How to simulate home automation system?
Software and hardware

There are two aspects of your design to be tested: software and hardware. We can further break the hardware down into the micro-controller and board (including the power-supply, USB and I/O pins) that come on the Arduino board and everything else - the PIR, lamps and relays.

Test, correct, retest

You can run a simulation by loading your test code onto the micro controller board, use a switch or button for the PIRs and LEDs for the lamps and relays. Draw up a test plan to function test each of your design requirements and then execute your plan. Correct any errors and retest until all is correct.

This will give you good confidence on those aspects of your system. Then you can spend the rest of the budget on your sensors and relays with low risk of failure.