Multi-objective optimization for voltage and frequency control of smart grids based on controllable loads

0 Introduction

Recently, the penetration of renewable-energy power generation into distribution networks has rapidly increased for environmental reasons [1,2].However, because of the randomness, intermittence, and volatility of the distributed power supply, severe are introduced into the system, including frequency collapse or voltage violation [3-5,10].

Meantime, the energy control capability has been enhanced following the use of smart end-user devices at the residential side.Therefore, voltage and frequency deviations can be controlled by these responsive enduser devices [6].In [8-11], except for diesel generation, the demand side is also expected to reasonably provide voltage and frequency support.Electric vehicle (EV) is one of the typical controllable loads that can provide adequate reactive power at the distribution level to compensate for power imbalance [12].To this end, EVs can be called upon to correct voltage deviations [13-16].Similarly, thermostatically controllable loads, e.g., electric water heaters, air conditioners, and ground-source heat pumps, are ideal candidates to participate in primary frequency control [17,18].The deviation in the system frequency is alleviated using electricity [14].In contrast to the traditional instant load-cutting operation [19,20], the power consumption of controllable loads can be controlled without affecting the consumer comfort, and the load demand smoothly and continuously changes.

However, because of complex intercoupling among the system frequency, nodal voltage, and active and reactive power [21-23], the aforementioned studies mainly focused on either voltage or frequency control.A multi-objective optimization framework for both voltage and frequency deviation correction has been lacking to fully employ the advantages of participating controllable loads, particularly in grids with high penetration of renewable-energy power.

Commonly, the solutions of multi-objective optimization problems are basically converted to a single-objective optimization problem by assigning weights to the targets [24,25].Moreover, the selection of weighting factors is related to the importance of each goal, which is a difficult decision if no prior information is available for the problem to be solved [25].Human factors greatly influence the weighting process; thus, this method suffers from the disadvantages of strong subjectivity [26].Furthermore, other methods such as the sequencing and center methods are also based on reducing the dimension [27,28].Similarly, the optimization process among the objectives is independent of one another [28,29].Thus, the results are very inconsistent because of the lack of information exchange; thus, operators suffer from the difficulty of making effective decisions, and the process takes much time.

Currently, an increasing number of researchers are aware that a great similarity exists between the multiobjective optimization and game problems [30,31].Game theory is an efficient method for solving multi-objective optimization problem [32,33].In the present study, to solve the aforementioned problem, our contributions are focused on the following areas.

(a) A multi-objective optimization model is designed to minimize the effects of fluctuating power outputs from renewable-energy power generations, which provides a guide for controllable loads to deal with voltage- and frequency-deviation events.

(b) An optimization strategy for obtaining the optimal values of nodal-power variations is developed based on the Nash game to overcome the deficiency of subjectivities in the traditional methods.

1 Modeling of voltage and frequency sensitivities to nodal-power variations

In this section, the design of the generator, load, and power-flow models to analyze the sensitivity information of the voltage and frequency to nodal-power variations is presented.

1.1 Generator model

According to the active power-frequency characteristic curve shown in Fig.1, the frequency deviation is considered in the generator model.

where is the active power specified at node i, ΔPGi is the deviation between the active power and specified value, KGi is the power-frequency characteristic coefficient of the generator at node i, and Δf is the deviation in the system frequency.

According to the generator reactive power-voltage characteristic curve shown in Fig.2, the relationship between the reactive-power variations and generator terminal-voltage variations can be expressed as

where ΔUGi is the generator terminal-voltage variations, ΔQG1 is the generator reactive-power variations caused by ΔUGi, and β is the reactive power-voltage characteristic coefficient of the generator at node i.

Meanwhile, the reactive output power is also associated with active-power deviation ΔPGi; therefore, the actual reactive output power of the generator is adjusted according to (3), i.e.,

whereis the reactive power specified at node i, ΔQGi is the deviation between the reactive power and specified value, and aQ and bQ are the coefficients of reactive-power generation control characteristics.

1.2 Load model

By considering the effects of the system frequency deviation and nodal voltage magnitude, the active and reactive power of the loads can be expressed as follows:

where is the load active power specified at node i,is the load reactive power specified at node i, Kpi and KQi are respectively the active and reactive-power-frequency characteristic coefficients of the load, ppi, pci, pzi, qpi, qci, and qzi are the active-and reactive-power-voltage characteristic coefficients of the load, Usp is the specified terminal voltage, and VLB is the voltage magnitude at the node connected to the load.

1.3 Power-flow model

The active and reactive power injected at bus i can be expressed in polar form as follows:

where Ui and U j are the voltage magnitude at nodes i and j, respectively, Gijij+jB is the element of the system admittance at row i and column j, and θij is the nodal phase difference between nodes i andj.

Once the power generation and consumption in the system become imbalance, the frequency deviates from the normal value.As discussed in the generator and load models, the active- and reactive-power mismatches are represented as functions of the state variables (Δf,θ,U), which are equal to zero when the system reaches a new stable equilibrium point.

These nonlinear power-flow equations are usually iteratively solved using the Newton-Raphson method until The relationship between the power mismatches and state variable modifications can be expressed by Jacobian matrix Jext.

where ΔP and ΔQ are the variable sets of the active- and reactive-power mismatches at each node, respectively, Δ(Δf) is the modification in the system frequency deviation, Δθ is the variable set of nodal phase modifications, and ΔU is the variable set of nodal voltage-magnitude modifications.

Alternatively, when the nodal active and reactive power is adjusted, changes occur in the system frequency and nodal voltage levels.In addition, the mathematical relationship can be obtained based on the power-flow functions.

where are the sensitivities of Δf to ΔP and ΔQ, respectively, are the sensitivities of θ to ΔP and ΔQ, respectively, and are the sensitivities of U to ΔP and ΔQ, respectively.

2 Multi-objective optimization framework for voltage and frequency control

A multi-objective optimization model is designed for controllable loads to participate in the voltage and frequency control, as presented in this section.

Design variables

Participation of responsive end-user devices in voltage and frequency control can be realized by supporting the nodal-power variations.Therefore, both ΔPi and ΔQi of node i are design variables.

Objective functions

The characteristics of randomness, intermittence, and volatility of distributed generations lead to generationconsumption imbalance, which aggravates the frequency and nodal voltage deviation in the system.

where Ui is the actual voltage magnitude at node i and U* is the standard voltage magnitude of the system.

According to the sensitivities of U to ΔP and ΔQ, ΔUi can be expressed as

Therefore, Ui is a function of design variables ΔPj and ΔQj ( j=1,2..., n).

where Ui0 is the voltage magnitude at node i before the control.

To minimize the frequency deviation in the system, another objective function can be expressed as

where f is the system frequency after the control and f * is the standard frequency magnitude.

Similarly, by deriving ΔUi, Δ Δ( f) can be expressed as follows with sensitivities of Δf to ΔP and ΔQ:

Then

Therefore, F2 is also a function of design variables ΔPj and ΔQj ( j=1,2..., n).

Constraint conditions

Because the nodal load power cannot be freely adjusted, constraints exist in both ΔPi and ΔQi, which are not allowed to exceed the upper limits.

3 Game analysis methods of multi-objective problem

3.1 Strategy space of game players

A large similarity exists between the multi-objective optimization and game problems.Therefore, the idea and method of the game theory are introduced into the multiobjective optimization to overcome the disadvantages of the traditional methods, such as reducing the strong subjectivity.

Players

P1 and P2 are considered as two participants in the multiobjective optimization problem.P1 wishes to minimize the sum of the nodal voltage-magnitude deviation, and P2 wishes to minimize the system frequency deviation.

Strategy space

The strategy space consists of design variables ΔPi and ΔQi proposed in the multi-objective optimization problem.

Utility functions

The utility functions of the players can be designed as the objection functions proposed in the previous section.

3.2 Nodal-power-variation optimization

Step 1: Finding the optimal solution of each single objective

Two single objective functions are optimized, and optimalsoluti on , where ,(1,2)i=are then obtained.

Step 2: Calculating the impact factor

Δji represents the impact factor of variable j on single objective i, which can be expressed as

For convenience, Δj(Δ j= {Δj1, Δj 2 }) includes the impact factors of variable j on each objective.

Then, the impact factors are analyzed using hierarchical clustering.

The impact factors are divided into two classes according to the number of optimization targets.The strategy sets of the corresponding participants are generated according to the average value of each class of impact factors on each objective function.

where S1 and S2 are the strategy sets for Players 1 and 2, respectively.

Step 3: Optimization process based on the Nash equilibrium model

Each player randomly selects the initial values of the design variables from their own strategy spaces which generate s(0) as the initial feasible strategy.

Let be the complement sets of in strategy set s(0).

For player is fixed.The particle swarm optimization (PSO) is used to solve best strategy for Player i; actually, other methods can also be used [34], which satisfies

Therefore

For two players, each player uses its own utility function as the goal and carries out single objective optimization in his own strategy space to obtain the best strategy in reply to the other player.

The union set of for each player forms new strategy set and then calculates norm between strategy sets s(1) and s(0).

(a) If ＞ε (ε is a fixed positive number), the convergence condition is not satisfied.Strategy set s(0) is updated as Then, the process returns, and step3 is repeated until the convergence condition is satisfied.

(b) If ≤ε, this condition signifies that the algorithm converges. represents the best strategy for Player 1 to reply to Player 2, and represents the best strategy for Player 2 to reply to Player 1.Then, the game ends.

According to the definition of Nash equilibrium in the game, if strategy of arbitrary Player i is the best strategy to reply to all other players, the following exists for any Sij∈Si :

Then, strategy set is called a Nash equilibrium.Hence, presented in this section is a Nash equilibrium, which is also the optimal solution of the multi-objective problem.

4 Case studies

The test system is an improved IEEE33 node distribution network shown in Fig.3, whose detailed information is presented in [35].It consists of 32 branches and three distributed generations (DG) with a capacity of 240 kW.The reference value of the nodal voltage is 12.66 kV, and the total load of the system is 3715 + j2300 kVA.

A variation in the distributed generation output power occurs due to external environment.In this part, the variation in the power output of each DG is assumed to increase by 10%.

Two specific objective functions are present with the participation of responsive end-user devices in the voltage and frequency control, as mentioned in Section 3.

We set the controllable load capacity of all nodes to ±20% of both their load active and reactive power.Therefore, the load power variations at each node are all design variables, which are listed as follows:

We then calculate their impact factors for every objective function.

For all vectors Δj( j=1,2,...,2 ×33), the first part represents the impact factor of Player 1, and the second part represents the impact factor of Player 2.Then, these design variables are analyzed using hierarchical clustering.The strategy spaces of Players 1 and 2 can be obtained, as shown in Fig.4.

Fig.4 shows that the longest distance between the impact factors of similarity decision is set as 0.95.Obviously, the design variables can be clustered into two groups.

According to the multi-objective optimization strategy proposed in Section 4, to obtain the best strategy for each player in the game iterations, the particle swarm population size is set to 50, the learning factors are set as c1 = c2=0.5, and the inertial weight is set as w=0.8.

The calculation processes of the voltage and frequency indexes converge within several generations, as shown in Figs.5 and 6, respectively.

During the final iteration of the game, the curve-analysis results verify that the voltage index converges to 13.0625, and the frequency deviation converges to almost zero.

Table 1 illustrates the results of the optimal active/reactive power variations in each node.The nodal voltages of the system before and after the control for the considered event are shown in Fig.7, which reveals a voltage profile closed to the standard line result with the application of the proposed strategy.

The comparisons of the voltage index and system frequency before and after the control are listed in Table 2.

The results listed in Table 2 illustrate that both index values have been optimized under the proposed control.In addition, the proposed method avoids the effect of the magnitude of the different objective functions and eliminates subjectivity.As a result, compared with the traditional method that is based on weighting targets, the proposed method not only improves the nodal voltage level but also markedly reduces the system frequency deviation.The effectiveness and rationality of the proposed control are then verified.

5 Conclusion

Existing approaches do not consider the potential powerconditioning ability of controllable loads.To deal with the voltage and frequency deviations caused by renewablepower sources, the multi-objective optimization model, which involves both nodal voltage and system frequency, is proposed based on responsive end-user devices.In addition, to obtain optimal nodal-power variations due to power-output disturbances from distributed generations, an optimization strategy is developed under the Nash game framework.Under the proposed strategy, the controllable loads are utilized to optimize the nodal-power variations.The simulation results indicate that the proposed control is effective when a sudden variation in the distributed generation output power occurs.Not only the nodal voltage level improves but also the system frequency deviation is markedly reduced.With the development of smart grids, the works presented in this paper can help in guiding the demand side support the voltage and frequency.

Acknowledgements

This work was supported by the National Key Research and Development Program of China (Basic Research Class) (No.2017YFB0903000) and the National Natural Science Foundation of China (No.U1909201).

Declaration of Competing Interest

We have no conflict of interest to declare.

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