AC-DC Power Systems Optimization with Droop Control Smooth Approximation

Piecewise linear droop controls are difficult to implement for several reasons. The non-differentiable nature of these functions poses challenges for control system design, tuning, and optimization. The control system design requires a solution to the complete representation of the power system including steady-state operation and dynamic controls formulated as a system of differential and algebraic equations, where non-differentiable functions add further complexity [17], [18].

The implementation of piecewise linear droop control in steady-state nonlinear power systems problems, such as power flow (PF) [19], optimal power flow (OPF) [20], and security-constrained optimal power flow (SCOPF) [21] is challenging. The PF problems typically use numerical solvers which only support twice differentiable functions in the model. In OPF and SCOPF problems, the typical method to encode piecewise linear droop control depends on integer variables. The inclusion of integer variables makes the overall problem with the ac/dc physics a mixed-integer nonlinear programming (MINLP) problem and therefore it is much more challenging to find optimal solutions [22].

To address these challenges, we propose a methodology to convert the piece-wise linear droop characteristics to smooth (differentiable everywhere) functions. We use a technique inspired by the machine learning literature, which entails encoding the piecewise functions through the ReLU¹¹1‘rectified linear unit’ function. After encoding the constraints in that fashion, we apply a substitution with a function that smoothly approximates the ReLU, i.e. the softplus function with a tuneable sharpness setting.

Overall, this process converts the nonsmooth droop characteristics into smooth but approximate constraints with tuneable approximation error, which can be efficiently solved using Newton’s method or derivative-based interior point solvers. Formulations are provided for different droop controls in ac-dc power systems and implemented in PF, OPF, and SCOPF problems to demonstrate the computational efficiency and accuracy in comparison to the mixed-integer formulations. In summary, the key contributions are:

• •

  A technique to derive smooth models encoding the attributes of piece-wise linear droop control including saturation. This enables more realistic SCOPF models that ensure better power-sharing, voltage regulation, and, system efficiency without expensive computation.

• •

  The implementation is made available, with support for different droop controls in ac-dc power systems in PF, OPF, and SCOPF problems demonstrating computation efficiency and accuracy in comparison to the mixed-integer formulations.

This paper is organized as follows. Section II describes the proposed smooth droop control functions. Sections III and IV present the simulation results and conclusions.

Consider the ReLU function $f(x)=\max\{0,x\}$, such that $x\in\mathbb{R}$, that is not differentiable at $x=0$. For any positive value $\epsilon>0$, $f(x)$ can be approximated by the smooth softplus function [23] with sharpness setting $\epsilon$,

with verifiable bounds on approximation error,

A generator responding in a contingency event $k$ adjusts its active power output by a common perturbation denoted by variable $\Delta_{k}$ according to its predefined (offline) participation factor $\alpha_{g}$ (droop slope) until it hits an operational bound $\underline{P_{g}}$, or $\overline{P_{g}}$ as shown in Fig. 1. This is referred to as the generator’s post-contingency frequency response. The active power output $P_{g,k}$ of a responding generator $g$ in contingency $k$ is given as,

A generator that is online in a contingency but is not selected to respond to that contingency maintains its active power output from the base case,

In a contingency event $k$, a responding generator $g$ tries to maintain its voltage magnitude $V_{i,k}$ equal to the base case voltage $V_{i,0}$ by adjusting its reactive power $Q_{g,k}$ unless it hits operational bounds $\underline{Q_{g}}$ or $\overline{Q_{g}}$ as shown in Fig. 2. This is referred to as PV/PQ bus switching control and modeled,

We convert conditional constraints (3) and (5) into continuously differentiable functions using smooth approximation (1). The constraint (3) can be written,

By applying (1), constraint (6) can be expressed,

The parameter $\epsilon>0$ can be used to regulate the degree of approximation accuracy as shown in Fig. 1, the approximation gap narrows for $\epsilon_{1}<\epsilon_{2}$ and further as $\epsilon\rightarrow 0$. Similarly, with generator-node mapping $\mathcal{T}^{g}\subset\mathcal{G}\times\mathcal{I}$, (5) can be stated,

where $V_{i,k}^{+}$ and $V_{i,k}^{-}$ represent the upper and lower margins of base case voltage $V_{i,0}$ from the bounds as,

The necessary conditions for (8) are the voltage magnitude $(\underline{V_{i}},\overline{V_{i}})$ and generator reactive power bounds at these buses. By applying (1), the constraint (8) can be expressed as,

The P–Vdc linear droop control as shown in Fig. 3 is expressed by a relation between dc voltage $V^{\text{dc}}_{d}$ at dc grid bus $d\in\mathcal{D}$ and the dc side power $P^{\text{dc}}_{c}$ of converter $c\in\mathcal{C}$, with the mapping $\mathcal{T}^{c}\subset\mathcal{C}\times\mathcal{D}$ and $\forall cd\in\mathcal{T}^{c}$,

where $k^{\text{droop}}_{c}$ is the droop coefficient, $P^{\text{dc*}}_{c}$ denotes the reference dc side power, and $V^{\text{dc*}}_{d}$ is the reference dc voltage. The $\zeta$ function returns the sign of the reference dc side power to represent both inverter and rectifier modes of operation.

The constraint (12) can be rewritten in the form of max functions explicitly for either of the sign conventions based on a given reference value,

Here, the converter dc power bounds are expressed as $\underline{x}=\max\{P^{\text{dc}}_{c}(\underline{V^{\text{dc}}_{d}}),\underline{P^{\text{dc}}_{c}}\}$, and $\overline{x}=\min\{P^{\text{dc}}_{c}(\overline{V^{\text{dc}}_{d}}),\overline{P^{\text{dc}}_{c}}\}$ using (12).

By applying (1), constraint (13) can be expressed,

The P–Vdc piecewise linear droop control with power deadband and voltage limits as shown in Fig. 4 can be defined,

The converter operates in a constant power mode when the dc voltage is between $\underline{\underline{V^{\text{dc}}_{d}}}$ and $\overline{\overline{V^{\text{dc}}_{d}}}$. When the voltage exceeds these limits, the droop control adjusts the power output of the converter until the voltage hits its bounds ${\underline{V^{\text{dc}}_{d}}}$ and ${\overline{V^{\text{dc}}_{d}}}$. Similar to (12), a smooth function for (15) with voltage limits is,

When $P^{\text{dc*}}_{c}>0$, $-{P^{\text{dc}}_{c}}^{\epsilon}(V^{\text{dc}}_{d})$ can be implimented. For the PF application, (12) and (15) are implemented without limits and thus, this five-step piecewise linear droop function is extended beyond voltage bounds. Furthermore, power limits can also be imposed on (16) similar to (13) and (14). Finally, the same smooth functions can be adapted for Q–Vac droop control.

The section compares the proposed smooth approximation-based NLP models with the MINLP models for the PF, OPF and SCOPF problems considering steady-state voltage regulation in ac-dc power systems using P–Vdc and Q–Vac droop controls and generator PQ response functions. Five scenarios with the following response functions are studied:

1. i)

   generator PQ droop;

2. ii)

   generator PQ, and converter P-Vdc droops;

3. iii)

   generator PQ, and converter P-Vdc w. deadband droops;

4. iv)

   generator PQ, and converter P-Vdc, Q-Vac droops;

5. v)

   generator PQ, and converter P-Vdc, Q-Vac w. deadband droops;

The numerical results, as presented in Table I, illustrate that the proposed smooth approximation-based NLP models significantly outperform the MINLP models in computational efficiency. While the objective function values remain largely consistent across the models, the incorporation of piecewise linear droop curves with deadbands in the MINLP models introduces additional binary variables (bvar), which increases the computation time. Notably, scenario v, characterized by the highest number of binary variables, exhibits the greatest computational burden. In the context of the SCOPF problem, the addition of each contingency substantially increases the number of continuous (cvar) and binary variables, rendering the problem increasingly challenging to solve. In the 67-bus case, in scenarios iii, iv, and v, MINLP SCOPF models could not be solved due to limited computational resources, whereas, the proposed NLP models efficiently solve these scenarios.

The proposed smooth approximation-based OPF models with and without P–Vdc only and P–Vdc, Q–Vac droop control are solved on case5 ac-dc power system with different load variations and the results are presented in Fig. 6. These results indicate that the implementation of P–Vdc and Q–Vac droop control substantially enhances voltage regulation. Furthermore, the findings underscore that the limited reactive power capacity of the HVDC converter necessitates allowing voltage drops only when the voltage reaches its bounds. This constraint highlights the importance of effectively managing reactive power within the system to maintain voltage stability under varying load conditions.

This section evaluates the impact of different P–Vdc droop slopes and the implementation of droop control with and without deadband on voltage regulation under varying load conditions in the case5 ac-dc power system as shown in Fig. 7. The droop control without a deadband employs a fixed droop coefficient, balancing the trade-off between voltage regulation stiffness and power-sharing accuracy. A small droop coefficient such as 0.003 in comparison to 0.007 improves voltage regulation but sacrifice power-sharing and vice versa. Typically, system operators allow a maximum 10% DC voltage variation, allocating 5% for the line drops, thus limiting the voltage range available for droop control and consequently the droop coefficient. However, the droop control without a deadband cannot operate the converter in constant power control. In contrast, droop control with a deadband maintains the converters in constant power control when the voltage is between 0.98 and 1.02 pu, therefore providing better power-sharing under higher loads and better voltage regulation under light loads.

The proposed smooth approximation allows efficient implementation of the P–Vdc, Q–Vac droop control, and generator PQ response functions in the SCOPF problem. As illustrated in Fig. 8, the P–Vdc, Q–Vac droop control for case5 ac-dc power system efficiently regulates voltage and manages power-sharing during various contingency events such as generator, ac transmission line, transformer, converter, and dc transmission line contingencies. Specifically generator 3, as shown in Fig. 8, is an offshore wind farm connected to the mainland via a HVDC system with no reactive power capability. The P–Vdc droop control ensures the converters operate in constant power control mode by leveraging the sufficient active power response from generators during contingencies, thus maintaining the base-case voltage throughout the events and achieving the final solution. Meanwhile, the Q–Vac droop control considers converters’ limited reactive power capacity, effectively regulating ac bus voltages and adhering to reactive power and voltage constraints.