Model predictive control for resilient frequency management in power systems

Abstract

A disturbance in a power system causes the frequency to deviate from its nominal value. The load and generation of the system are strategically adjusted to restore the synchronous frequency. This paper introduces novel shrinking-horizon model predictive control (MPC) technique, which employs a centralized controller for managing the load-frequency of a single-area power system and distributed controllers for multi-area systems. The controller optimally changes generation settings and sheds non-critical loads to make the frequency and tie-line power deviation zero. In contrast to existing approaches that use an approximate first-order transfer function model, this paper presents a structure-preserving linear state-space model for power systems. This model takes into account frequency and voltage dependencies of both load and generation, allowing for more accurate representation of power system behavior. During rescheduling, the controller minimizes additional cost associated with changes while satisfying various operational and physical constraints. The paper conducts several case studies using IEEE test systems to devise corrective action plans for both under- and over-frequency scenarios. The proposed controller’s robustness is tested against disturbances originating from renewable energy sources. Furthermore, a performance comparison is drawn between the proposed controller and existing control techniques. The comparative analysis indicates that the proposed approach consistently outperforms other controllers in terms of dynamic performance improvement, measured by parameters such as settling time, overshoot, undershoot, and error reduction. It is observed that the maximum undershoot and the settling time of the frequency is the least for centralized control. Comparative studies indicate that under different circumstances, the proposed approach outperforms other controllers in dynamic performance improvement, measured by settling time, overshoot, undershoot, and error reduction. The net rescheduling cost obtained by the proposed centralized MPC, cooperative distributed MPC, and non-cooperative distributed MPC-based controllers is 38, 25, and 24% lower than the conventional PI controller used for LFC of IEEE 39-bus system. This validates the suggested controllers’ cost-effectiveness.

Appendix

1.1 Partial derivatives and variables used in Sect. 3

Partial derivative used in (18):

$$\begin{aligned}{} & {} \frac{\partial P_i}{\partial b_j} \approx {\left\{ \begin{array}{ll} V_i \sum _{k = 1, a(k) \ne a(i)}^n B_{ik}, \text { if } j = a(i),\\ - V_i \sum _{k = 1, k \in N_j}^n B_{ik}, \text { if }j \ne a(i) \end{array}\right. } \end{aligned}$$

(44a)

$$\begin{aligned}{} & {} \frac{\partial P_{d_i}}{\partial (\Delta f_{a(i)})} = P_{d0_i}k_i(a_i + b_iV_i + c_iV_i^2) \end{aligned}$$

(44b)

where $N_j$ is the set of buses in area j.

Partial derivatives needed for (19):

$$\begin{aligned} \frac{\partial P_{ik}}{\partial \delta _j}&= {\left\{ \begin{array}{ll} -V_ib_{ik},&{} \text {if}\;j = i\\ +V_ib_{ik},&{} \text {if}\; j = k,\\ 0,&{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

(45a)

$$\begin{aligned} \frac{\partial P_{ik}}{\partial b_j}&= {\left\{ \begin{array}{ll} -V_ib_{ik}, &{}\text {if}\; j = a(i)\\ +V_ib_{ik}, &{}\text {if}\; j = a(k).\\ 0,&{}\text {otherwise}. \end{array}\right. } \end{aligned}$$

(45b)

Sub-matrices defined in (20): For $i = 1, 2,\ldots ,n$ and $j = 1, 2,\ldots ,n$:

$$\begin{aligned} {E_1}_{ij} = {\left\{ \begin{array}{ll} -B_{ij}, &{}\text {if}\; j \notin \mathcal {B}_s\\ \frac{1}{V_i}\left\{ \frac{P_{max_i}}{R_i} + P_{d0_i}k_i(a_i + b_iV_i\right. &{}\\ \quad \left. + c_iV_i^2)\right\} ,&{}\text {if}\; j \in \mathcal {B}_s; j = a(i)\\ 0, &{}\text {otherwise. } \\ \end{array}\right. } \end{aligned}$$

(46a)

For $i = 1, 2,\ldots ,n$ and $j = 1, 2,\ldots ,n_a; j \ne p$:

$$\begin{aligned} {E_2}_{ij} = {\left\{ \begin{array}{ll} \sum _{k = 1, a(k) \ne a(i)}^n B_{ik}, &{}\text {if}\; j = a(i)\\ -\sum _{k \in N_j}^n B_{ik}, &{} \text {if}\; j \ne a(i)\\ \end{array}\right. } \end{aligned}$$

(46b)

$E_3$ is an $n \times n$ dimensional diagonal matrix with ${E_3}_{ii} = \frac{P_{d0_i}}{V_i}(b_i + 2c_iV_i)$. For $t = 1, 2,\ldots ,n_a; t \ne p$ and $j = 1, 2,\ldots ,n; j \notin \mathcal {B}_s$:

$$\begin{aligned} {E_4}_{tj} = {\left\{ \begin{array}{ll} -b_{(t)ik}, &{} \text {if}\; j = i\\ +b_{(t)ik}, &{} \text {if}\; j = k\\ 0, &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

(46c)

For $t = 1, 2,\ldots ,n_a; t \ne p$ and $j = 1, 2,\ldots ,n_a; j\ne p$:

$$\begin{aligned} {E_5}_{tj} = {\left\{ \begin{array}{ll} -b_{(t)ik}, &{} \text {if}\; j = a(i)\\ +b_{(t)ik}, &{} \text {if}\;j = a(k)\\ 0, &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

(46d)

For $t = 1, 2,\ldots ,n_a; t \ne p$, tj , ${E_6}_{tj} = 2g_{(t)ik}$, if $j = i$; 0 otherwise. $E_7$ and $E_8$ are $n \times n$ dimensional diagonal matrices, such as ${E_7}_{ii} =\frac{1}{V_i}$, and ${E_8}_{ii} = -\frac{1}{V_i}(a_i + b_iV_i + c_iV_i^2)$, respectively. For $t = 1,2,\ldots ,n_a; t \ne p$ and $j = 1,2,\ldots ,n_a; j \ne p$, ${E_9}_{tt} = \frac{1}{V_i}$, for $t = j$. For $i,j = 1, 2,\ldots ,n$

$$\begin{aligned} {E_{10}}_{ij} = {\left\{ \begin{array}{ll} \frac{Q_{d0_i}}{V_i}k_i'(a_i' +b_i'V_i + c_i'V_i^2), &{}\text {if}\; j \in \mathcal {B}_s; j = a(i)\\ 0,&{} \text { otherwise. } \\ \end{array}\right. } \end{aligned}$$

(46e)

Matrices $E_{11}, E_{12}$ and $E_{13}$ are same as $M_4$, $M_7$ and $M_8$, respectively.

1.2 System parameters for single-area system

$R =0.05$ Hz/pu, $D = 0.8$ pu/Hz, $H = 5$ pu s, $T_g = 0.2$ s, $T_t=0.5$ s.

1.3 System parameters for three-area system

System parameters used for three-area system is listed in Table 12.

Table 12 System parameters for three-area system

1.4 System parameters for RESs

WTG: $\rho =1.225$ kg/m$^3$, $A=1648$ m$^2$, $R_b=22.9$ m, $w_r=3.14$ rad/s, $C_1=-0.6175$, $C_2=116$, $C_3=0.4$, $C_4=0$, $C_5=5$, $C_6=21$, $C_7=0.1405$, $k_{wt}=1$, $\tau _{wt}=1.5$ s.

SPVG: $\eta =10$%, $A_s=4084$ m$^2$, $T_a=25\,^\circ$C.