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Problems associated with energy distribution, consumption and management are undoubtedly some of the most significant problems that energy utilities face globally. For instance, when development takes place, the demand for electrical power and in particular domestic electrical energy also increases. Thus improvement of energy distribution policies becomes important for utilities and energy decision making agencies. The authors had earlier [1] [2] provided a mixed strategy 2-player game model for a residential energy consumption profile for winter and summer seasons of the year using a dual-occupancy high-rise (11-storey) building located within the Polytechnic of Namibia, Windhoek. The optimum energy values and the corresponding probabilities obtained from the model extend the usual simple statistical analyses of minimum and maximum energy values and their associated percentages. The time-block and the week-day strategies depict critical probabilistic values worth considering for decision purposes, especially, the necessity and justification for a dual tariff regime for the residential and workplace residents of the building as against the existing institutional uniform energy tariff policy. However, this paper presents extended results of post-optimality analyses for the winter and summer seasons, and thus provides the optimal range of energy values over which the energy consumption can change without changing the optimal tariff estimate parameters obtained from the mixed strategy of critical energy game values. The post-optimality analyses also provide extended information on the mixed strategy of non-optimal week-day solutions obtained from the game model, hence validating one of the essential roles of sensitivity analysis, namely, investigation of sub-optimal solutions. From application point of view, the post-optimality model provides a useful tool for Utilities, especially for identifying flexibility range of optimal break-even energy values for consumers, such as in the informal settlements where metering is rather a challenge to determine varied or non-uniform tariffs.

Post-optimality Analysis (or Sensitivity Analysis) is concerned with the propagation of uncertainties in mathematical models. It belongs to a broader area of Perturbation Analysis [

Specifically, our earlier game model solutions focused on the optimal week-day strategies (or equivalently identified the optimal days of use of energy) and the game value (which was proposed as a uniform tariff estimate parameter). By nature of the game model, there was no information on the energy consumption for the non-optimal week days. The post-optimality analysis model in this paper fills the gap, providing information on the “sub-optimal” solutions which are earlier characterized as “non-optimal” in the game model [

We earlier as in [

Subject to

where

A mixed strategy solution with respective probabilities and value of the game were obtained. However, in this paper, a direct linear programming problem (LPP) approach is employed for the above model to confirm our earlier game model optimal solution and to further derive the associated post-optimality results.

In the following LPP model to be solved, the decision variables

Defining the variables x1, x2, x3, x4, x5, x6 and x7 as the energy decision variables for the days of the week, namely for Monday, Tuesday…, Sunday, our LPP model is as in Equations (4) and (5):

Subject to

The coefficients in the above model are the energy readings taken from the dual-occupancy high-rise building (stated in the Abstract).

The optimal solution for the above model is as follows

And

Theoretically, given the expected value of the game defined by Equation (6) in our previous game model [

where T is the time strategy vector,

Thus, we have

Moreover, the column player’s (week day) optimal mixed strategy is obtained by the formula in Equation (8):

where

The comparative results are provided in

It is observed that the above probabilities coincide with those earlier obtained from our game model [

The above gives on dividing by

LPP Model | Game Model (2012) | |
---|---|---|

Optimal Objective | ||

Variables | ||

Probabilities | ||

0 | 0 | |

0 | 0 | |

0 | 0 | |

0.002333756 | 0.093275856 | |

0.001966939 | 0.078614866 | |

0.015540878 | 0.621139861 | |

0.005178361 | 0.206969423 |

The 5^{th} degree polynomial regression plots for the optimal range of values (with the week day mean plot for the winter season) are as in

It is observed that the four-day optimal energy consumption days for the season (Thursday-Sunday) as obtained in our game model show fairly stable consumption profile from the post-optimality results (i.e. the values showing very close proximity).

Superposed plots with the mixed-strategy game model tariff estimate value are shown in

The following time-block optimal range of values in

Using Equation (6), we obtain the corresponding post-optimality energy ranges by computing the values using Equation (9).

where

Thus the ranges of energy values are as follows:

where

Following similar procedure as in the winter model, we have the summer linear programming model given by Equations (10) and (11):

Subject to

Days | Thu | Fri | Sat | Sun | |
---|---|---|---|---|---|

Time | Probabilities | 0.0933 | 0.0786 | 0.6211 | 0.2070 |

09.00 | 0.1208 | 44.1326 | 42.7080 | 39.21589 | 39.3082 |

09.30 | 0.1460 | 38.4818 | 37.7524 | 39.86168 | 41.7991 |

10.00 | 0.2311 | 36.4835 | 38.6264 | 40.30111 | 41.0489 |

20.00 | 0.5021 | 41.0025 | 40.5711 | 40.02678 | 39.0969 |

Game Value | 39.9681 |

From | Till | |
---|---|---|

Objective | 0.0250199331361483 | 0.0250199331361483 |

Variables | ||

−∞ | 1.03569583315223 | |

−∞ | 1.04397756219379 | |

−∞ | 1.00819201095926 | |

0.994672313337416 | 1.01503531275992 | |

0.991059745754735 | 1.00782112455324 | |

0.986527203511101 | 1.00571855368084 | |

0.98833593943488 | 1.02232335482563 |

The optimal solution is as follows:

From | Till | |
---|---|---|

Objective | 0.0250199331361483 | 0.0250199331361483 |

Variables | ||

0.99418022 | 1.007099534 | |

0.990632894 | 1.004957712 | |

0.9958438 | 1.00606894 | |

0.990470928 | 1.004768121 |

and

For the summer season, the value of the game is obtained using Equation (12) below:

Thus, we have

The comparative table of results for the summer season is as follows in

It is also observed that the above probabilities coincide with those earlier obtained from our game model [

Running the LPSolve Sensitivity Analysis routine for the problem, we have the results as presented in

On dividing by

LPP Model | Game Model | |
---|---|---|

Optimal Objective | ||

Variables | ||

Probabilities | ||

0 | 0 | |

0.009060915 | 0.319548198 | |

0 | 0 | |

0 | 0 | |

0.000330919 | 0.011670423 | |

0.009666755 | 0.34091416 | |

0.009296804 | 0.327867221 |

Days | Tue | Fri | Sat | Sun | |
---|---|---|---|---|---|

Time | Probabilities | 0.3195 | 0.0117 | 0.3409 | 0.3279 |

7.30 | 0.0695 | 39.65638 | 40.34272 | 34.27103 | 31.8429 |

9.30 | 0.1784 | 33.41002 | 31.67203 | 38.42097 | 33.92427 |

19.00 | 0.2106 | 36.835885 | 34.58144 | 34.28885 | 34.77837 |

20.00 | 0.5415 | 34.704865 | 36.06655 | 34.73525 | 36.33819 |

Value | 35.2666 |

Variables | From | Till |
---|---|---|

Objective | 0.0283553930690763 | 0.0283553930690763 |

−∞ | 1.02422635412771 | |

0.981537627468657 | 1.06062733133012 | |

−∞ | 1.03629009420123 | |

−∞ | 1.03412262609405 | |

0.982292128971773 | 1.02150606857908 | |

0.969765665592283 | 1.03416157676571 | |

0.974735091792164 | 1.0145564767584 |

When the seasonal mean and post-optimality values in

The following time-block optimal range of values in

Using Equation (6), we obtain the corresponding post-optimality energy ranges by using Equation (13).

where

Thus the ranges of energy values are as follows:

Solving a linear programming problem usually provides more information about an optimal solution than merely the values of the decision variables. Associated with an optimal solution are shadow prices (also referred to as dual variables or marginal values) for the constraints. The shadow price on a particular constraint represents the change in the value of the objective function per unit increase in the right hand-side value of that constraint. Thus duality in linear programming is essentially a unifying theory that develops the relationships between a

From | Till | |
---|---|---|

Objective | 0.025019933136 | 0.02501993313 |

Variables | ||

0.997025303 | 1.030001202 | |

0.952937943 | 1.014548607 | |

0.971988805 | 1.001191668 | |

0.998253965 | 1.002466773 |

given (primal) linear programming problem and another related (dual) linear programming problem stated in terms of variables with this shadow-price interpretation

The related dual problems to the week-day models for the two seasons have the time-blocks as the dual variables. In

From the foregoing model results, the following conclusions are drawn:

1) The direct LPP models and their associated post-optimality analyses for the two seasons validate our earlier optimal mixed strategy game values, both for the week-day and time-block optimal values.

2) The post-optimality analysis for each of the seasons provides additional information on the non-optimal week-day mixed strategies, specifically, the analysis shows the significant maximum energy values which are

not explicit in the game model results as summarised in

3) In each of the seasons, the tariff energy estimator obtained from the mixed-strategy model lies within the post-optimality ranges. This is clearly seen in the plots in

4) While the Time-Block post-optimality results may not be of significant application from utility point of view, however, they equally validate our mixed-strategy game model values, and also specifically show the range of energy values within the optimal time-blocks that were not explicit from the game model.

5) It is noteworthy to see that the time-optimal solution of the dual problem model reveals small deviations from each of the season’s means, globally.

We had earlier [

Utilities, i.e. agencies in charge of distribution of electricity commonly face a serious challenge of determining a tariff policy as noted in [

Week-Day | Winter Post-Optimality Week-Day Energy Range | Summer Post-Optimality Week-Day Energy Range |
---|---|---|

Tariff Estimator = 39.9681 | Tariff Estimator = 35.2666 | |

Monday | ||

Tuesday | ||

Wednesday | ||

Thursday | ||

Friday | ||

Saturday | ||

Sunday |

utility break-even analysis. This requires a post-optimality (or sensitivity) analysis as presented in this paper and reflected in the abstract.

The authors wish to acknowledge the partial support of this research work by the Polytechnic of Namibia’s Institutional Research and Publication Committee (IRPC).