A nonlinear control method for price-based demand response program in smart grid

doi:10.1016/j.ijepes.2015.07.024

International Journal of Electrical Power & Energy Systems

Volume 74, January 2016, Pages 322-328

https://doi.org/10.1016/j.ijepes.2015.07.024 Get rights and content

Highlights

•
Demand response is formulated as a nonlinear power management system.
•
A distributed power control algorithm is developed based on quadratic pricing function.
•
The condition is established to guarantee the equilibrium of the power control algorithm is socially optimal.
•
The influence of the additive disturbances on the power control algorithm is shown.
•
Implementation details are given for the power control algorithm.

Abstract

This paper proposes a price-based demand response program by the nonlinear control method. The demand response program is formulated as a nonlinear power management system with price feedback. We give the conditions of the price parameters for both the global asymptotic stability of the system and the social welfare optimality of the equilibrium point. Furthermore, the system is shown to be input-to-state (ISS) stable when there are additive disturbances on the power measurements and the price, and the discrete-time implementation of the power management system is given. Simulation results demonstrate the balance between supply and demand and the stability of the system with and without disturbances.

Introduction

Smart grid is an intelligent power system that integrates advanced control, communications, and sensing technologies into the power grid [1]. In smart grid, demand response can motivate customers to shift their loads from on-peak to off-peak periods [2]. It is widely accepted that demand response is a more cost-effective way than providing enough generation capabilities to meet the peak load [3], [4], [5], [6], [7]. In general, there are two categories of demand response programs: incentive-based programs and price-based programs. The incentive-based programs include the direct load control program, the emergency demand response program, and the ancillary services market. For the price-based programs, the utilities can change the power consumption of customers by pricing, such as time of use (TOU), critical peak pricing (CPP), extreme day CPP (ED-CPP), extreme day pricing (EDP), and real-time pricing (RTP) [8]. Smart grid increases the opportunities for demand response by providing real-time data to providers and customers. In smart grid, the price can be provided to the customers in real time. For example, the electricity provider announces electricity prices on a rolling basis in the RTP program, and the price for a given time period (e.g., an hour) is determined and published before the start of the period (e.g., 15 min beforehand).

There exist a number of literature on the price-based demand response programs. Different demand response programs were developed based on game theory [9], [10], [11], stochastic optimization [12], [13], intelligent optimization [14], and dual decomposition method [15], [16]. The social welfare maximization was achieved by optimizing the individual utilities of the customers in the demand response program based on dual decomposition. Then, a distributed power control algorithm was proposed for demand response with communication loss [17]. The works mentioned above assumed that the price is adjusted according to a pricing algorithm instead of an explicit pricing function. Recently, a linear pricing function was developed to achieve the balance between supply and demand for smart grid [18], [19], and a nonlinear pricing function was used to design a distributed demand response algorithm [20]. Nevertheless, few works are devoted to the social optimality of the distributed power control under nonlinear pricing function and the influence of the disturbances on the power control algorithm.

In this study, we use a quadratic pricing function and establish the conditions on the social optimality of the distributed power control algorithm. Due to the unavoidable disturbances on power systems, we further consider the distributed power control with additive disturbances on the power measurements and the price. The differences between our work and the other smart grid algorithms are shown in Table 1. To the best of our knowledge, the social optimality of the distributed power control under the nonlinear pricing function and the influence of the disturbances on the power control algorithm have not been studied. The main contributions are as follows.

•
The price-based demand response program is formulated as a nonlinear power management system.
•
The condition is established for the equivalence of the equilibrium point of the system and the optimal solution of a social welfare maximization problem.
•
The proof of the stability is given for the power management system with and without disturbances on the power measurements and the price.

The rest of the paper is organized as follows. In Section 2, the demand response program is formulated as a nonlinear power management system. In Section 3, the conditions of the price parameters are established for both the global asymptotic stability of the system and the social welfare optimality of the equilibrium point. In Section 4, the input-to-state (ISS) stability is shown for the system with disturbances on the power measurements and the price. The discrete-time implementation of the power management system is proposed in Section 5, and the simulation results are given in Section 6. Finally, conclusions are summarized in Section 7.

Section snippets

System model

As shown in Fig. 1, we consider a smart power system consisting of one electricity provider and N customers. The operation cycle of the power system is divided into several time slots. In each time slot, the electricity provider decides the electricity price and announces it to the customers. Then, the customers manage their power consumption according to the announced price.

Stability and optimality

In this section, we will study the stability of the power management system (2), (3). Before the proof, the definition of global asymptotic stability is given.

Definition 1

Stability [24]

Let $x = 0$ be an equilibrium point for $\dot{x} = f (x)$ with $x (0) = x_{0}$ . The equilibrium point $x = 0$ of $\dot{x} = f (x)$ is said to be globally asymptotically stable if $\lim_{t \to \infty} ‖ x (t) ‖ = 0$ for all initial conditions $x_{0}$ .

Theorem 1

The power management system (2), (3) are globally asymptotically stable if $a > (N - 2) (2 b \sum_{i \in N} x_{i} + c) .$

Proof

Let $ϕ (x) = \dot{x}$ , where $ϕ (x) = (ϕ_{1} (x), \dots, ϕ_{N} (x))$ and $\dot{x} = ({\dot{x}}_{1}, \dots, {\dot{x}}_{N})$ .

Power management system with additive disturbances

In reality, the power measurements and the price are not accurate due to the errors in the two-way communications between the electricity provider and the customers. It is necessary to study the impact of disturbances on the power management system. As shown in Fig. 4, $d_{1}$ and $d_{2}$ denote the additive disturbances on the price and the total power consumption, respectively. Then, the power control algorithm with disturbances is denoted as ${\dot{x}}_{i} = k_{i} (ω_{i} - {ax}_{i} - p (x) + d_{1}), i \in N,$ and the electricity price with

Discrete-time implementation

From the viewpoint of implementation, the discrete-time counterparts of the power control algorithm are considered. The discrete-time control algorithm without disturbances is given as $x_{i} (m + 1) = x_{i} (m) + μ (ω_{i} - {ax}_{i} (m) - p (m)),$ $p (m + 1) = b {(\sum_{i \in N} x_{i} (m))}^{2} + c \sum_{i \in N} x_{i} (m),$ and the discrete-time control algorithm with disturbances is denoted as $x_{i} (m + 1) = x_{i} (m) + μ (ω_{i} - {ax}_{i} (m) - p (m) + d_{1}),$ $p (m + 1) = b {(\sum_{i \in N} x_{i} (m) + d_{2})}^{2} + c (\sum_{i \in N} x_{i} (m) + d_{2}) .$

In practice, the electricity provider sets the electricity price according to the forecast demand and announces

Numerical results

In the simulations, we consider a residential power system composed of ten customers and one electricity provider. The power supply Q is varying from 10 kW to 42 kW. The parameters a and b are set to 3.3 and 0.01, respectively. The step size $μ$ is 0.07. The willingness parameter $ω_{i}$ is randomly selected from [20], [25]. The electricity prices in different time slots are shown in Fig. 7 for the power management system with and without disturbances. We observe that the disturbances cause errors to

Conclusion

This paper uses a nonlinear control method to generate a price-based demand response program. The demand response program is formulated as a nonlinear power management system, and the stability is shown for the system with and without disturbances. It is shown that the power management system can match supply with demand when there are no disturbances, and the disturbances will result in the errors in electricity price and the matching errors between supply and demand. This further degrades the

Acknowledgement

This work was supported in part by National Natural Science Foundation of China under Grants 61503324, 61573303, 61473202, 61172095 and 61203104 and in part by Project Funded by China Postdoctoral Science Foundation under Grant 2015M570233.

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View full text

A nonlinear control method for price-based demand response program in smart grid

Highlights

Abstract

Introduction

Section snippets

System model

Stability and optimality

Stability [24]

Power management system with additive disturbances

Discrete-time implementation

Numerical results

Conclusion

Acknowledgement

Electric J

Energy Econ

Electric J

Renew Sustain Energy Rev

Electric Power Syst Res

Int J Electric Power Energy Syst

Int J Electric Power Energy Syst

Int J Electric Power Energy Syst

Int J Electric Power Energy Syst

Introducing dynamic demand response in the LFC model

IEEE Trans Power Syst

Autonomous demand-side management based on game-theoretic energy consumption scheduling for the future smart grid

IEEE Trans Smart Grid