Sure! Here is the test case generator for the provided problem statement:

```python
import random
import string

def tcgen():
    N = random.randint(1, 2 * 10**5)
    
    S = []
    for _ in range(N):
        length = random.randint(1, 10)
        S_i = ''.join(random.choices(string.ascii_lowercase, k=length))
        S.append(S_i)
    
    return N, S

# Example usage
if __name__ == "__main__":
    N, S = tcgen()
    print(N)
    for s in S:
        print(s)
```

This generator creates a random number `N` (the number of lottery draws) within the specified range and generates `N` random strings `S_i` with lengths between 1 and 10, consisting of lowercase English letters. You can run this script to generate test cases for the problem.

Here's a brief explanation of each part:

1. `N = random.randint(1, 2 * 10**5)`: This line generates a random integer `N` in the range [1, 200,000].

2. `S = []`: Initializes an empty list to store the strings.

3. The `for` loop runs `N` times, each time generating a random string `S_i`:
    - `length = random.randint(1, 10)`: Determines the length of the string `S_i`.
    - `S_i = ''.join(random.choices(string.ascii_lowercase, k=length))`: Generates a random string of the determined length using lowercase English letters and appends it to the list `S`.

4. `return N, S`: Returns the generated values `N` and `S`.

You can run the script and it will print an example test case to the standard output.