1. Introduction
With the rapid increase in energy consumption and environmental pollution, renewable energy power generation is becoming more and more popular and various kinds of distributed generations are connecting into the power grid. Meanwhile, the incorporation of renewable energy units gives rise to the increasing need for resource capacity or ancillary services if there exists major forecast uncertainty. Meanwhile, flexible loads in the demand side can provide different kinds of ancillary services, such as frequency regulation, load following, and other services. The widely adopted flexible loads for these services include thermostatically controlled loads (TCLs) and plug-in electric vehicles (PEVs), which can respond the power dispatch or electricity price timely. To address these concerns, there have been several valuable studies on the coordination and interaction of traditional units, renewable units, and flexible loads of source–load systems [
1,
2,
3,
4].
Renewable power generation has gained much attention and been in an increasing trend, which is beneficial to environment and economics. Especially, wind and photovoltaic generation are thought to be the most developed renewable sources worldwide. However, the power produced by these renewable energies largely depends on natural environmental conditions, such as wind speed or illumination intensity, which are stochastic and cannot be precisely predicted.
The uncertainty of renewable power may pose new challenges for power system operation and control, especially during times of high penetration [
5]. To provide a flexible and comprehensive consideration of the forecast error of renewable power [
6], a common solution is to restrict wind power and abandon light power so as to protect the power systems. Another solution is utilizing the various optimization methods. The look-ahead dispatch method was considered in [
7,
8], which has proved to be an effective strategy to reduce the power imbalance caused by the injection of renewable power [
9]. By taking into account the uncertainty of renewable power, robust optimization [
10,
11], stochastic optimization [
12,
13], and chance-constrained stochastic optimization [
14,
15] have been the most popular methods to account for uncertainties in power generation.
On the other hand, the demand response of power grids has already switched from traditional mode of load curtailment to the mode of dynamical response without interfering with users’ comfort levels [
16]. DR optimization has received much attention. The authors in [
17] proposed a hierarchical demand response architecture to control and coordinate the performance of various DR category resources. Demand response controllability was investigated in [
18] for the unit commitment model with limited predictability and residential DR resources. By considering the site selection and the incentive price, an optimal strategy for responsive loads was proposed in [
19] for source–network–load system. Online demand response for nondeferrable loads was investigated in [
20] with rechargeable battery and renewable energy. By the method of the alternating direction method of multipliers, a hierarchical robust distributed optimization was proposed for demand response services [
21].
As for the TCL agents, it has been shown that TCLs can provide power-balancing reserves when aggregated due to their thermal energy storage capacity [
22,
23]. The authors in [
24] demonstrated that TCLs can be considered as rapid DR activation loads for power system control operations. By using switching-rate actuation, the authors in [
25] studied the demand response of TCLs and household refrigerators. Distributed load following was investigated in [
16] for aggregate TCL loads. The authors in [
26] presented a mean-field model for analysis and control of the aggregate demand of heterogeneous TCLs. By using a stochastic Markov decision process and distributed robust optimization, ref. [
27] internalized the exogenous uncertain dynamics of TCLs.
Literature on demand response and EV charging scheduling has proved that EVs will become the main demand response resource in the near-future. Many researches have concentrated on charging optimization and control problems. The authors in [
28,
29,
30] investigated the optimal energy management and control problems of smart home with PEVs and photovoltaic arrays. By designing a smart-charging scheme for PEV, it was shown in [
31] that the aggregate EVs are able to reduce the peak demand or peak shaving. A fair demand response strategy for EVs was proposed in [
32] for a cloud-based energy management service in a given time period. The authors in [
33] proposed a charging load model for an electric vehicle charging station, which could be integrated to distribution systems so as to obtain the optimal charging decisions for demand response provision.
Both aggregate TCLs and aggregate PEVs are large-scale flexible loads in power grids, which can be involved in the DR program together to share the power imbalance. By considering the uncertainty in renewable energy generation, load consumption, and load reserve capacities, a chance-constrained optimal power flow model was proposed in [
34,
35] to procure minimum cost energy. The authors in [
36] proposed a chance-constrained optimal power flow model to schedule the power production of both generators and controllable electric loads. By the method of stochastic model predictive control, the authors in [
37] investigated the optimal power dispatch and control for power grids with renewable energy resources and EVs. Based on the mixed-integer linear programming method, intelligent DR for industrial energy management was designed in [
38] by considering TCLs and EVs.
Despite there being several related studies conducted for TCLs and PEVs, there are still some challenging problems that remain unsolved. According to different DR incentive strategies, such as price response, policy response, or control response, the actual customers’ responses are varied and diverse. As for the control response loads, the basic control model of the objective is needed. The individual models of the TCL and PEV are mature while the aggregate models and the aggregate control strategies of these loads are immature. Motivated by the above observations, this paper intends to investigate the optimization dispatch and demand response control of source–load systems with uncertain renewable power injection and flexible TCL and PEV load agents. To the best knowledge of the authors, most of the demand response problems are solved by various kinds of optimization models and methods; few published literature have investigated this problem by aggregate control algorithms. This paper aims to fill this gap and solve the demand response optimal power allocation and response problems of source–load systems via aggregate control models and strategies for flexible loads.
The main contributions are summarized three-fold: (1) A probabilistic controllable interval is introduced to the chance-constrained look-ahead optimization, which can cope with the uncertainty of both the renewable power generation and the flexible load response; (2) compared with discrete-time on/off control of TCLs, a continuous-time setpoint temperature regulating control algorithm based on aggregated models is proposed to guidance the power change of the TCL agent; (3) a time-varying charging power control algorithm based on the saturation function is proposed for the PEV agent such that the aggregate PEVs can follow the reference power trajectory.
An outline of the remainder of the paper is organized as follows.
Section 2 states the problem formulation and the optimization and control framework.
Section 3 provides the chance-constrained look-ahead programming model for the source–load system with the injection of renewable power and flexible TCL/PEV agents.
Section 4 describes the aggregate model for TCL/PEV agents and designs the corresponding control algorithms for the optimal power profile tracking.
Section 5 shows the effectiveness of the proposed optimization and control algorithm on a modified IEEE 39 bus system.
Section 6 discusses the optimization and control framework and draws the conclusions.
Some abbreviations are provided in the following before the main results.
Acronyms | Full Name |
DR | Demand response |
TCL | Thermostatically controlled load |
TCL | Plug-in electric vehicle |
SAA | Sample average approximation |
TOU | Time-of-use |
SoC | State-of-charge |
2. Problem Formulation
Consider the coordination optimization problem of a source–load system, where the source of the system includes the traditional generating units and renewable power (mainly wind power and photovoltaic power) and the load of the system includes rigid load and flexible load. The rigid loads, such as lighting and computers, are always uncontrollable but can be predicted. The flexible loads, such as thermostatically controlled loads and plug-in electric vehicles, can be controlled by the corresponding control signals. On the other hand, the renewable power injection is always a random variable because the wind speed and the ambient temperature and illumination are always random. Therefore, how to balance the power production and consumption with the maximal social welfare is a crucial problem among units and flexible loads.
This paper intends to solve this problem by setting up a chance-constrained look-ahead programming model for the source–load system and designing two kinds of demand response control algorithms for TCLs and EVs. Specifically, flexible loads are aggregated as a load agent, which can be involved in the electricity market to participate in the load bidding. The terminal DR loads are controlled by the load agent by issuing the corresponding control signals. The schematic diagram of the optimization dispatch is shown in
Figure 1.
Based on the prediction of the rigid load, the actual renewable power injection, and the day-ahead power generation plan, the look-ahead optimization dispatch with chance constraints can be solved by the sample average approximation (SAA) method [
39,
40]. Furthermore, the generating units respond with optimal generating instructions and the flexible load agents achieve the optimal power profile by demand response control of massive-terminal, small, controlled loads. The detailed control models and the control algorithms of TCL agents and PEV agents will be discussed in the following section.
3. Chance-Constrained Look-Ahead Optimization
This paper considers the joint real-time economic dispatch problem for generating units and flexible load agents by considering the uncertainty of renewable energy power generation. The objective of the power scheduling is to maximize social welfare, i.e., maximizing both generating units and load agents:
where
is the total social welfare with respect to the real-time power variable
(day-ahead scheduling plus intraday corrective scheduling);
is the time-of-use (TOU) power price;
and
are the welfare functions of the
ith generating unit and the
jth load agent, which are given as follows:
For the units, the cost function
usually can be approximated by a quadratic convex function
, where
,
, and
are predetermined constants and
is the generated power. For the load agents, the utility function
, often assumed to be the convex utility function with the zero initial value, is a quadratic utility function that can be described by [
41,
42]
where
and
are predetermined constant coefficients.
Considering the randomness of the actual renewable power, the following chance constraint with a controllable interval is involved:
where
is the controllable confidence interval of the source–load system, which is often set to be smaller than the actual operation interval because of the response uncertainty of the flexible DR loads;
is the renewable power injection with the random wind power variable
and the random photovoltaic power
;
is the prediction value of the rigid load in the system; the probability
is required to be at least
.
Other considered inequality constraint conditions for such an optimization problem are given as follows:
for
, and
;
and
are the lower and upper bounds for
ith generating unit;
and
are the time-varying lower and upper bounds for
jth load agent;
and
are the lower and upper ramping rates of the units and load agents.
By the sample average approximation method [
39,
40], the optimal power scheduling for units and flexible load agents can be obtained by solving the look-ahead optimization (
1)–(
5) with chance constraint. In the following, the design of the demand response control algorithms for the aggregate TCL agents and PEV agents will be provided.
Remark 1. The power variability and uncertainty of the renewable power are handled by the probabilistic chance-constrained optimization, where the probability distributions of the wind power and the PV power are assumed to be mutually independent. Then, the joint probability density function can be derived by the probability theory. Then, the sample of the renewable power can be generated by its probability distribution. The power balance constraint is transformed to be a probability confidence interval with a predefined confidence level , which is able to cope with the volatility of the renewable power. On the other hand, the inherent uncertainty of flexible TCLs and PEVs is absorbed by the reserve capacity of the system, i.e., the controllable confidence interval of the optimization constraint (4) can be set smaller than the actual operation interval of the source–load system. 5. Case Study
This section validates the performance of the proposed chance-constrained optimization and demand response control architecture through numerical simulation on a modified IEEE 39-bus test system, the unifilar diagram system structure is given in
Figure 4. Suppose that the loads under bus nodes B11, B23, B28, and B32 are flexible controllable loads, which are managed by the corresponding load agents TCL/PEV A1/2, and loads under the bus nodes B27 and B29 are fixed loads. On the other hand, the generators G1∼G6 are slow units and generators G7∼G10 are AGC units.
The flexible controllable loads serve as demand side resources, which can provide active power regulation services together with generation units. Six slow units and the four flexible load agents are dispatched by the dispatch center based on the chance-constrained look-ahead dispatch. The cost coefficients and capacities for all the participants are given in
Table 1. We considered the optimization dispatch and demand response control of a summer working day, where the TCLs (mainly air conditioners) and PEVs are controlled in real-time.
Furthermore, the optimization period is 15 min and the look-ahead period
; the DR control sampling period for real-time control is 20 s. The coupling time-scale relationship is given in
Figure 5.
In the test system, we assume all TCLs in the same agent are homogeneous—that is, with the same thermal capacitance
C, thermal resistance
R, output cooling energy
, energy transmission efficiency
, and preferred setpoint
. The number of TCLs in each agent and the initial proportion of the off TCLs
and other parameters are given in
Table 2. Suppose the ambient temperature is 24∼38
C, given in
Figure 6 and
= 34
C. The predicted rigid load and the TOU price are provided in
Figure 6 as well.
On the other hand, the renewable power includes wind power and photovoltaic power, where the wind power is a random variable bounded by its rated output power. Suppose there are 50 wind generators in the wind farm with a rated power of 1.5 MW for each generator and 60 photovoltaic panels in the system. The photovoltaic power is closed related to the ambient temperature and illumination; its value is provided in
Figure 6 as well. In the simulation, wind power and photovoltaic power are mixed together, the renewable power interval is shown in
Figure 7, and a sampled wind power curve is distributed in the power interval.
The controllable interval of the source–load system is set to be
of the maximal regulation capacity 100 MW,
MW, and the chance-constraint probability
. By solving the chance-constrained look-ahead optimization, the optimal power trajectories for units and load agents are given in
Figure 8 and
Figure 9, respectively.
As can be seen from
Figure 8 and
Figure 9, the power generation follows the load fluctuation and load agents gain the corresponding regulation capacity as well. If all the TCLs and PEVs are not involved in the demand response program, then the extra power generation will be compensated by the generating units. Next, the simulation of the demand response control is illustrated.
As for the TCL agents, the state-space dimension
, the sampling period
is set to be 20 s, and the initial temperature of the TCLs follows a uniform distribution
C; the comfort temperature intervals for the two agents are given with
and
separately. By setting
and
, and running the system (
6) with control input (
8), the reference power and the actual aggregate power are given in
Figure 10 and
Figure 11. The relative error curves are provided in
Figure 12, and are smaller than
.
As for the PEV agents, the SoC interval is set to be
, and the interval was divided into
subintervals. Furthermore,
means the EVs with the SoC value lower than
can only enter but not exit and EVs with the SoC value upper than
can enter or exit freely. The sampling period
s as well. The initial centralization of PEV
and
for
. The hourly transport flows for the incoming EVs of two PEV agents in time period from 07:00 to 19:00 are given in
Figure 13. According to the survey of the daily trip lengths, over
of the PEVs return to the first SoC discretization segment and the transport flow coefficient
is set to be
and
for
.
The output transport flow of PEVs can be calculated by Equation (
15), where
is set to be
—that is,
PEVs charged that can leave while choosing not to leave at this time. Here,
denotes the concentration of PEV whose SoC has reached the target charging area.
We set
in the saturation function,
, and the initial value
in the controller. Then, by system (
14) with the control input (
16), the optimal power profile and aggregate power of PEV agent are given in
Figure 14 and
Figure 15 for agents PEV
and
.
As can be seen from
Figure 14 and
Figure 15, the PEV agents follow the optimal charging power profile well since the maximal charging power of the transport flow of PEVs in the simulation is much larger than its actual charging power. The relative error curves are provided in
Figure 16, which are smaller than
.
Finally, the total response deviation curve
is shown in
Figure 17. As can be seen from the figure, the deviation is distributed in the regulation interval and the statistical probability is
, which satisfies the chance-constraint probability.
As can be seen from the simulation results and the relative tracking errors, the demand response control performance of the TCL and PEV agents is acceptable as long as the optimal power curves are solvable, which shows that the proposed demand response control algorithm can realize the demand response power tracking of flexible load agents. By participating in the demand response program, the owner of the TCL or PEV can receive some compensation with lower electricity costs. The proposed dispatch and control framework can be applied not only in bulk power systems but also in microgrids since the structural design is similar.
Remark 5. In the upper layer of optimization calculation, the renewable power curve for 15 min of data is derived by the cubic spline interpolation method. In the lower layer of control procedure, the control interval (20 s) for the flexible TCL and PEV loads is much smaller than the renewable power injection sampling period (1 h), which allows the DR loads sufficient time to track the uncertainty of the power injection.
6. Discussion
Compared with the traditional mode of power generation following load, the renewable power injection and the flexible loads in the demand side could participate in the interactive operation of power grids. The chance-constrained look-ahead optimization and demand response control algorithm proposed in this paper are effective for the coordinating operation of the source–load system, which can be applied in practice.
(1) Implementation: In the proposed dispatch optimization and control framework, units and load agents report their basic information to the dispatch center; then, the optimization can be solved in the dispatch center. The optimal dispatch plan is returned to the units and agents, and the units achieve the scheduled power by their own control algorithms. As for load agents, the control system of agents are equipped with the aggregate models for TCLs and PEVs, based on the error feedback algorithms; the corresponding control signals for the terminal TCLs and PEVs are generated and then broadcasted to them in a centralized way. The controllable power interval can be fulfilled by AGC units and flexible loads with price compensation. In order to protect the power grid, the power interval for the optimization calculation can be set conservatively, such as of the actual regulation capacity.
(2) Drawbacks: As for the chance-constrained optimization, there may be no optimal solution for the optimization. The confidence interval can be set larger and the confidence probability can be set smaller in the actual optimization, even when there is no solution for the optimization. On the other hand, since the aggregate models are approximate models, the actual control response errors are unavoidable. It has been shown in literature that the accuracy of the model is related to the dimension of the state-space model, i.e., the higher the dimensionality of the model, the higher the accuracy of the model. Conversely, the higher the dimensionality of the model, the higher the computation complexity. Therefore, a moderate choice for the dimension is feasible for the actual application. Meanwhile, since the proposed demand response control algorithm for PEVs is based on the time-varying charging power instead of on/off charging control, the EVs are assumed to be always available. Therefore, the chargeability of the EVs and user comfort levels have not been considered sufficiently in the manuscript. If the EV needs to be charged to the desired SoC with the desired minimal time instant, the EV can be charged at the maximal charging power directly.