Introduction to Artificial Neural Networks and Fuzzy Systems



Introduction to Artificial Neural Networks and Fuzzy Systems

ECE/CS/ME 539

Class Project

Instructor: Yu Hen Hu

Student: Yong Sun

Engine Operating Parameter Optimization using Genetic Algorithm

IINTRODUCTION

With more-and-more stringent emission standards, future diesel engine technologies will need to incorporate advanced combustion strategies with optimized engine operating parameters for achieving low emissions while maintaining fuel economy and power density. Genetic algorithms (GA) are being used by engine researchers to optimize engine design and operating parameters to achieve these goals.

In this study, a micro-Genetic Algorithm (μGA) code was coupled with a 3D engine Computational Fluid Dynamic (CFD) code KIVA to optimize six engine operating parameters. A merit function was defined based on three engine performance parameters to evaluate the merit of the combination of the parameters. Several penalty functions were also defined based on the physical constraints of the engine. The results were used as inputs for the development of a multi-layer perceptron (MLP) configuration. The MLP network can be used to predict the engine performance based on the engine operating parameters.

GENETIC ALGORITHM

Genetic algorithm overview

Genetic algorithms are global search techniques based on the mechanics of natural selection which combine a “survival of the fittest” approach with some randomization and/or mutation. Unlike local search techniques, which are typically highly dependent on the starting point and have constraints such as solution continuity and differentiability, as a global search method, GA place few constraints on the solution domain and are thus much more robust for ill-behaved solution spaces. In addition, GA tends to converge to a global optimum for multi-modal functions with many local extrema. GA can also handle high dimensional problems and problems which require high parameter resolution. These features make GA suitable for the optimization of engine design and operating parameters.

Micro-genetic algorithm (μGA)

The micro-Genetic Algorithm (μGA) is a “small population” Genetic Algorithm (GA) that operates on the principles of natural selection or “survival of the fittest” to evolve the best potential solution (i.e., design) over a number of generations to the most-fit, or optimal solution. In contrast to the more classical Simple Genetic Algorithm (SGA), which requires a large number of individuals in each population (i.e., 30 – 200), the μGA uses a μ.population of five individuals (Krishnakumar 1989). This is very convenient for three-dimensional engine simulations, for example, which require a large amount of computational time. The small population size allows for entire generations to be run in parallel with five (or four, as will be discussed later) CPUs. This feature significantly reduces the amount of elapsed time required to achieve the most-fit solution. In addition, Krishnakumar (1989) and Senecal (2000) have shown that the μGA requires a fewer number of total function evaluations compared to SGAs for their test problems.

Gen4μGA code

The μGA used in the present study is based on the GA code of Carroll (1996) and can be outlined as follows:

1. A μ-population of five designs is generated randomly.

2. The fitness of each design is determined and the fittest individual is carried to the next generation (elitist strategy).

3. The parents of the remaining four individuals are determined using a tournament selection strategy. In this strategy, designs are paired randomly and adjacent pairs compete to become parents of the remaining four individuals in the following generation (Krishnakumar 1989).

4. Convergence of the μ-population is checked. If the population is converged, go to step 1, keeping the best individual and generating the other four randomly. If the population has not converged, go to step 2.

Note that mutations are not applied in the μGA since enough diversity is introduced after convergence of a μ-population.

A summary of the gen4 μGA code used in this study is presented in Fig. 2. In this code, the user is allowed to initialize one individual as the present baseline design, while randomly sampling the parameter space to determine the other four. The merit function evaluation part can be considered as a “black box” operation, where 3D CFD code KIVA is coupled with the gen4 code and the merit function of each individual is evaluated.

The gen4 code can be run in either serial or parallel mode. In this study, the parallel mode was used because the parallel option allows for much more rapid population evaluation since each generation is run in “parallel” on four CPUs (although the μGA includes five individuals per generation, the present implementation only requires four to be calculated because one of the individuals is the optimum individual from all previous generations).

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Figure 1. Flow chart of the gen4 μGA optimization code.

APPROACH, RESULTS AND DISCUSSION

Coding

The gen4 code uses a binary string to represent a set of design parameters (Goldberg 1989). To extend the analogy with nature, the binary string is commonly referred to as a chromosome, while its features are known as genes.

The precision of each of the design parameters included in a chromosome is based on the desired range of the parameter (i.e., its minimum and maximum values) and how many bits are used in its representation. Thus, the precision π of a parameter Xi is given by (Homaifar et al. 1994):

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where λi is the number of bits in its representation. Thus if m possible values of a parameter are desired, its binary representation will include λi=ln(m)/ln(2) bits.

In this study, six engine operating parameters was optimized, intake valve closure (IVC) timing, start of injection (SOI) timing, the spray angles of two pulses, the dwell between the two pulses, and the fraction of fuel injected in each pulse. The six parameters are denoted as A, B, C, D, E and F for simplicity. The range and resolution of the six parameters are listed in Table 1.

Table 1 Ranges and resolutions of the parameters

|Parameter |Baseline |range |Resolution |# of possible values |λi |

|A |-143 |-143~-83 |4 |16 |4 |

|B |0.44 |0~1 |0.067 (1/15) |16 |4 |

|C |0 |0~75 |5 |16 |4 |

|D |60 |48~141 |3 |32 |5 |

|E |125 |48~141 |3 |32 |5 |

|F |0.5 |0.0~1.0 |0.143 (1/7) |8 |3 |

As a example, consider the coding of the parameter F with Xmax=1, Xmin=0 and λ=3. Thus, according to the above equation, π=1/7. The resulting binary representations for the eight numbers are presented in Table 2.

Table 2: Real number and binary representations for parameter F.

|real number |binary representation |

|0 |000 |

|1/7 |001 |

|2/7 |010 |

|3/7 |011 |

|4/7 |100 |

|5/7 |101 |

|6/7 |110 |

|1 |111 |

Function evaluation

The fitness, or objective function (merit function), is the “figure of merit” for each individual chromosome, and thus determines its probability of taking part in reproduction, In this study the merit function is defined based on three output variables related to the engine performance, NOx emissions, Unburned Hydrocarbon (HC) and Break Specific Fuel Consumption (BSFC). They are denoted as X, Y and Z for simplicity. The merit function is defined as:

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From this definition, the lower X, Y and Z are, the higher merit function value an individual has.

Several penalty functions have also been applied to the merit function based on the peak in-cylinder pressure, exhaust temperature and pressure, maximum rate of pressure rise, soot emissions misfire and the wall-film amount at Exhaust Valve Opening (EVO). The purpose of applying these penalty functions is to avoid unphysical cases which may have high merit function but is not practicable.

Reproduction (selectioin)

A reproduction operator combines the relative fitness of a generation’s chromosomes with some randomness in order to determine parents of the following generation. Tournament selection strategy is used in the present gen4 code. It can be summarized in the following way:

1. The present generation is first “mixed up” such that the order of individuals is completely random.

2. The fitness of individual 1 is compared with the fitness of individual 2. The individual with the higher fitness is chosen as “parent 1.”

3. The fitness of individual 3 is compared with the fitness of individual 4. The individual with the higher fitness is chosen as “parent 2.”

4. Parents 1 and 2 are used in the crossover operation.

The above process is repeated until the desired number of children is produced.

Another strategy used in conjunction with the tournament selection strategies in the gen4 code, is the so-called “elitist approach” (Goldberg 1989). Elitism automatically forms a child in the new generation with the exact qualities of the parent with the highest merit. This approach guarantees that the highest fitness in a given generation will be at least as high as in the previous generation.

Crossover

The crossover operation is the process of mating the selected strings in the reproduction phase in order to form the individuals of a new generation. In the gen4 code, uniform crossover is used. In this method, a random number between zero and unity is chosen at each position in the first parent string. If the random number is larger than a crossover probability, which is set to 0.5, the bit is set to the second parent’s value, otherwise it remains the same as the first parent’s value.

Convergence and restarting

While global optimality is not guaranteed with genetic algorithms, the population of solutions will evolve over successive generations so that the fitness of the best individual increases toward the global optimum. In the gen4 code, convergence is defined as the progression towards chromosome uniformity. Thus, for the genetic algorithm operators described above, a gene may be considered to be converged, if 95% of that particular gene in the entire population shares the same value. A population is then converged when all of the genes are converged.

Since multi-dimensional engine simulations require a large amount of CPU time, a method is needed to stop the search process. In the gen4 code, a maximum number of generation is defined. The code also has a “restart” feature. If the search has not found a satisfying optimum within the specified maximum number of generations, the process can be restarted without any loss of information. If the maximum merit function does not change for a sufficient number of generations, an optimal, or near-optimal design is thought to be indicated and the program can be stopped by the user.

Running the gen4 code

A summary of the source files of the gen4 code is listed in Tables 3. A baseline individual and its merit is given in init.ga file and the ranges and resolutions of the six parameters are given in input.ga file. This file also specifies the maximum generation and the restart flag.

Table 3 Summary descriptions of gen4 source files

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The gen4 code was modified to communicate with the KIVA code. Gen4 code generate the six input parameters (A~F) of 4 citizens and sends them to the KIVA code. The KIVA code calculates the outputs (X, Y and Z) and evaluates the merit of each individual and sends them to the gen4 code. The gen4 code was also modified to record the process of convergence. The information of all individuals in all generations is recorded and written to a perform.ga file.

Results and discussion

In this study, 351 generation was run to get the ‘optimum’ case. Since there are 16×16×16×32×32×8=33,554,432 possible combinations of the six parameters and running the KIVA code is also time consuming (2 hours for each generation), GA may not able to find the global maximum. However, GA was able to find a satisfying case with great improvement compared with the baseline case and the maximum merit was observed to not change for more than 200 generations. Thus, it is considered that an ‘optimum’ case was found. Figure 2 shows the history of the maximum merit.

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Figure 2 Maximum merit as a function of generation number

Table 4 Comparison between the baseline and optimum case

| |Input |Output |Merit |

|A |B |C |D |E |F |X |Y |Z |Merit | |Baseline |-143 |0.440 |0 |60 |125 |0.5 |11.7 |139.7 |1.386 |-5.7 | |Optimum |-139 |0.267 |15 |90 |141 |0.857 |2.3 |1.2 |1.333 |125.4 | |Table 4 shows the comparison between the baseline and optimum case. It can be seen that significant improvement has been achieved after the optimization using theμGA algorithm.

MULTI-LAYER PERCRPTRON NETWORK

After running GA optimization, we have a set of data with inputs and outputs of all the individuals. Next, a Multi-Layer Perceptron (MLP) network is setup to correlate the six inputs and three outputs. The codes are modified from the bp code used in class. The subroutines modified are attached.

30% of the data was selected randomly and reserved for testing. Only 70% of the data was used for training. The inputs were scaled to [-5,5] and outputs [0.2, 0.8] by each column. The MLP network has 2 layers and the hidden layer has 8 neurons. The hidden layer uses hyperbolic tangent activation function and output layer uses sigmoidal function. The parameters used in the code are as below:

Input layer dimension: 6

Output layer dimension: 3

Learning rate: 0.1

Momentum constant: 0.8

Nepoch: 2500

Epoch size K: 64

Nck: 50

The final training error is 0.0090 and the testing error is 0.0127.

The weight matrix of the hidden layer is:

-0.0928 0.9782 -0.7673 0.3791 5.9811 1.4702 -1.3212

1.5729 0.3189 -0.3103 -1.8411 -4.5168 1.1668 0.0312

1.8331 -0.5381 2.1315 1.1248 -0.1184 0.4692 -0.2042

2.9580 -0.0267 -0.5908 -0.2677 3.3509 0.0241 1.0066

4.4803 1.4442 -3.8054 -0.5408 0.7064 0.6601 0.1934

-2.0629 0.7693 -2.3483 0.0138 -0.5446 -0.0094 1.7161

1.4502 -0.2288 -1.4001 -0.7772 4.8072 0.0577 -2.5117

-2.1469 2.2675 2.4541 3.2225 0.1221 -2.6403 -0.6212

The weight matrix of the output layer is:

0.2365 0.3723 0.1336 0.3715 -0.4441 -0.1610 -0.1951 0.0984 -0.0809

1.3772 0.4357 -0.5307 0.0127 -0.8271 -0.3923 0.5913 0.1959 0.4667

0.5020 0.0197 -0.1090 0.0477 -0.4352 -0.1570 0.1923 0.0680 0.7139

Different number of MLP layers and number of neurons in each layer have been tried, but the training and testing error are quite similar. Different learning rate and momentum have also been tried, and the results are also not changed so much.

CONCLUSIONS

1. Genetic algorithm is a global optimization tool and can be used for engine design and operating parameter optimization effectively and efficiently.

2. Optimization results can be used to train the MLP network, and the developed network can be used to predict engine performance based on the inputs of the six engine operating parameters.

REFERENCE

1. Carroll, D. L., “Chemical Laser Modeling with Genetic Algorithms,” AIAA Journal, 34, 338, 1996.

2. Krishnakumar, K., “Micro-Genetic Algorithms for Stationary and Non-Stationary Function Optimization,” SPIE 1196, Intelligent Control and Adaptive Systems, 1989.

3. Senecal, P.K., and Reitz, R.D., “Simultaneous Reduction of Engine Emissions and Fuel Consumption Using Genetic Algorithms and Multi-Dimensional Spray and Combustion Modeling,” SAE 2000-01-1890, 2000.

4. Senecal, P.K., “Numerical Optimization Using the Gen4 Micro-Genetic Algorithm Code”, ERC Document, 2000

5. Goldberg, D. E., Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading, MA, 1989.

6. Homaifar, A., Lai H. Y. and McCormick, E., “System Optimization of Turbofan Engines using Genetic Algorithms,” Appl. Math. Modelling, 18, 1994.

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