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Linear quadratic Gaussian (LQG) control is considered to be a foundation of modern optimal control theory [3]. The LQG scheme employs the following linearised state- space plant model in the power system [181]:

𝒙̇ = 𝑨𝒙 + 𝑩𝒖 + 𝜞𝒘 (5. 2)

𝒚 = 𝑪𝒙 + 𝑫𝒖 + 𝒗 (5. 3)

where,

w : process noise.

v : measurement (sensor) noise.

The LQG scheme employs the linearized state space model of the system to find the control input u(t) signal that minimises the quadratic cost function in equation (5. 4):

𝐉_𝐤= 𝐥𝐢𝐦

𝛕→∞𝐄 ∫ (𝐱

𝐓_{𝐐𝐱 + 𝐮}𝐓_{𝐑𝐮)𝐝𝐭} 𝛕

𝟎 (5. 4)

In this design the objective is to minimize the energy of the controlled output (states’ deviation) and the energy of the input signal (control effort). The values of the diagonal elements of Q are selected to penalize the corresponding states when deviating from their steady-states values. Similarly, the values of the diagonal elements of R are set in order to penalize the corresponding system inputs [3].

For the LQR problem, the solution of equation (5. 4) can be written in the form of standard state feedback law presented in equation (5. 5). The LQR controller gain is computed by solving the associated Algebraic Riccati Equation (ARE) as shown in equation (5. 7) based on the cost function presented above [168], [177] .

𝒖(𝒕) = −𝑲𝒙(𝒕) (5. 5)

𝑲 = 𝑹−𝟏_𝑩𝑻_{𝑿 (5. 6)}

where, K is a constant state feedback matrix and XTX=≥0 is the unique positive semi- definite solution of the algebraic Riccati equation (5. 7):

𝑨𝑻_{𝑿 + 𝑿𝑨 − 𝑿𝑩𝑹}−𝟏_𝑩𝑻_{𝑿 + 𝑸 = 𝟎 (5. 7)}

For solving the LQR problem all states must be measurable and since in most cases the states of the system are not available an observer is required to estimate the unavailable ones. Consequently, the LQG, as shown in Figure 5.9, is the combination of an optimal LQR state feedback gain with an optimal state estimator (Kalman filter) [168], [177]. The design of the LQG controller scheme is intricate, especially within large power systems. As the weighting factors are affecting states of the system, if these states are involved in modes that do not require altering, their damping can be adversely affected. Moreover, the indirect access to the targeted modes makes the controller tuning process prohibitively complex as the size of the system becomes larger [3, 10].

In order to provide direct alteration of the targeted modes without any adverse effect on the other modes a re-formulated LQR cost function in terms of modal variables is used in the MLQG controller scheme (Figure 5.9). The LQR control problem in the modal formulation is shown in equation (5. 8) [3, 10].

𝑱_𝒌= 𝐥𝐢𝐦 𝛕→∞𝑬 ∫ (𝐱 𝐓_(𝐌𝐓_𝑸 𝒎𝐌)𝐱 + 𝐮𝐓𝐑𝐮)𝐝𝐭 𝛕 𝟎 (5. 8)

where, Qm and R are appropriately chosen weighting matrices such that M is a real

matrix that provides mapping between system modal variables z and state variables x, as in equation (5. 9):

𝒛(𝒕) = 𝑴𝒙(𝒕) (5. 9)

The modal variables z are directly related to the system modes and the real transformation matrix M is associated with the matrix of right eigenvectors Φ as M = Φ−1. It should be mentioned that the weighting matrices Qm and R are diagonal

matrices. Similar to the LQG scheme, in matrix R, the value of the diagonal elements should be set to penalize the corresponding controller’s outputs. Each diagonal element of this matrix, (Qm), is directly related to a modal variable zi and consequently, with the

equivalent mode eλit. In the MLQG scheme, the controller solely affects the modes of interest, by shifting them to the left in the complex plane, while keeping the locations of other modes unchanged. Additionally, each mode can be moved independently only by tuning the value of the corresponding element in the Qm matrix. In this research, the

tuning process has been implemented using optimization method. Normally, each element is tuned to the lowest value which increases damping of the targeted mode within the required range and high gain values should be avoided [168], [177].

In order to design an MLQG controller, selection of the weighting matrices is a critical step. The corresponding weights in the Qm matrix to the modes of interest are set to non-

zero while keeping the other elements as zero. The non-zero elements will be tuned to achieve being the required range.

The standard LQG feedback control law can be written as:

𝒖(𝒕) = −𝑲𝒙̂(𝒕) (5. 10)

where, 𝒙̂ is an estimate of the states x which can be calculated by (5. 11) 𝒙̂̇ = 𝑨𝒙̂ + 𝑩𝒖 + 𝑳(𝒚 − 𝑪𝒙̂) + 𝑳𝒗 (5. 11)

where, L is a constant estimation error feedback matrix, which is obtained by solving the Algebraic Riccati Equation (ARE) associated with the cost function described by equation (5. 12):

𝑱_𝑳 = 𝑬 {∫ (𝒙𝑻 𝑻_{𝑾𝒙 + 𝒖}𝑻_{𝑽𝒖)𝒅𝒕} 𝟎

}

𝑻→⧝𝒍𝒊𝒎 (5. 12)

where, W and V are weighting matrices which, in this study, have been tuned using Loop Transfer Recovery (LTR) as described in the next section.

5.5.1 Application of Loop Transient Recovery (LTR)

As can be seen in Figure 5.9, the combination of an optimal LQR state feedback gain with an optimal state estimator (Kalman filter) is used in the LQG controller, which both have good individual robustness properties [3, 181, 182] . Nevertheless, by combining these two loops to form the LQG controller, individual robustness properties are lost [3]. A commonly used method to recover these robustness properties for designing of LQG is the loop transfer recovery (LTR) method. This recovery can be used at either plant input or output [3, 183]. The optimal choice of the matrix L in equation (5. 11) is calculated by solving the ARE associated with the cost function (5. 12). The weighting matrices W and V can be calculated as in equations (5. 13) and (5. 14) [177]:

𝑾 = 𝜞𝑾_𝟎𝜞𝑻_{+ 𝒒𝑩𝜽𝑩}𝑻 _{(5. 13)}

𝑽 = 𝑽_𝟎 (5. 14)

where 𝑾_𝟎 and 𝑽_𝟎 are estimates of the nominal model noise, and 𝜽 is any positive definite matrix.