This study investigates measurement-induced phase transitions (MIPTs) in quantum spin chains using matrix product states (MPS) with the Time-Dependent Variational Principle (TDVP) algorithm. The TDVP method approximates quantum evolution as a classical non-linear process with a conserved charge, enabling efficient numerical analysis of MIPTs. The error rate in this approximation exhibits a phase transition in the monitoring strength, detectable via scaling analysis with low bond dimensions. The method allows for the determination of critical parameters in many-body quantum systems. The study shows that the error rate decays as $1/\log \chi$ for unitary evolution but changes to exponential decay for $\gamma > \gamma_c$, indicating a phase transition. This transition is characterized by a charge-sharpening (CS) transition, distinct from the entanglement transition, detected through charge fluctuations in local subsystems. The CS transition occurs at a smaller measurement rate than the entanglement transition, with the latter showing sub-extensive scaling in magnetization variance. The study uses two models: the XXX and J-XXX spin chains. The TDVP method is applied to simulate time evolution with fixed bond dimensions, projecting the Hamiltonian evolution onto the MPS manifold. The error rate and charge fluctuations are analyzed to detect phase transitions. The results show that the critical measurement rate for the CS transition is smaller than that for the entanglement transition, consistent with Kosterlitz-Thouless (KT) scaling. The study also compares results with exact diagonalization (ED) simulations, finding that the TDVP method provides accurate critical parameters with faster convergence in bond dimension. The CS transition is detected through the scaling of magnetization variance, showing a transition from extensive to sub-extensive behavior at $\gamma \geq \gamma_\#$. The TDVP method is shown to be effective in detecting MIPTs in systems of arbitrary dimensions and sizes, offering a powerful tool for studying quantum many-body systems. The results highlight the importance of methods like TDVP in probing quantum phase transitions, particularly in systems with continuous time evolution and measurement-induced dynamics.This study investigates measurement-induced phase transitions (MIPTs) in quantum spin chains using matrix product states (MPS) with the Time-Dependent Variational Principle (TDVP) algorithm. The TDVP method approximates quantum evolution as a classical non-linear process with a conserved charge, enabling efficient numerical analysis of MIPTs. The error rate in this approximation exhibits a phase transition in the monitoring strength, detectable via scaling analysis with low bond dimensions. The method allows for the determination of critical parameters in many-body quantum systems. The study shows that the error rate decays as $1/\log \chi$ for unitary evolution but changes to exponential decay for $\gamma > \gamma_c$, indicating a phase transition. This transition is characterized by a charge-sharpening (CS) transition, distinct from the entanglement transition, detected through charge fluctuations in local subsystems. The CS transition occurs at a smaller measurement rate than the entanglement transition, with the latter showing sub-extensive scaling in magnetization variance. The study uses two models: the XXX and J-XXX spin chains. The TDVP method is applied to simulate time evolution with fixed bond dimensions, projecting the Hamiltonian evolution onto the MPS manifold. The error rate and charge fluctuations are analyzed to detect phase transitions. The results show that the critical measurement rate for the CS transition is smaller than that for the entanglement transition, consistent with Kosterlitz-Thouless (KT) scaling. The study also compares results with exact diagonalization (ED) simulations, finding that the TDVP method provides accurate critical parameters with faster convergence in bond dimension. The CS transition is detected through the scaling of magnetization variance, showing a transition from extensive to sub-extensive behavior at $\gamma \geq \gamma_\#$. The TDVP method is shown to be effective in detecting MIPTs in systems of arbitrary dimensions and sizes, offering a powerful tool for studying quantum many-body systems. The results highlight the importance of methods like TDVP in probing quantum phase transitions, particularly in systems with continuous time evolution and measurement-induced dynamics.