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The Conservation Voltage Reduction (CVR) is a technique that aims to achieve the decrease of power consumption as a result of voltage reduction. The customer is supplied with the lowest possible voltage level compatible with the stipulated level by the regulatory agency. International Standards ANSI C84.1-2006 and IEEE std 1250-1995 specify the range of supply voltage to electronics equipment from 0.9 to 1.05 pu of nominal voltage. To analyse the CVR effect in distribution systems with different load characteristics (residential, commercial, industrial or a combination of these), mathematical load models are used. Typically, these equipment/load models are used to analyse load aggregation without any consideration of its nonlinearity characteristics. Aiming to analyse the nonlinear characteristics and its consequences, this paper presents a discussion of the neglected variables as well as the results of a set of measurements of nonlinear loads. Different mathematical models are applied to obtain them for each load. Using these models the load aggregation is evaluated. It is presented that although the models show adequate results for individual loads, the same does not occur for aggregated models if the harmonic contribution is not considered. Consequently, to apply the load model in CVR it is necessary to consider the harmonics presence and the model has to be done using only the fundamental frequency data. The discussion about the causes is done and the models are compared with the measurements.

Conservation Voltage Reduction (CVR) is a method of energy reduction consumption resulting from a reduction of feeder voltage [

The load model for CVR analysis is a mathematical representation of the relationship between the active and reactive power consumed and load feed voltage; in other words, it describes the load behavior when it is connected on power grid considering the voltage variation. Such models can be done without thermal cycles or static [

Loads without thermal cycles consume energy in a time-invariant manner, with the exception of voltage variations. Specifically, there is no control feedback loop. As an example, a light bulb will consume energy when turned on, as a function of voltage in a fixed manner. In contrast, a load with a thermal cycle, such as a water heater, will have a varying duty cycle dependent on the supply voltage, Schneider [

The load model, according to Price et al. [

Some researchers found in literature demonstrate the importance of load model application on Conservation Voltage Reduction (CVR) [

None of those publications consider the nonlinear characteristics of the load. The contribution of the harmonics in the load behavior has to be considered in the analysis. This paper evaluates three model static load techniques as used in literature, without any consideration of nonlinearities and including their analyses. This paper is divided as following: Firstly three CVR model techniques were presented. These techniques can be applied considering or not the harmonics components presence. Then, in the second part the approach differences were shown. To obtain the model using the measured data, it was presented the parameter estimation technique. It is desired to maintain the physics aspects of the model; then the ellipsoid algorithm (using restrictions) and its application were presented. Finally the used methodology was shown as well as the results. The comparison has been done and the conclusions were shown.

End-user load models are mathematical functions used to describe the behavior of commercial or residential loads considering the active and reactive power as function of both variables: voltage and sometimes frequency. The static models used to represent the load are performed in two main types: by an exponential model and by polynomial models, also known as ZIP models [

In exponential models the relationship between the consumed power and the feed voltage is given by an exponential function, as in Equation (1) for the active power and in Equation (2) for the reactive power [

where

When both coefficients

In polynomial models, ZIP model, the load is represented by following some physical characteristics. Some equipment, such as electronic ballasts, have constant power consumption under voltage variation. On the other hand, others behave as a combination of two or three of the following characteristics: Constant Impedance (Z), Constant Current (I) and Constant Power (P).

The expressions for active and reactive power (P_{i}, Q_{i}) of the ZIP model are presented by Equations (3) and (4) respectively. For both Equations, the coefficients have to obey the constraints presented in Equation (5). The relationship between power consumption and voltage magnitude is given by a polynomial Equation [

where:

^{th} load.

^{th} load.

Schneider et al. [

where:

^{th} load.

^{th} load.

In any ZIP model, when the load is represented as Constant Impedance, it means that the power varies directly with the square of the voltage; in the case of the Constant Current, the power varies directly with the voltage; and, lastly, if the load is modeled as Constant Power, the power does not vary when the voltage varies [

None in all presented models is mentioned about harmonic components in any reference although its presence affects the data. Once measurements are used to obtain the parameters for each model, and as the components suffer influences from harmonics, it is necessary to discuss how to consider its influence before start using the data.

The apparent power measured by the equipment gives us two parcels called P and Q. For sinusoidal source supplying a linear load, these parcels are [

where:

V: RMS voltage.

V_{1}: Fundamental voltage.

V_{1}: Fundamental current.

S: Apparent power.

P_{1}: Fundamental active power.

Q_{1}: Fundamental reactive power.

P_{M}: Measured active power.

Q_{M}: Measured reactive power.

Considering nonsinusoidal source supplying a nonlinear load, however, the apparent power assumes a different value including the harmonics effect. The measured active power P_{M} now has also the contribution of the harmonic active power P_{H}. Similarly the measured reactive power Q_{M} is not only composed by Q_{1} but also by a sum of other parcels, namely D_{V}, D_{I} and D_{H}. This measured power should no longer be called reactive power, but nonactive power, [

consequently,

where:

V_{1}: Fundamental voltage.

V_{H}: Harmonic voltage.

I_{1}: Fundamental current.

I_{H}: Harmonic current.

S: Apparent power.

P_{1}: Fundamental active power.

Q_{1}: Fundamental reactive power.

P_{M}: Measured active power.

Q_{M}: Measured reactive power.

D_{V}: Voltage distortion power.

D_{I}: Current distortion power.

P_{H}: Harmonic active power.

D_{H}: Harmonic distortion power.

N: Nonactive power.

The parameter estimation is a procedure which uses some samples from measurements to calculate one or more unknown parameters. The used measurements (samples) in the estimation process are subject to errors, so that the estimated parameters also have associated errors [

where

The statistical criteria used in the parameter estimation may be quoted the maximum likelihood, where the goal is to maximize the probability that the estimated state variable is the actual value of the state variable vector [

In the maximum likelihood criterion, it is necessary to estimate the variable x that maximizes the likelihood of measuring

where:

^{th} measurement.

^{th} measurement.

^{th} measurement.

In one matrix approach, this Equation can be written as in (15):

where: R: covariance matrix of the measurements, as can be seen in (16).

This algorithm consists in finding a solution to the optimization problem through a succession of increasingly smaller ellipsoids, starting from the initial ellipsoid containing the optimal point [

Given the constrained optimization problem described in (17) [

where:

The ellipsoid algorithm can be described by recursive Equations (19), (20) and (21), which generate a sequence of

where m is the gradient or sub-gradient of the function or of the most violated constraint, according to the rule described in (22) and (23) [

_{opt}) and its objective function value (f_{opt}), and iii. the algorithm parameters (for example the tolerance for calculate the gradient numerically (tol)) are initialized. After that, the variable receives the value of the largest constraint function at x. Then the objective function value at the point x is calculated. If r is negative, the algorithm calculates the objective function gradient at point x (f_{x}) and checks whether the value of f_{k} is smaller than the optimal value achieved so far. If f_{k} is smaller than f_{opt} (optimal function value), x_{opt} and f_{opt} receive the value of x_{k} and f_{k}, respectively, otherwise the algorithm goes to the next step. If r is bigger than or equal to zero, the algorithm calculates the gradient of the most violated constraint. In the next step, the algorithm computes the variables to generate the next ellipsoid based on the computed gradient vector. This process is repeated until the convergence is achieved.

The data measurements were performed three times for each load and at different times. During the experiments, the voltage has been changed in steps. An energy analyzer was used to collect data. Every 5 minutes the input voltage was increased by five volts starting the measurement at 110 V and ending at 130 V. The nominal voltage is 127 V. Samples were collected every second. For each load, it was used at least 4500 measurements. These data were used in the following models.

For the exponential model, the optimization problem is given as in Equation (24) [

where:

For ZIP model [

where:

The same formulation is presented for the ZIP reactive power model [

For the ZIP_mod model [

where:

The model proposes to minimize the active and reactive power functions simultaneously through a weighted sum, that is, both power functions are minimized with the same weight. Often the objective functions (active and reactive power) have different magnitude order. Therefore, one function can be minimized faster than the other. To solve this problem, the optimal objective function was calculated for the active power (

This paper uses the Ellipsoid optimization technique for parameter estimation obeying the model constraints described on Equations (24), (25) and (26).

Knowing load mathematical models, it is possible to obtain a model of a group of loads by simply adding the single models, that is, the procedure used for linear loads (superposition theorem). In order to analyse the effect of the harmonics in the load models and in their aggregation, an experiment is carried out with six different loads, and the analysis is divided in two phases.

1) During the first phase, each equipment has been tested for different voltage values in the range of 0.9 and 1.05 pu. The measurements have been made repeatedly and the variables S, S_{1}, P, P_{1}, N, Q_{1} have been registered. Using the two groups of variables (S_{1}, P_{1}, Q_{1} and S, P, N) the mathematical models have been obtained by following the procedure previously mentioned. Consequently, two representations (i. only the fundamental frequency components (S_{1}, P_{1}, Q_{1}) and ii. harmonics contributions (S, P, N)) have been calculated for each of the three models (Exponential, ZIP and ZIP_mod) and for each equipment.

2) In the second phase, six pieces of equipment were connected simultaneously and the measurements for the same variables shown in phase 1 have been collected. Six models have been adjusted, two representations for the two groups of variables (i. only the fundamental frequency components (S_{1}, P_{1}, Q_{1}) and ii. harmonics contributions (S, P, N)) for each method (Exponential, ZIP and ZIP_mod).

Following the methodology, the experiments have been done and the results are presented in this section.

_{1} varies when the voltage changes in steps. They are almost the same, as the active harmonic power P_{H} is very low. Otherwise _{M} (that in fact is N) and Q_{1}. _{1} is more evident. Using these data, the models, as a function of voltage, Exp, ZIP and ZIP_mod, are obtained.

_{1}, Q_{1}). As the active power suffers little influence of harmonics, the calculated parameters (

Load | Abbreviation |
---|---|

Compact Fluorescent Light (16 W) | CFL |

Incandescent Light Bulb (100 W) | Inc. |

Venti-Delta Fan (45 W) | Fan |

LG LCD―Liquid Crystal Display (30 W) | LCD |

Personal Computer (100W) | PC |

CCE Television (Cathod Ray Tube) (75 W - MAX) | TV |

Load | Exponential model parameters (V_{o} = 127V) | |||||||
---|---|---|---|---|---|---|---|---|

Reference parameter P | Reference parameter P_{1} | Reference parameter N | Reference parameter Q_{1} | |||||

P_{o} | P_{o} | Q_{o} | Q_{o} | |||||

CFL | 0.0100 | 15.64 | 0.0007 | 15.64 | 0.9957 | −21.85 | 0.2684 | −10.37 |

Inc | 1.3424 | 102.24 | 1.3419 | 102.16 | 0.0000 | 0.00 | 0.0001 | 0.00 |

TV | 0.0015 | 49.45 | 0.0001 | 50.21 | 0.0456 | −49.10 | 0.0001 | −2.48 |

Fan | 1.7390 | 137.57 | 1.7358 | 137.33 | 1.4688 | 59.91 | 1.8514 | 47.64 |

LCD | 0.0002 | 30.71 | 0.0004 | 31.11 | 0.5800 | −23.09 | 0.0000 | −3.11 |

PC | 0.0002 | 90.79 | 0.0016 | 92.09 | 1.1222 | 74.13 | 0.0001 | 7.33 |

For Modified ZIP model, the results are presented in a different way. As the Equations of active and reactive power are coupled, the results are presented in _{1}, P_{1}, Q_{1} and S, P, N respectively. It can be seen that the results are strongly affected by the presence of harmonics.

In the second phase the modeled loads, used in phase I, are turned on in the same time, and measurements of total active and nonactive power are done. Using these data, the models for the group of equipments data are obtained (Collective Model).

Finally, the model obtained by summing up the individual models is calculated (Sum of Models) by applying superposition theorem used in Equation (28).

, (28)

Load | ZIP model parameters (V_{o} = 127V) | |||||||
---|---|---|---|---|---|---|---|---|

Reference parameter P | Reference parameter P_{1} | |||||||

P_{p} | I_{p} | Z_{p} | P_{o} (W) | P_{p} | I_{p} | Z_{p} | P_{o} (W) | |

CFL | 0.90 | 0.08 | 0.01 | 15.55 | 0.93 | 0.07 | 0.01 | 15.65 |

Inc | 0.30 | 0.00 | 0.70 | 102.22 | 0.30 | 0.00 | 0.70 | 102.14 |

TV | 0.93 | 0.05 | 0.02 | 49.53 | 0.85 | 0.13 | 0.02 | 51.23 |

Fan | 0.12 | 0.00 | 0.88 | 137.53 | 0.12 | 0.00 | 0.88 | 137.29 |

LCD | 0.95 | 0.02 | 0.03 | 30.84 | 0.85 | 0.09 | 0.06 | 31.27 |

PC | 0.98 | 0.00 | 0.01 | 91.00 | 0.97 | 0.02 | 0.02 | 92.70 |

Load | Reference parameter N | Reference parameter Q_{1} | ||||||

P_{q} | I_{q} | Z_{q} | Q_{o} (Var) | P_{q} | I_{q} | Z_{q} | Q_{o} (Var) | |

CFL | 0.00 | 1.00 | 0.00 | −21.85 | 0.72 | 0.28 | 0.00 | −10.37 |

Inc | - | - | - | 0.00 | - | - | - | 0.00 |

TV | 0.29 | 0.58 | 0.13 | −45.55 | 0.72 | 0.08 | 0.21 | −1.95 |

Fan | 0.25 | 0.00 | 0.75 | 59.89 | 0.07 | 0.00 | 0.93 | 47.62 |

LCD | 0.68 | 0.00 | 0.32 | −23.10 | 0.56 | 0.38 | 0.06 | −3.22 |

PC | 0.00 | 0.88 | 0.12 | 74.11 | 0.86 | 0.12 | 0.02 | 7.19 |

Load | ZIP_mod (V_{o} = 127V) | ||||||
---|---|---|---|---|---|---|---|

Reference parameters S_{1}, P_{1} and Q_{1} | |||||||

P% | I% | Z% | Pᶿ | Iᶿ | Zᶿ | Sn | |

CFL | 0.90 | 0.00 | 0.09 | 5.79 | 6.13 | 3.97 | 21.03 |

Inc | 0.0000 | 0.3550 | 0.6450 | 0.00 | 0.00 | 0.00 | 102.38 |

TV | 0.8315 | 0.0143 | 0.1541 | 6.05 | 1.86 | 1.36 | 60.26 |

Fan | 0.0415 | 0.3287 | 0.6299 | 3.79 | 0.33 | 0.36 | 157.96 |

LCD | 0.7300 | 0.2655 | 0.0045 | 6.00 | 0.46 | 0.65 | 33.56 |

PC | 0.5121 | 0.4203 | 0.0676 | 0.30 | 5.95 | 2.42 | 112.49 |

Load | Reference parameters S, P and N | ||||||

P% | I% | Z% | Pᶿ | Iᶿ | Zᶿ | Sn | |

CFL | 0.48 | 0.29 | 0.23 | 5.88 | 5.14 | 4.28 | 33.16 |

Inc | 0.2529 | 0.1165 | 0.6306 | 0.00 | 0.00 | 0.00 | 103.41 |

TV | 0.8064 | 0.0020 | 0.1916 | 5.38 | 1.95 | 1.66 | 102.76 |

Fan | 0.0017 | 0.3033 | 0.6950 | 5.94 | 0.74 | 0.27 | 153.65 |

LCD | 0.8334 | 0.0476 | 0.1190 | 5.77 | 5.48 | 4.63 | 40.85 |

PC | 0.5871 | 0.1544 | 0.2585 | 0.27 | 0.63 | 1.96 | 152.63 |

Exponential model parameters (V_{o} = 127V) | |||||||
---|---|---|---|---|---|---|---|

Reference parameter P | Reference parameter P_{1} | Reference parameter N | Reference parameter Q_{1} | ||||

P_{o} | P_{o} | Q_{o} | Q_{o} | ||||

0.8122 | 443.22 | 0.8089 | 445.40 | 2.2265 | 129.85 | 2.2334 | 38.04 |

ZIP model parameters (V_{o} = 127V) | |||||||
---|---|---|---|---|---|---|---|

Reference parameter P | Reference parameter P_{1} | ||||||

P_{p} | I_{p} | Z_{p} | P_{o} (W) | P_{p} | I_{p} | Z_{p} | P_{o} (W) |

0.57 | 0.00 | 0.43 | 443.11 | 0.57 | 0.00 | 0.43 | 445.29 |

Reference parameter N | Reference parameter Q_{1} | ||||||

P_{q} | I_{q} | Z_{q} | Q_{o} (Var) | P_{q} | I_{q} | Z_{q} | Q_{o} (Var) |

0.21 | 0.15 | 0.64 | 131.43 | 0.14 | 0.28 | 0.58 | 37.07 |

ZIP_mod model parameters (V_{o} = 127V) | ||||||
---|---|---|---|---|---|---|

Reference parameters S_{1}, P_{1} and Q_{1} | ||||||

P% | I% | Z% | Pᶿ | Iᶿ | Zᶿ | Sn |

0.0615 | 0.7927 | 0.1458 | 0.61 | 6.17 | 2.21 | 592.76 |

Reference parameters S, P and N | ||||||

P% | I% | Z% | Pᶿ | Iᶿ | Zᶿ | Sn |

0.4998 | 0.0313 | 0.4689 | 6.24 | 0.02 | 0.65 | 489.70 |

V_{i} = 110V | V_{i} = 130V | |||||||
---|---|---|---|---|---|---|---|---|

P_{m} or P | P_{1} | Q_{M}, N | Q_{1} | P_{m} | P_{1} | Q_{m} or N | Q_{1} | |

Measured values | 396.47 | 398.45 | 93.81 | 28.33 | 454.78 | 456.84 | 140.34 | 40.82 |

Exponential model output (Equation (1) and (2) with the data of | ||||||||

Collective model^{a} | 394.40 | 396.52 | 25.88 | 27.60 | 451.71 | 453.89 | 37.54 | 40.08 |

Sum of models^{b} | 378.00 | 380.28 | −19.74 | 28.28 | 435.35 | 437.47 | −11.04 | 41.06 |

ZIP model output (Equation (3) and (4) with the data of | ||||||||

Collective model^{a} | 395.18 | 397.30 | 107.80 | 30.32 | 452.29 | 454.47 | 135.91 | 38.34 |

Sum of models^{b} | 377.30 | 379.55 | 30.98 | 28.79 | 436.12 | 439.95 | 45.88 | 41.29 |

Modified ZIP model output (Equation (6) and (7) with the data of | ||||||||

Collective model^{a} | 395.29 | 395.88 | 94.61 | 27.57 | 452.22 | 454.12 | 135.85 | 40.00 |

Sum of models^{b} | 380.41 | 377.59 | 20.79 | 27.18 | 435.52 | 441.11 | 48.08 | 42.28 |

^{a}These values were obtained using the model built using data for all equipment turned on simultaneously; ^{b}These values were obtained using the model built using the sum of individual models.

^{*}Corresponding author.

This paper demonstrated the application of optimization technique for the treatment of ellipsoid constraints of three static load models widely used in the literature. This technique was used in order to obtain real models for the loads, so that the result achieved can produce physical meaning. All the mathematical formulations, as well as the methodology used to achieve the results are available in the initial sections.

For most of the loads, the three models showed similar results, except for electronic loads such as compact fluorescent lamp, monitor, computer and television. Such loads showed a reduction in power consumption with the increase in voltage level. Thus, it was evident that among the three physical models presented, only the ZIP_mod can faithfully represent the load, once the other models do not show curves with negative slope.

The literature presents these models and suggests using the superposition theorem to obtain the model which represents a group of loads by simple summing up the individual models. This paper shows that the results of the obtained models, which do not filter the data (use only the fundamental frequency), cannot be used for load aggregation. In other words, to apply this kind of load model it is necessary to work with only the filtered data of the load, otherwise the final model will present very significant errors.

W.D. Caetano thanks CAPES (Coordination of improvement of Higher Education Personnel) for financial support.

William DouglasCaetano,Patrícia Romeiro daSilva Jota, (2016) Load Static Models for Conservation Voltage Reduction in the Presence of Harmonics. Energy and Power Engineering,08,62-75. doi: 10.4236/epe.2016.82006