Voltage Stability Studies for Distribution Networks: Assessing Load Dynamics

We model a two-node distribution circuit, with an infinite bus at Node 1, and a load connected at Node 2. The two-node distribution circuit is presented in Fig 1.

We use phasor representation throughout. We denote by $V_{1}$ and $\delta_{1}$ the voltage magnitude and voltage phase angle at Node 1, and by $V_{2}$ and $\delta_{2}$ the voltage magnitude and voltage phase angle at Node 2. We denote by $R$ and $X$ the line resistance and reactance respectively, and by $P_{2}$ and $Q_{2}$ the real and reactive power respectively at Node 2. Assuming $\delta_{1}=0$ and $R=X$, the algebraic constraints defining power flow between Node 1 and 2 are given by:

We use the generic dynamic load model presented in [5]. The load is modelled as a linear system $\dot{x}=Ax+B$, with the entries of $A$ and $B$ varying non-linearly with the voltage $V_{2}$. Equation (3) defines the state-space load model, and (4)–(7) define the steady state and transient load characteristics that are used as entries in the $A$ and $B$ matrices.

The system is at equilibrium when:

and when (1) is satisfied for $P_{2}$ and $Q_{2}$ given by (8) and (9):

Observe that the stability of a given equilibrium point will depend on both the algebraic constraint from (1) and the load dynamics from (3). Due to the system’s non-linearity, we choose to investigate stability via simulation rather than analytically.

Equation (1) in the DL model can have up to two positive, real voltage solutions. This is a potential source of ambiguity, so we formulate a DC benchmark model which always has a unique voltage solution. In Fig. 2 we present the circuit representation of this benchmark model, re-constructed from [11].

This circuit has a source voltage $V_{1}$, similar to the infinite bus in the previous model, and a line resistance $R$. The ‘load’ is comprised of a capacitor in parallel with a tunnel diode. The capacitor has linear dynamic behaviour, and the tunnel diode has static non-linear behaviour. Together, they give nonlinear, dynamic behaviour. The system’s behaviour is given by:

where $h(V_{2})=aV_{2}^{5}+bV_{2}^{4}+cV_{2}^{3}+dV_{2}^{2}+eV_{2}$. The parameters of $h(V_{2})$ are chosen such that the system has three equilibrium points, where an equilibrium point is a voltage satisfying the equation:

Example circuit characteristics are provided in Fig. 3, with the three equilibrium points marked. Due to the slope of $h(V_{2})$ at the points of intersection with the Node 2 load line, the lower and upper equilibrium points are stable, and the middle equilibrium point is unstable. The stable equilibrium points each have well-defined regions of attraction, separated by the unstable middle equilibrium voltage. Furthermore, under a disturbance that causes a high voltage stable equilibrium point to be lost, the dynamics allow the system to transition from a high voltage to a low voltage equilibrium point. Conceptually, this benchmark model behaves as we would expect a physical power system to behave.

Here, we present our investigation into the system response to a range of initial voltages for $V_{2}$. In Fig. 4 we present our simulation results for the benchmark system with parameters specified in Table I.

From the three trajectories presented in Fig. 4 we can observe the regions of attraction for each stable equilibrium point. We observe that under certain initial $V_{2}$, the system converges to one stable equilibrium rather than the other.

In Fig. 5, we present simulation results of the benchmark system’s disturbance response, again using the parameters in Table I. A disturbance to both the source voltage $V_{1}$ and line resistance $R$ causes the system’s P-V curve to shrink inwards. This results in the loss of the high voltage stable equilibrium point. Consequently, the voltage decays to the new low voltage stable equilibrium point. Not only do these results demonstrate what we might call ‘complete’ voltage collapse, but the model uses physical state variables and has a physical interpretation, which is useful.

We present our simulation results that investigate the system response to a range of initial load states, $x$ and $y$. In Fig 6, we present our simulation results obtained using the parameters displayed in Table II.

Specifying initial states $x$ and $y$ does not specify a unique system voltage and power, due to (1) having multiple voltage solutions. Therefore, in addition to specifying initial states $x$ and $y$, we also have to select which solution of (1) to use. If we select the high-voltage solution at each timestep, then we get trajectories that converge to the low voltage equilibrium at $V_{2}=0.0002889$ (see Fig. 6a), and if we select the low-voltage solution at each timestep, we get trajectories that converge to the high voltage equilibrium at $V_{2}=0.9612$ (see Fig. 6b). Therefore, this model is limited if we want to investigate its dynamic response from all possible initial conditions.

We simulate the system response to a load level increase by applying a step increase to $P_{0}$ and $Q_{0}$. We let the maximum voltage solution drive the simulation for three distinct cases. In Case 1, the post-disturbance system still has two equilibrium voltages. In Case 2, there is only one post-disturbance equilibrium voltage, at $V_{2}=0$. In Case 3, there are no post-disturbance equilibrium points. The parameters for each case are presented in Table III. In Fig. 7 we present the simulation results.

In Fig. 7a, we observe that if the post-disturbance system still has a high voltage equilibrium, then the load dynamics move the system to that new operating point.

In Fig. 7b, we observe that even when we can analytically calculate a single equilibrium point at $V_{2}=0$, we cannot reach it. The simulation stops prematurely because the system reaches a point where there are no longer real, positive voltage solutions to the algebraic constraint.

In Fig. 7c, we observe after the disturbance, the load dynamics pull the system voltage down, but the simulation ends prematurely because the system reaches a point where there are no longer real, positive voltage solutions to the algebraic constraint. Again, this result is unsatisfactory because it doesn’t indicate whether the voltage goes to $0$, or what behaviour occurs beyond this point. Furthermore, the use of abstract states $x$ and $y$ rather than physical states like $V$, $P$, and $Q$ makes providing a physical interpretation of the results more difficult.

Lastly, we examine the load state space for the system with parameters given in Table II and observe that if there exist two positive real solutions to (10), and we use initial states $x$ and $y$ that lie within the valid region shown in Fig. 8, the system will always converge to either the high voltage equilibrium (if driven by the maximum voltage solution) or the low voltage equilibrium (if driven by the minimum voltage solution). In this sense, each equilibrium point is stable. However, this feature is likely to be a product of the specific choices for (4)–(7). Alternative load characteristics should be simulated to determine the conditions under which unstable equilibrium voltages could exist in this system.

The fundamental limitation of the DL model is its inability to model behaviour across the entire power-voltage domain. By comparing the DL model to the benchmark model, we can gain important insight into the study of load dynamics and their impact on voltage stability in distribution networks. First, we observed that the concepts of stable equilibria and regions of attraction from control theory applicable to power systems. The DL model has a ‘stable’ operating region, which is the region where we get two real positive voltage solutions to (1). However, we cannot simulate trajectories that enter or exit this region, and therefore we cannot improve our understanding of what physically happens in the ‘unstable’ regions. This contrasts with the benchmark model, where the load dynamics actually move the system from an acceptable operating point to another, unacceptable operating point.