Consider the following OCV-SOC model: V0(s) = k0 + k1s + k2s 1 + k3s 2 + k4s 3 + k5s 4 + k6 ln(s) + k7 ln(1 s) (1) where k0 = 9.082, k1 = 76.604, k2 = 103.087, k3 = 18.185, k4 = 2.062, k5 = 0.102, k6 = 141.199, and k7 = 1.117. These parameters were computed for SOC values s [0, 1] after they were linearly scaled to s 0 [.175, .825].

find the SOC of the given battery corresponding to a measure OCV of 3.75 V. Assume that there is voltage measurement error and we need to find out the resulting SOC error. It is given that the voltage measurement error is zero-mean with standard deviation 10 mV.

(a) Use the following Monte-Carlo simulation approach to find out the standard deviation of the error in SOC error

- Generate 1000 'voltage measurements' using the above-mentioned measurement error (zero-mean, standard deviation of 10 mV). The Matlab command '3.75 + 0.01*randn(1000,1)' will generate these 1000 measurements. The standard deviation of these measurements is 0.01 V (or 10 mV). - Compute the SOC corresponding to each of these 1000 voltage measurements

- Compute the SOC corresponding to OCV = 3.75 V

- Compute the SOC computation error corresponding to each of the 1000 measurements

- Compute the standard deviation of the SOC computation error

(b) Repeat the above for SOC estimation corresponding to an OCV of 3.95 V

(c) Repeat the above for SOC estimation corresponding to an OCV of 3.6 V

(d) Repeat the above for SOC estimation corresponding to an OCV of 3.3 V

(e) Consider the following statement: "Given the same voltage measurement, the error in computed SOC varies depending on the SOC region" Do you agree or disagree with the above statement? justify your answer.

4. An inequality developed by Russian mathematician Chebyshev gives the minimum percentage of values in ANY sample that can be found within some number (k 1} standard deviations from the mean. Let P be the percentage of values within .14: standard deviations of the mean value. Chebyshevs inequality states that for ANY distribution, P 100* (i 1/k:)2. (a) For any distribution, what does Chebyshevs inequality say about the percentage of values that are within 2, 3. 5. 10 standard deviations of the mean? (b) For each of the distributions below, determine the percentage of observations within 2 and 3 standard deviations of the mean value. Comment on how these percentages compare to the percentage found using Chebyshevs inequality. i. A standard normal distribution (X N N(0,1)) ii. An exponential distribution with A = 2 (X N exp(2)) iii. A Poisson distribution with A = 2 X ~Pois(2) iv. A binomial distribution with n, = 10 and 'p = 0.2. X m binom(10..45) 4. An inequality developed by Russian mathematician Chebyshev gives the minimum percentage of values in ANY sample that can be found within some number ( 1) standard deviations from the mean. Let P be the percentage of values within k standard deviations of the mean value. Chebyshevs inequality states that for ANY distribution, P 100 * (1 -1/k)2. (a) For any distribution, what does Chebyshevs inequality say about the percentage of values that are within 2, 3, 5, 10 standard deviations of the mean? (b) For each of the distributions below, determine the percentage of observations within 2 and 3 standard deviations of the mean value. Comment on how these percentages compare to the percentage found using Chebyshevs inequality. i. A standard normal distribution (X ~ N(0,1)) ii. An exponential distribution with A = 2 (X ~ exp(2)) iii. A Poisson distribution with A = 2 X ~Pois(2) iv. A binomial distribution with n = 10 and p = 0.2. X ~ binom(10,0.45)1. What is the meaning of the central limit theorem for means, and the central limit theorem for proportions? When and how are they used? Compare and contrast the central limit theorem for means vs. the central limit theorem for proportions. Include at least one similarity, and one difference.6. A 3-state Markov chain has the following transition matrix: 0.6 0.3 0.17 P = 0.2 0.7 0.1 0.5 0 0.5 Using Xo = (0.6, 0.0, 0.4), find X1 and X2. 0.4 0.6 07 7. A 3-state Markov chain has transition matrix P = 0 0.6 0.4 0.6 0.2 0.2 Find the steady state vector for this Markov chain