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If the cause of your hypertension is Graves disease only, your pattern is very common.  My TSH did not move for two years while my Free T3 and Free T4 immediately fell to the middle of the range and stayed there while my initial and continuing dosage was 10 mg of Methimazole.  I was concerned about my TSH not rising and did more research.  I found an article in an Endocrine Medical Journal about a study that was done in Italy that showed that hypertensive patients benefited from adding Regular L-Carnitine to their treatment and I tried that.  Before I tried it, I got tested for my Carnitine levels (Total Carnitine and Free Carnitine) and I was deficient.  I was also tested and found to be deficient in Vitamin D (vitamin D is essential for good thyroid functioning) and I was low on Magnesium so I added all of those supplements and my TSH began to rise.  I then experimented with different carnitines and added Acetyl L-carnitine and my TSH just shot right up.  While this is a very useful supplement for this purpose, it also carries risks because with the use of this combined with meds, it causes you to go very hypo very fast so you need a lot more monitoring blood tests and a lot of adjustments to both drug dose and supplement dose if you want to avoid going hypo.  My Frees fell to bottom of range and stayed there, never went out of range altogether but it sure was a fine balancing act.  I let my doc adjust my meds dose and I adjusted the supplements.  He decreased by small amounts and that worked well for me.  While it is a balancing act, I feel fine from my hyper perspective and have been able to keep all my levels in range though it still requires monitoring about every 6 weeks.  If pregnancy is desired and you are trying to achieve this, then the only medication that won't cause problems for the fetus is an alternate drug called Propylthiouracil or PPU for short is usually used.