Voltage Stability Margin’s Solution Using Nonlinear Programming Method Based on Dynamic Power Flow

INTRODUCTION

---

As the economy develops at a high speed, more and more electric power is needed. With the limitation of the environment, economy, construction costs and other factors, the speed of power network construction is lower than that of power source construction. So, power system often operates under the condition of heavy load and some hidden troubles of safety and stability exist. Voltage stability has a great influence on power system stable operation and voltage stability margin namely load margin is an important index to reflect the voltage stability. Dispatchers can know the statement of voltage stability by voltage stability margin and choose the appropriate voltage regulation strategy (Craig and Tapan, 2003; Hong et al., 1997). Therefore, whether the result of voltage stability margin is practical or not will influence the correctness of dispatchers’ decision and then that will be related to the safe and stable operation of the whole power system.

There are many methods to compute the voltage stability margin’s index at present and the most common methods are continuation power flow (Canizares and Alvarado, 1993; Ajjarapu and Christy, 1992) and nonlinear programming method (Van Cutsem, 1991; Irisarri et al., 1997; Rosehart et al., 2005). Continuation power flow obtains the system’s P-U curve by computing power flow repeatedly and determine the voltage stability margin according to the position of voltage collapse critical point in the P-U curve. But the method is not comprehensive enough in considering the system safety constraints, its computational efficiency is not high enough and the position of critical point obtained is not accurate enough (Chiang et al., 1995; Jean-Jumeau and Chiang, 1993). By using nonlinear programming method, the solution of critical point can be transformed into load optimization problem and then the optimization problem can be solved through establishing mathematical model and choosing the appropriate optimization method (Van Cutsem, 1991; Parker et al., 1996). The method can consider many kinds of system constraints flexibly and analyze and research in different emphasis points according to the actual situation. So the result of voltage stability margin computed by using nonlinear programming method is more practical and useful for dispatchers.

Normal nonlinear programming method computes the voltage stability margin under specified load increasing modes by introducing a load growth factor into equality constraints. Guo et al. (1999) computes the voltage collapse critical point by using normal nonlinear programming model and predictor corrector primal dual interior point method under load increasing modes at different nodes, such as at a single node, at an area or at all nodes. Using normal nonlinear programming model can compute the system voltage stability margin under many load increasing modes. The output of generators is decided by calculation results of the optimization algorithm, instead of combining with the practical growth mode of generators’ active power outputs. When considering the growth mode of generators’ active power outputs, if only the load increment is distributed to generators, the specified slack node will undertake all of the unbalanced active power produced by network loss. Thus, on the one hand, the results of voltage stability margin will be affected by the selection of slack node. On the other hand, when load increases nearby the voltage collapse critical point, the increment of network loss is significant. If the network loss is only undertaken by the specified slack node, that will affect the accuracy of voltage stability margin’s results.

This study considers the growth mode of generators’ active power outputs based on the normal nonlinear programming model and builds the nonlinear programming model combined with dynamic power flow (Ramanathan et al., 1986), named dynamic nonlinear programming model. The model can distribute the unbalanced power produced by load growth to many generators reasonably and results of voltage stability margin calculated are not affected by the selection of slack node. Meanwhile, interior point method is selected to solve the nonlinear programming problem, because of its low computational complexity, simpleness and good convergence (Granville et al., 1996). Tests on IEEE-30 system and Liaoning power system verified the advantages and effectiveness of the method.

METHODOLOGY

Load margin: Now indexes of voltage stability evaluation mainly include the state index and margin index (Chiang et al., 1997). The margin index has many advantages, such as its definite physical meaning, the linear relationship between its magnitude and the distance between the system operating point and the voltage collapse critical point and so on. Dispatchers can know the statement of the system clearly and take effective measures in advance to avoid voltage collapse by the margin index. So the margin index is widely used in the system operation.

The P-U curve is shown in Fig. 1. The abscissa represents active load and the ordinate is node voltage.

Fig. 1:	P-U curve

Expression of load margin is as follows:

(1)

The expression of load margin percentage is as follows:

(2)

where, P₀ represents active load of the current operating point, P_max represents active load of the voltage collapse critical point.

Model of load margin’s solution using normal nonlinear programming method: Load margin’s solution using normal nonlinear programming method can ensure that the results satisfy the constraints rigorously. So, the determination of constraints is a key link of the method and the difference of constrains will have large influence on the results. The mathematical model of load margin’s solution using normal nonlinear programming method includes three parts which are objective function, equality constraints and inequality constraints.

The objective function is to make the factor of load growth λ maximum. The expression is as follows:

(3)

Because the interior point method is a method to obtain the minimum, the maximum of λ should be transformed into the minimum of -λ. Where λ obtains its maximum is the voltage collapse critical point.

The equality constraints are balance equations of power flow which are modified. The expression in polar coordinates is as follows:

(4)

where, P_Li0, Q_Li0 represent, respectively active power, reactive power of load i in the ground state, λ represents the factor of load growth, K_LPi, K_LQi represent, respectively load growth coefficients of active power, reactive power. P_Li and Q_Li from the balance equations of power flow are modified appropriately in Eq. 4. The sign of K_LPi and K_LQi reflects growth directions of active and reactive load and the magnitude of K_LPi and K_LQi reflects the proportion of load increment of each node to the total load increment.

The inequality constraints include mainly the constraints of generators’ active power outputs, generators’ reactive power outputs and node voltage amplitude. The expression is as follows:

(5)

where, S_A, S_R represent, respectively the sets of active sources and reactive sources, S represents the set of all nodes, represent, respectively the upper limit and lower limit of generators’ active power outputs, represent, respectively the upper limit and lower limit of generators’ reactive power outputs, represent, respectively the upper limit and lower limit of node voltage amplitude. In practical applications, other inequality constraints can be added flexibly according to the need so that various conditions can be considered in the model.

When K_LPi and K_LQi are both equal to 1, λ is equal to the load margin percentage. Under other load growth modes, the load margin percentage L should be computed according to Eq. 6:

(6)

where, P_Limax represents the load of node i at the voltage collapse critical point, S_D represents the set of load growth nodes.

Model of load margin’s solution using dynamic nonlinear programming method: In the normal calculation of power flow, the unbalanced power of the system will be balanced by a slack node. That is not exactly accord with the practical operation of power system. When much unbalanced power exists in the system, the results of power flow calculation may not reflect the practical distribution of system’s power flow exactly. In the calculation of dynamic power flow, unbalanced power will be shared by the generators which still have regulating ability and that is accord with the practical operation of power system. So, the slack node is no need in the calculation of dynamic power flow and the results are more practical.

Assuming that the expression of unbalanced power in the system is as follows:

(7)

where, P_Gi and P_Li represent respectively active power output and active load of node i under the current state, P_Loss is represents total network loss. The unbalanced power will be shared by the generators which have regulating ability according to dynamic power flow. So, the balance equation of active power flow turns into follows:

(8)

where, β_i represents the distribution coefficient of system’s unbalanced active power shared by generators and

is satisfied. If there is no generator on node i or the generator has no regulating ability, β_i = 0.

Combining with the theory of dynamic power flow, this study adds an equation constraint to the model of normal nonlinear programming method. The equation constraint is as follows:

(9)

where, P_Gi0 is represents the active power output of generator in the ground state,

represents the total unbalanced active power, including the increment of active load and network loss. After the equation constraint is added, when the load increases, the optimization variable P_Gi will satisfy not only the balance equation of active power flow but also some unbalanced power required.

The value of the distribution coefficient β_i can be determined by the concrete application. β_i can be determined by frequency response characteristics of generators, some economic criteria, distribution of unit capacity, distribution of unit capacity margin and so on. Because data information of the example used in this paper is limited, β_i is determined by the distribution of unit capacity margin. The expression is as follows:

(10)

Meanwhile, the active power outputs of generators will not decrease when the load increases according to the practical operation of power system. So, this study, turns the inequation constraints of active power outputs into follows:

(11)

Therefore, the objective function of the improved nonlinear programming model based on the theory of dynamic power flow is not changed. The equation constraints are composed of Eq. 4 and 9. The inequation constraints are changed by replacing the constraint of active power outputs in Eq. 5 with 11. The improved model is applied in IEEE-30 system and Liaoning power grid in this study and load margin of the two systems is calculated by means of the interior method in the optimization toolbox of MATLAB.