (and a suboptimum decoding) usually replace the s/s-MAP ones (Figure 18).
\nOne discrete-time joint equalization and decoding approach has been introduced in [52] and is shown in Figure 19.
σ2 is given in Eq. (11). If we consider only the equalization loop in Figure 19, we notice that it describes exactly the dynamical behavior of discrete-time recurrent neural networks [19, 25, 33, 54–56]
\nRemark 11. If
Motivated by the expected improvement of the energy efficiency by analog implementation compared with the conventional digital one, we map in this section the joint equalization and decoding structure given in Figure 19 to a continuous-time framework. s/s MAP DEC in Figure 19 is replaced by a suboptimum decoding algorithm: the belief propagation. Moreover, equalization and decoding loops in Figure 19 are replaced by RNN and HORNN as discussed previously in Sections 4-A and 6-A, respectively. The introduced structure serves as a computational model for an analog hardware implementation and does not need any training.
\n\nFigure 20 shows a novel continuous-time joint equalization and decoding based on recurrent neural network structures. The dynamical behavior of the whole system is described by the following differential equations:
Continuous-time joint equalization and decoding based on a recurrent neural network structure for real-valued symbol alphabet (for complex-valued ones, cf. Remark 3). L⌣(t)\n\n, x⌣(t)\n\n are the soft output of the decoder and the equalizer, respectively. (a) The equalization part. (b) The decoding part.
\n
\nEq. (26a) and Figure 20(a) describe the continuous-time vector equalization, cf. Eqs. (1), (5), (7), (8).
\nEq. (26c) and Figure 20(b) describe the continuous-time belief propagation, cf. Eqs. (16), (18), (20).
Comparing Figure 20 with Figures 7 and 15, we notice that
The function
The functions
The novel structure in Figure 20 is general and stays valid for the following cases:
\nContinuous-time separate equalization and decoding based on a recurrent neural network structure for real-valued symbol alphabet (for complex-valued ones, cf. Remark 3). x⌣(t)\n\n is the soft output of the equalizer.
\n
Separate equalization and decoding: In this case, Figure 20(a) is modified such that no feedback from decoder to equalizer is applied. This is shown in Figure 21(a). Only at the end of the separate equalization and decoding process, the output is given as
Equalization and decoding take place separately at the same time.
Successive equalization and decoding: only after the end of the equalization process,
Coded transmission over an AWGN channel: In this case,
Uncoded transmission over an “interference-causing” channel: In this case,
The diagonal elements in
Unequal diagonal elements in ϒe (and ϒd) represent some kind of continuous-time asynchronicity [46]. Asynchronicity in discrete-time RNNs is desirable since it provides the ability to avoid limit cycles, which can probably occur in synchronous discrete-time RNNs [54, 57].
\nAssuming
From a dynamical point of view, the case
Remark 12. We notice that the parameters of the transmission model (block transmit matrix, symbol alphabet, block size, channel coding scheme) are utilized to define the parameters of the continuous-time recurrent neural network structure in Figure 20 such that no training is needed. This represents in practice a big advantage especially for analog hardware. However, to enable different coding schemes and symbol alphabets, either a full connectivity or a vector and interneuron connectivity controls are needed. Both structures are challenging from a hardware implementation point of view.
\nRemark 13. For the ease of depiction, Figures 20 and 21 assume that one transmitted block contains exactly one codeword. This is not necessarily the case in practice. As an example, if one transmitted block contains two codewords, one RNN and two parallel HORNNs will be needed. On the other hand, if one codeword lasts over two transmitted blocks, two parallel RNNs and one HORNN is needed.
\nWe simulate the dynamical system as given in Eq. (26) and Figure 20 based on the first Euler method [58]. We assume: BER vs. Eb/N0\n for evolution time equals 20⋅τ\n. Continuous-time (joint) equalization and decoding. BPSK symbol alphabet.Figure 22.
BPSK modulation scheme (symbol alphabet
Each transmitted block contains one codeword, cf. Remark 13.
\n
Channel coding scheme: An LDPC code with
Multipath channels [60]:\n
- Proakis-a abbreviated in the following by its channel impulse response
- Proakis-b abbreviated in the following by its channel impulse response
The impulse response of
The block transmission matrix R is a banded Toeplitz matrix of the autocorrelation function of the channel impulse response [61]. The following cases are considered:
Uncoded transmission over AWGN channel. The bit error rate can be obtained analytically and is given as
Coded transmission over AWGN channel and continuous-time decoding at the receiver (HORNN-belief propagation).
Uncoded transmission over (the abovementioned) multipath channels and continuous-time equalization at the receiver (RNN-equalization).
Coded transmission over (the abovementioned) multipath channels. We distinguish between joint equalization and decoding (Figure 20) and separate equalization and decoding (Figure 21). In the latter case, equalization is performed firstly, and consequently the decoding.
The evolution time for the whole system in all cases is 20 ⋅ τ, i.e., all simulated scenarios deliver the same throughput. For separate equalization and decoding, the evolution time of the equalization equals the evolution time of the decoding and equals 10 τ. The simulation results are shown in Figure 22. We notice the following:
Joint equalization and decoding outperforms the separate one, which is a fact we know from the discrete-time case. Our proposed model in Figure 20 is capable of “transforming” this advantage to the continuous-time case.
For the channel
For the channel
If only equalization performance is considered, we compare between “Uncoded & EQ” curves and “Uncoded BER” curves. In Figure 22(a), the vector equalizer based on continuous-time recurrent neural networks is capable to remove already all interferences caused by the multipath channel
Remark 14. Interleaving and antigray mapping often encountered in the context of iterative equalization and decoding can be easily integrated in the proposed model in Figure 20. Antigray mapping will influence the functions
Joint equalization and decoding is a detection technique which possesses the potential for improving the bit error rates of the transmission at the cost of additional computational complexity at the receiver. Joint equalization and decoding is being considered only for the discrete-time case. However, for high data rates, the energy consumption of a digital implementation becomes a limiting factor and shortens the lifetime of the battery. Improving the energy efficiency revives the analog implementation option for joint equalization and decoding algorithms, particularly taking advantage of the nonlinearity of the corresponding algorithms.
\nContinuous-time recurrent neural networks serve as promising computational models for analog hardware implementation and stand out due to their Lyapunov stability (the proved existence of attracting equilibrium points under specific conditions) and special suitability for analog VLSI. They have often been applied for solving optimization problems even without the need for a training. The drop of the training is particularly favorable for analog hardware implementation.
\nIn this chapter, we introduced a novel continuous-time recurrent neural network structure, which is capable to perform continuous-time joint equalization and decoding. This structure is based on continuous-time recurrent neural networks for equalization and continuous-time high-order recurrent neural networks for belief propagation, a well-known decoding algorithm for low-density parity-check codes. In both cases, the behavior of the underlying dynamical system has been addressed, Lyapunov stability and simulated annealing are a few examples. The parameters of the transmission system (channel matrix, symbol alphabet, block size, channel coding scheme) are used to define the parameters of the proposed recurrent neural network such that no training is needed.
\nSimulation results showed that the superiority of joint equalization and decoding preserves, if this is done in analog according to our proposed model. Compared with the digital implementation, the analog one is expected to improve the energy efficiency and consume less chip area. We confirmed this for the analog hardware implementation of the equalization part. In this case, the analog vector equalization achieves an energy efficiency of a few picojoule per equalized bit, which is three to four orders of magnitude better than the digital counterparts. Additionally, analog hardware implementation aspects have been discussed. We showed as an example the importance of the interneuron connectivity, especially pointing out the challenges represented either by the hardware implementation of a massively distributed network, or by the routing of the signals using (de)multiplexers.
\nSustainable urban development motivates investments in environment-friendly and user-centered Public Transport (PT) services. Three trends towards next generation PT systems are observed, namely 1) introduction of greener vehicles such as electric/hybrid busses (e-busses), 2) focus on high service quality (e.g. increased ride comfort via mitigation of stop-and-go driving) and 3) reduction of emissions and operating costs related to fuel/energy consumption and equipment wear and tear. These trends however bring new challenges. The first challenge is posed by different operational characteristics and constraints of e-busses, e.g. they need to periodically recharge batteries at e-charging stations placed in selected stops and terminals. This brings additional constraints into PT operations and its cost dynamics. The existing approaches lack the required degree of modeling detail necessary to capture the complex interactions emerging between bus operations and charging infrastructure. The second challenge is how to guarantee comfort- and cost-effective operations without negatively impacting general traffic performance. Relying solely on strategies such as Transit Signal Priority (TSP), which prioritize PT vehicles at signalized intersections, might cause congestion effects that could backfire on the PT system itself.
The main contribution of this work is that we jointly address constraints and control capabilities of all entities of the PT ecosystem, which consists of signal control, (e-)busses, and e-bus charging infrastructure. The developed methods combine cooperation and negotiation between all actors thanks to connectivity, in order to effectively achieve mutual goals. Thanks to bus real-time positioning systems (Automatic Vehicle Location, AVL) and vehicle-to-infrastructure communication (Signal Phase and Timing, SPaT), multi-objective optimization is employed to determine bus dispatching time, operating speeds, dwell time plans, e-bus charging schedules, and TSP requirements. Regarding the interaction between busses and e-charging infrastructure, the objective is to minimize electricity costs and adhere to the planned bus dispatching times. From the online/operational perspective, the problem is to model and optimize a connected and cooperative system with a set of heuristic tools and actions, such that real-time system disturbances can be addressed, in order to maximize the adherence to the offline plans. For example, busses can use information on upcoming green times to adapt their speeds or hold at a stop in order to avoid stopping at signals. Consequently, stop-and-go is mitigated in an efficient and non-invasive way.
This chapter is structured as follows. Chapter 2 provides an overview of the e-bus eco-system, and the integrated design approach we developed in this work. Chapter 3 focuses on the integrated scheduling and charging problem at the planning phase, in particular considering a hybrid fleet of electric and hybrid busses. Chapter 4 deals with the operational phase, and in particular it shows the benefits of the cooperative ITS-based control strategies. Finally, chapter 5 provides an outline and the potential future research directions for this research.
The quality and service level of bus systems often rely on the interaction of different lines, in order to provide optimal frequencies and hence acceptable waiting times for the users, and to offer sufficient capacity to accommodate the demand, measured in terms of passenger flows. These flows vary across the network due to the variability of the demand, which differs depending on the origin and the destination of the users, and in time. To match the demand with the supply, bus operators aim to manage efficiently their fleet of vehicles, identifying at any time the most opportune vehicle type and the number of vehicles to be assigned to a line, together with their dispatching times. This decision has consequences on the way lines run smoothly and provide a certain level of service quality to the passengers, as well as it impacts the operational costs (Figure 1). In this study we consider design decisions (node density, network density and line density) as given.
Integrated design of bus systems.
Allocating a small number of vehicles limits the service frequency, which affects the waiting time. However, increasing this number will have a negative impact on the operating costs, since more vehicles and drivers will need to be employed. Too many vehicles on the other hand may result in under-utilizing vehicle capacities, reducing the marginal profits for the operators. Hence, optimally allocating fleet resources in the network is a fundamental planning problem that impacts both operators’ costs and passengers’ experience.
Emerging trends in green PT systems offer new benefits: e-busses reduce emissions, energy use, noise as well as offer smoother rides. There are three types of e-busses—hybrid electric, plug-in hybrid electric, and battery electric. The last two are able to recharge their batteries from an electric power grid via an opportunity charging—a bus periodically charges at bus stops or terminals. This allows to downsize battery and extend bus range to a desirable value. E-bus systems are currently moving from pilot projects to small-scale deployments with single line/operator with very few charging stations. The potentials and needs of large-scale e-bus systems have been investigated by the EU’s Zero Emission Urban Bus System (ZeEUS) project [1] as well as Volvo’s City Mobility Program [2]. More recent EU projects investigated the impact of fleet mix and configuration parameters to the operation costs [3].
When introducing e-busses, additional costs need in fact to be accounted for, since current battery-capacitated e-busses need to be recharged multiple times a day (e.g. a Volvo 6700 bus can perform a trip in full electric mode for around 30 km, and each vehicle can run distances of a few hundreds of km each day). Current opportunity charging technologies allow a bus to recharge up to 80% in a matter of 6-10 min, while novel flash charging technologies can recharge in less than a minute, but it extends the range of only few more kilometers. An example is the TOSA system in Geneva, a single line that uses both opportunity (3-4 min with low power) and at bus stops e-charging (15-second each 1-1.5 km with high power) [4]. Given the costs of fast and flash charging, bus operators charge their e-busses overnight, when the cost of electricity is lowest, and then use opportunity charging stations, typically located at line terminals, to recharge during the short resting times of the drivers. Flash charging are up to date very rarely implemented, given the very high costs of the relatively small gain in terms of range extension.
The charging infrastructure creates a strong link between infrastructure planning and bus operations [5]. The location and charging operations in fact influences the dispatching times of the vehicles, and in turn irregularities in the operations with recurrent phenomena of bus bunching may result in busses queuing at the charging station, with additional propagation of delays and overall degradation of service levels. Therefore, past research focused on developing a proper system design including strategic locations of e-charging stations [6, 7]. Energy efficiency was also addressed via energy management strategies for the engine [8], and regenerative breaking technologies [9], and taking into account environmental policies such as zero-emission zones [10].
In this study we contribute to this stream of research by focusing on the problem of integrating vehicle scheduling and dispatching times with recharging needs and operations of the e-bus fleet. In particular, we consider the problem of managing a mixed fleet of vehicles, which will be likely to be the case for the next years to come, since full electrification will require heavy investments in the electrical grid and current batteries and chargers are considered a relatively immature technology to completely replace combustion engines. We show in Chapter 3 that optimally assigning vehicle types in the network will provide benefits for both service quality (mitigation of delays due to charging) and operating costs (more e-busses used in daily operations are likely to bring lower energy consumption costs).
The main PT service quality objective is expressed in terms of punctuality (for schedule-based operations) and/or headway regularity (for headway-based operations). Current methods are based upon in-vehicle support systems, managing holding strategies and preferential signal control (TSP) and providing PT vehicles with preferential treatment at intersections via temporary traffic signal timing adjustments [11, 12]. For schedule-based operations, holding strategies (delaying departure of a bus from a bus stop until the scheduled time) ensure punctuality by managing slack times (extra “backup time” inserted into schedules) [13]. The problem of existing methods is that they slow down busses due to the fact that they add delays to the planned trip time [14]. They also address isolated lines and ignore any disturbances observed in real-world PT operations [15]. Headway-based operations are more difficult to control, as the strategies need to account for several busses [16, 17] and multiple interacting lines [18]. Thus, additional ITS systems such as Automated Vehicle Location (AVL), Automated Passenger Count (APC) and a central coordination entity are used to control busses in real time [15, 19].
The core reliability objective is also supported by TSP strategies capable of providing conditional priority. However, since TSP influences the traffic flow reliability [20] its acceptance is limited. Future improvements of TSP exploiting AVL can be achieved. This allows previously unfeasible continuous exchange of information between vehicles and traffic signals [21], allowing cooperation bus-signal through e.g. speed advisory [22, 23]. Such systems are one of the few ITS applications that would provide benefits even at early stages of CV technology [24].
Recent advances in V2I communication enable developing a new promising efficiency-oriented class of driving support systems aiming at improving driving efficiency, comfort and reducing unnecessary stops at signals [25]. Opposite to signal control, which uses CV technology to collect information about the approaching vehicles, in V2I-based systems vehicles use signal control information to optimize their own speeds accordingly. The two SPaT-based DASs researched in literature are the Green Light Optimal Speed Advisory (GLOSA) and Green Light Optimal Dwell Time Advisory (GLODTA). GLOSA provides vehicles with speed guidance, while GLODTA advises additional holding at bus stops. Their main advantage is that these systems improve bus performance with respect to traffic signals, but, unlike TSP, they are non-intrusive (i.e. do not influence signal timings). The two V2I-based advisory systems can be combined to mutually increase their effectiveness [26] and they can be combined with traditional holding strategies. These integrated controls have been shown to meet both objectives of service regularity and reducing the number of stops, as well as they reduce the number of TSP requests [27, 28].
In this work we adopt a cooperative system approach, following the C-ITS paradigm, reinforced by an energy-aware decision support system. This approach allows to manage the interplay between PT ecosystem actors (vehicles, signals, and e-infrastructure). Secondly, it enables joint optimization and coordination of actions carried out by the different actors, in order to achieve system goals.
Figure 2 provides an overview of the eCoBus integrated system developed in this project. The core module consists of collecting static input, namely the location of charging stations, lines timetables, together with the characteristics of the fleet (number of e-busses and hybrid vehicles), the characteristics of the lines (trip lengths) and of the signal infrastructure. We also assume to collect in real time trip times through AVL technology, battery states from the busses, status of each charging (occupied, available) and to have a good estimate of the passenger arrivals at stops (via e.g. APC information). These are input to the scheduling and charging optimization module, which is presented in detail in Section 3, whereas the driver advisory system combining holding and C-ITS based control and TSP are used at the operational phase to manage the vehicles in real time. The integrated system is shown to provide significant benefits both for planning objectives (better use of the fleet and the charging infrastructure, lower operations costs), and management goals (lower trip time variability and passenger costs, less fuel or energy consumed, less use of TSP requests). These benefits will be showcased in simulations using realistic scenarios in the next sections.
The eCoBus integrated management ecosystem.
Vehicle scheduling problems in public transportation have been approached as part of the “full operational planning process” [29]. From a modeling perspective, these problems are usually formulated as Mixed-Integer Linear Programs (MILP), under the name of Single/Multi-Depot Vehicle Scheduling Problem (SDVSP/MDVSP) [30]. The impact of electrification on bus scheduling problems has been recently taken into consideration by researchers, e.g. [31, 32], in preparation and support towards widespread Public Transport electrification. In this Section we present results stemming from our own recent research efforts [33, 34, 35] concerning the development of mixed-fleet vehicle scheduling models and algorithms tailored to the ongoing electrification of the bus fleet in the City of Luxembourg.
Compared to combustion, a fleet of electric or partially electric busses brings novel challenges to transit planning. Within the four decisional stages as discussed in [29] (line planning, timetabling, vehicle scheduling and crew rostering), electrification especially influences scheduling. The problem faces an increase in complexity, as recharging operations must be included without introducing disturbances in the existing schedule, to both ensure that busses have sufficient charge to perform trips and to avoid conflicts at the charging infrastructure. When handling a mixed fleet, optimal scheduling policies should therefore seek to take as much advantage as possible from both coexisting technologies.
In this Section we introduce mathematical models and methodologies to address the problem of scheduling a mixed fleet of conventional and electric busses. We begin by introducing the offline, planning stage optimization problem related to both single and multi-terminal instances. Subsequently we discuss a potential extension towards online, reactive rescheduling in the presence of disturbances, such as delays. Finally, we present a multi-terminal case study based on the city of Luxembourg, in the eponymous country.
We formulate the problem of assigning a mixed fleet of
Throughout this Section we adopt the assumption that full charging of e-busses happens within a single time step, as will be detailed later. Table 1 introduces the meaning of each variable and parameter, as well as their domain.
Var. | Domain | Explanation |
---|---|---|
1 if trip j is initiated by e-bus i at time t, 0 otherwise | ||
1 if trip j is initiated by h-bus h at time t, 0 otherwise | ||
1 if e-bus i is being recharged at charging station m at time t, 0 otherwise | ||
Total energy in kWh that e-bus i has at time t | ||
Total energy in kWh required to perform trip j, considering e-bus as a means of transport | ||
Preferred departure time step for trip j | ||
Duration of trip j in time steps | ||
Slack variable, necessary to ensure that constraint (12) does not violate the domain of | ||
Total battery capacity in kWh for all electric busses | ||
Minimum battery charge in percentage for each electric bus | ||
1 if e-bus i is not available to perform any trip nor recharge at time t, 0 otherwise | ||
1 if h-bus h is not available to perform any trip at time, 0 otherwise |
Problem variables and parameters.
The formulation’s objective function, in Eq. (1), is that of minimizing the total operational cost:
Trip costs c and
Energy component
Constraint (3) ensures that an e-bus can be assigned to at most one trip at a time, or recharge in at most one charger at a time, only in those time steps in which the bus is available. Constraint (4) implies that an e-bus which initiates a trip j whose duration in time steps is greater than one cannot be used to perform any other trip nor recharged throughout the entire duration of trip j. Constraints (5) and (6) enforce similar dynamics for trips being performed by hybrid busses. Note that matrices
By supplying a set of lines with accompanying timetables, the model can be employed to determine the optimal scheduling for a mixed-fleet of e-busses and h-busses. Parameters such as fleet size, fleet composition (% of electrics, % of hybrids), charging stations’ availability and capacity are supplied exogenously.
In order to correctly represent realistic Public Transport services, we improve and extend the model showcased in the previous section to appropriately represent multi-terminal schedules featuring deadheading trips.
For each trip j we therefore introduce a departure terminal
We discretize time in consecutive time steps
In this work, we allow deadheading trips for electric busses only. Deadheading is, in fact, critical to optimize usage of electric busses, which have cheaper operational costs, and to optimize their charging dynamics, allowing them to move to terminals equipped with charging stations when needed, while it is not strictly necessary for optimal dispatching of hybrid/conventional combustion busses. The model could anyway be easily extended to consider deadheading for hybrid busses.
The updated formulation’s objective function is as follows:
Cost vectors
Energy component
Constraints (17)–(21) avoid conflicts in the usage of resources. Constraints (22)–(23) model when the scheduled trips should be executed. Constraints (24)–(30) control the charging and discharging dynamics and ensure that the trip execution is consistent with battery status. Constraints (31)–(41) control the location dynamics of each bus.
The optimization model described in the previous section is aimed at determining full day bus schedules in the Public Transport planning stage, i.e. assuming a specific trip timetable and considering no deviations arising from operations. However, the model has been designed with the explicit objective of enabling the application of time-based decomposition schemes, for the sake of scalability in seeking solutions at the planning stage. In this Section we showcase how this decomposable nature can be further exploited, in combination with a Model Predictive Control scheme, to compute real-time rescheduling in case of major disruptions arising from operations (e.g. delays due to overcrowding, bunching, congestion,…).
The two models described earlier can be rather straightforwardly decomposed along the time variable
These constraints communicate the status of the fleet along different time-lapses, informing the later time periods on both availability and battery status of the busses as a result of the scheduling decisions performed in the earlier time periods.
By correctly configuring the frequency of time decomposition (at each time step), the width of the time lapses (chosen equal to the desired prediction horizon) and the appropriate bus availability data (busses are marked available only after the effective trip completion, rather than following a pre-determined trip duration), a Model Predictive Control application can be devised, as shown in Figure 3.
MPC scheme.
The proposed model and solution framework have been implemented in Matlab™, employing IBM’s ILOG Cplex 12.7 as optimization software. We validated our multi-terminal model against a real-life instance arising in the city of Luxembourg, considering several urban bus lines, as shown in Figure 4. Four of the terminals are currently equipped with two opportunity charging stations each. We employ our model on two different sets of tests: one addressing a subset of bus lines (lines 1, 16, 9 and 14, comprising 536 daily trips across 5 bus terminals, 2 of which are equipped with chargers), and one representing the complete instance (10 bus lines, 1034 daily trips across 12 terminals, 4 of which are equipped with chargers).
Case study: 10 bus lines in the City of Luxembourg.
The results shown in Figures 5 and 6 show consistently that, as the fleet transitions towards full electrification, the overall operational cost decreases and the number of total recharges increases accordingly.
Bus lines, 536 trips – Total operational costs and recharge operations (left); distinct cost factors (right).
10 bus lines, 1034 trips – Total operational cost and recharge operations (left); distinct cost factors (right).
It is interesting to note that the rate at which operational costs decrease and the total amount of recharging operations increase both exhibit an inflection point: in the set of tests addressing all the 10 lines, the gradient decreases at about 30% of electrified fleet, while in the reduced problem addressing 4 bus lines it becomes actually flat at about 70% of electrified fleet.
These results showcase that a diminishing returns effect might arise when approaching full electric operations. The effect is however less impactful in the full-scale scenario, implying that complex instances might lead to larger potential gains to be attained through electrification.
Operation is the last pillar, following design and planning. The nature of public transport operations is stochastic, with disruptions occurring due to irregularities in travel times and variation in passenger demand. Thanks to the advances in Intelligent Transportation Systems (ITS), the performance of a transit network can be monitored in real time, and corrective actions can be applied to restore the targeted level of service. All different applications have widened the spectrum of real time control strategies that can be deployed [13]. Until now, C-ITS Driver Advisory Systems have exclusively focused on assisting vehicles traverse signalized intersections and reducing the number of TSP requests, disregarding the consequences of their control actions to the regularity of the transit line [27]. The regularization of a line is the main objective of many real time strategies for public transport, with holding to be one of the thoroughly investigated in literature and applied in practice [17, 36]. We investigate how C-ITS can complement holding strategy and achieve a synergy to address both the objectives of regularity and the mitigation of the number of stops at signalized intersections.
We combine two DAS, namely GLOSA and GLODTA, with a rule-based holding criterion at stops prior to signalized intersections, to provide a pair of holding time and speed advisory or a holding time to achieve both objectives. The combined controllers are presented in the following sections, followed by the results obtained from a real-world case study.
The first regularity based advisory system is R-GLOSA. At the bus stops applied, it instructs a vehicle to be held to regulate the operation and depart with the speed needed to traverse the next green phase. After the arrival of a vehicle at a bus stop prior to a signalized intersection and the completion of dwell time, its position subject to the preceding and the succeeding vehicle is checked. If the headway from the preceding vehicle is short enough, then the vehicle will be held until the consecutive headways are even. We use the same rule-based holding criterion with [36], which regulates the departure time of a vehicle and limits the maximum allowed headway based on the planned headway.
After holding time is calculated, the departure time from the stop is updated and the expected arrival to the first downstream signalized intersection is estimated. The expected arrival time at the first signalized intersection downstream
After the expected arrival time is calculated, information of the signal timing and phasing are transmitted, to estimate if the vehicle will stop or not by the time of the arrival at the intersection. If the current indication is red then the remaining time for red
In case of green, the vehicle should either accelerate to catch the current phase or wait for the next green phase. Therefore, two candidate speeds can be recommended, one for the estimated arrival time
where
In case of two candidate speeds, the one respecting the speed limits is selected. If both speeds are within the speed limits,
R-GLODTA is the second hybrid controller, combining holding and GLODTA. In principle, holding and GLODTA are using the same control logic, by extending the time at stop to achieve their objectives to restore regularity and mitigate stops at traffic lights respectively. Therefore, with this controller, the prolongation of dwell time at stops aims to satisfy both objectives. After the vehicle arrives at the stop and completes dwell time, two candidate holding times are calculated to restore regularity. Then, the expected arrival time to the next signalized intersection is estimated using Eq. (43).
If the expected arrival time is during green phase, then no GLODTA time is needed. In contrast, if the vehicle is expected to arrive during red, then the waiting time at traffic light
The waiting time at the traffic light corresponds to the GLODTA time
The hybrid controller can work as holding or GLODTA alone depending on the current performance and needs of the system. If both candidate holding times (for regularity and GLODTA) meet the criteria, then the shorter time is selected. If with both holding times, the vehicle is expected to arrive during red, then the holding time with the less estimated remaining time at the traffic light is selected and the controller counts simply as a regularity controller:
In case of on time or late arrival, the vehicle will depart after
The two hybrid controllers are tested for one of the busiest lines of the city of Luxembourg, AVL Line 16. Line 16 is the backbone of the bus network of the city of Luxembourg. As depicted in Figure 7, The line consists of 19 stops, among which there are stops in the city center, the central business district of Kirchberg and the new activity zone of Cloche d’Or at the south. Additionally, Line 16 connects the central railway station, the airport and the Kirchberg multimodal transport hub. The line is running in high frequency and double articulated busses are used. In addition, the busses run in dedicated lanes and are equipped with AVL technology. We assume that all traffic lights have the same signal program with cycle of 120 s (80 green and 40 red) with the red indication first at the simulation environment. No coordination has been considered between signals.
Line 16 in Luxembourg City.
Two case studies, one for each of the newly introduced controller, were conducted. In both cases, a do-nothing scenario is used as a benchmark scenario. In addition, the hybrid controllers are compared with a holding strategy and the individual application of GLOSA and GLODTA. Moreover, different levels of TSP are put into test. For the R-GLOSA scenarios, three different levels are tested. The first level, referred as weak TSP, the scenario in which both green extend and green recall are up to 5 s. With strong TSP, green phase can be modified by 15 s In the R-GLODTA scenarios only strong TSP is tested. Lastly, in the R-GLODTA scenarios, the hybrid controller is combined with GLOSA and TSP.
The main performance indicators used in this study are the adherence of headway of the line as well as the total trip time and its variability. Moreover, we will also analyze the delay at the signalized intersections and the times the vehicles managed to pass through a green phase. Finally, for the performance of the joint controller, the number of times requested is given and the share or each sub-controller are recorded. In summary, these are the performance indicators selected for the study:
Regularity indicators: Coefficient of variation of headways; bunching;
Passengers’ cost indicators: in-vehicle time; waiting time at stops;
Link performance indicators: stop frequency and delay at traffic light, average speed and running time;
Controller performance: share of control requests and of controller choice.
All regularity indicators are summarized in Table 2. It is clear from the results that the control schemes, the objective of which is to regulate the operation, dominate the regularity indicators. The coefficient of variation of headway and the level of bunching are chosen are regularity indicators. It should be noted that R-GLOSA has a minor difference from holding control since it is based on the same criterion to calculate holding time. The additional gaining comes from the speed recommendation given by the GLOSA part of the controller. Among strategies there are no significant differences in waiting time of passengers at stops. The independent application of the two DASs has no effect on system’s regularity. Both have the same performance with the benchmark scenario. The regularity indicators remain unchanged regardless the TSP strength and similar to the do-nothing scenario. R-GLOSA manages to integrate the performance of holding strategy in terms of regularity and GLOSA in terms of cycle time. The cycle time with R-GLOSA is better than weak TSP and results to the least variable cycle time among all strategies, giving the operator the opportunity to administer more efficiently the available resources and construct a more robust schedule.
CV Line | Bunching | Waiting time [s] | In vehicle time [s] | Cycle time [s] | Cycle time deviation [s] | |
---|---|---|---|---|---|---|
NC | 0.599 | 0.372 | 302.98 | 204.74 | 4096.91 | 415.61 |
HOLDING | 0.486 | 0.269 | 302.38 | 211.90 | 4042.55 | 415.61 |
GLODTA | 0.628 | 0.382 | 302.40 | 212.49 | 4166.16 | 505.49 |
GLOSA | 0.597 | 0.351 | 302.63 | 200.66 | 4050.49 | 480.16 |
RGLOSA | 0.466 | 0.254 | 302.30 | 212.73 | 4042.09 | 394.56 |
TSP5 | 0.607 | 0.378 | 303.14 | 204.00 | 4060.26 | 472.07 |
TSP10 | 0.590 | 0.358 | 302.22 | 203.25 | 4013.18 | 472.07 |
TSP15 | 0.613 | 0.370 | 301.45 | 198.51 | 4012.75 | 490.94 |
Regularity performance indicators.
The performance indicators for the links are documented in Table 3. It is worth noting that R-GLOSA reports the highest frequency of stops at traffic lights. However, the total average delay at traffic lights is comparable to strong TSP, which has the best performance in these two indicators. GLOSA and GLODTA perform better than holding in reducing the number of stops and the delay at traffic signals. The running time on the signalized links is also lower, meeting the objectives of both GLODTA and GLOSA. R-GLOSA reduces the running time at signalized links at the same level of weak TSP. The average speed of the vehicles increases only at the scenarios with TSP.
Frequency of stop at traffic lights | Total average delay at traffic lights [s] | Running time | Average speed [km/h] | |
---|---|---|---|---|
NC | 0.309 | 1778.8 | 2821.0 | 18.8 |
HOLDING | 0.302 | 1751.0 | 2817.0 | 18.8 |
GLODTA | 0.237 | 947.6 | 2790.6 | 19.1 |
GLOSA | 0.305 | 942.3 | 2808.3 | 19.0 |
RGLOSA | 0.374 | 465.2 | 2828.7 | 18.6 |
TSP5 | 0.223 | 1265.9 | 2781.0 | 19.2 |
TSP10 | 0.152 | 876.5 | 2757.2 | 19.4 |
TSP15 | 0.076 | 435.5 | 2738.2 | 19.7 |
Link performance indicators.
Figure 8 shows the trade-off between the average delay at traffic lights and the additional time due to control. When holding is applied, the travel time increases and the additional delay at signalized intersections is not taken into account. TSP heavily prioritizes PT neglecting the impact on regularity by increasing bunching. Obviously, the application of TSP or GLOSA do not introduce any control delay at stops. GLODTA and GLOSA results to similar performance as with intermediate TSP. In contrast to TSP and GLOSA, holding is not causing any delay at traffic lights but increases significantly the additional time added due to control at stops. The delay of R-GLOSA is similar to the one holding, but delay at traffic signals is significantly reduced to the level of strong TSP. Therefore, the savings obtained in running time can compensate the additional delay at stops. The results can vary subject to the chosen holding criterion.
Tradeoff between waiting time at traffic light and holding time at stop.
In Figure 9, the coefficient of variation (CV) of headway of all R-GLODTA case study scenarios is depicted. Strategies that target the mitigation of stops at traffic lights neglect the regularity of the line. Between GLODTA or TSP scenarios can be found with minor differences compared to the benchmark scenario, reporting high level of variability which propagates along the line. On the other hand, holding, All the R-GLODTA scenarios show significant improvement on maintaining the propagation of headway low. R-GLODTA outperforms holding and its performance improves further with weak TSP. Although R-GLODTA with GLOSA performs better than GLODTA and TSP, the combination is not the most effective compared to R-GLODTA and TSP.
Coefficient of variation of headway per stop.
Regularity performance indicators at line level are summarized in Table 4. Similarly to the results in terms of coefficient of variation per stop, R-GLODTA outperforms the other strategies with minor differences from holding and R-GLODTA with TSP. GLOSA has a significant impact on the regularity of the line This can be explained by the fact the GLOSA adjusts the speed in order to traverse green. Acceleration and deceleration can shorten the headway between consecutive vehicles and cause platoons. Again, R-GLODTA has the lowest level of bunching between all scenarios. Passenger indicators are also recorded during simulation. As expected, differences between strategies can be observed in in-vehicle times. The additional time added due to control actions increases the time passengers spend on board. The higher in-vehicle time can be compensated with a more robust travel time and the overall improved performance of the line.
CV of Headway | Bunching | Waiting time [s] | In vehicle time [s] | |
---|---|---|---|---|
NC | 0.59 | 0.37 | 300.03 | 204.87 |
GLODTA | 0.62 | 0.37 | 300.98 | 211.2 |
HOLDING | 0.48 | 0.27 | 300.08 | 212.71 |
R-GLODTA | 0.44 | 0.20 | 299.96 | 215.00 |
R-GLODTA + TSP | 0.42 | 0.19 | 301.9 | 212.36 |
R-GLODTA + GLOSA | 0.43 | 0.21 | 301.64 | 226.26 |
TSP | 0.62 | 0.38 | 302.75 | 202.77 |
Regularity performance indicators.
One of the objectives of the proposed scheme is the mitigation of stop and go at signalized intersections, therefore the performance of each scenario at a link level is assessed. The results are summarized in Table 5.
Stop at traffic light frequency per segment | Total waiting time at traffic light per segment [s] | Total running time [s] | Average speed [km/h] | Times GLOSA triggered per segment | Number of TSP requests per segment | |
---|---|---|---|---|---|---|
NC | 5.6 | 113.9 | 2160.3 | 18.8 | 0.0 | 0.0 |
GLODTA | 4.3 | 60.7 | 2135.7 | 19.0 | 0.0 | 0.0 |
HOLDING | 5.4 | 109.2 | 2154.2 | 18.8 | 0.0 | 0.0 |
TSP | 1.3 | 26.1 | 2084.6 | 19.7 | 0.0 | 4.1 |
R-GLODTA | 4.7 | 69.4 | 2132.4 | 19.0 | 0.0 | 0.0 |
R-GLODTA+TSP | 2.9 | 52.7 | 2115.9 | 19.3 | 0.0 | 1.7 |
R-GLODTA+GLOSA | 4.7 | 49.6 | 2172.1 | 18.7 | 2.1 | 0.0 |
Link performance indicators.
Unquestionably, providing unconditional signal priority to PT can reduce dramatically the number of stops at signals and the corresponding delay at signalized intersections. However, this reduction will potentially penalize the rest of the traffic. R-GLODTA shows slightly increased number of stops compared to GLODTA alone. This can be explained by the fact that the combined controller prioritizes regularity over stopping at signals. Therefore, it will not exchange holding for regularity to secure passing during green. Weak TSP improves substantially the performance of R-GLODTA in terms of frequency of stops and delay at intersections. Speed adjustment with GLOSA transfers waiting time at traffic lights to running times to the links. A GLOSA advices to decelerate in order to arrive at the intersection during green, prolongs the running time between stops. All R-GLODTA scenarios result in lower total running time compared to an independent application of GLODTA or holding but higher than TSP, but they compensate with their regularity indicators, especially bunching. Among scenarios the differences of the speed are marginal.
We compare the number of TSP requests between the TSP and the R-GLODTA with TSP scenarios. The number of TSP requests is halved with R-GLODTA and with the combination of weak TSP can achieve comparable results with TSP in reducing stop and go actions at traffic lights while it contributes to the regularity of the line.
A final analysis is performed to check how many times the strategies are adopted in the simulated scenarios. Table 6 shows the share of each control decision, i.e. when each control was needed. Fixing regularity is prioritized over reducing stops at traffic lights. Controlling actions are reduced when R-GLODTA is combined with TSP. R-GLODTA aims to address both objectives and the number of independent applications of holding or GLODTA. On the other hand, the combination with TSP or GLOSA reinforces the objective of GLODTA. The need of holding alone intensifies in these scenarios to restore regularity. With GLOSA, holding is triggered more than half of the times a controller was requested. If the changes of speed do not account for the sequence of vehicles, undesired phenomena as formation of platoons are more likely to occur and impact the performance of a bus line.
Control request | Controller choice | |||
---|---|---|---|---|
GLODTA | Holding | R-GLODTA | ||
R-GLODTA | 61% | 38% | 42% | 19% |
R-GLODTA + TSP | 58% | 37% | 49% | 14% |
R-GLODTA + GLOSA | 62% | 37% | 51% | 13% |
Controller frequency.
This chapter has presented an integrated approach to manage electrified bus systems using Cooperative ITS. We first discussed the challenges and opportunities brought by next generation public transport systems, which require to manage the system in an integrated way. Then we introduced novel optimization methods for joint bus scheduling and charging, and real-time operational control strategies. Results in realistic simulations show how the integrated systems achieves cost effective, reliable and energy efficient operations.
The authors acknowledge Marcin Seredynski (Volvo E-Bus Competence Center), Erika Picarelli and Andrea D’Ariano(University of Rome Tre). This project has been carried on under to the FNR-CORE Grant C16/IS/11349329 (eCoBus).
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